Advanced Topics in PhysicsVelocity Speed and Rates of Change

Photo credit Dr Persin Founder of Lnk2Lrn

Advanced Topics in Physics - Calculus Application

The Rate of Change of a Function

Consider a function x = f(t)

t

x

A

bull ( t f(t) )

bull ( t + t f(t + t)) B

t t + + t

f(t + t)

f(t)

The Slope of the Secant Line AB

A

bull (t f(t) )

bull ( t + t f(t + t)) B

t t + + t

f(t + t)

ft)

y f(t+t) - f(t)m = =

x (t+ t) - t

Gives us the Gives us the average rate of average rate of change of position versus timechange of position versus time or the average velocity vor the average velocity vavgavg of of

an object an object

More useful is More useful is instantaneousinstantaneous rate of changerate of changePosition

x

tTime

1630rsquos Descartes and Fermat discover the general rule for the slope of tangent to a polynomial using the Limit as t 0

Reneacute Descartes

ldquoI think therefore I amrdquo

Pierre de Fermat

xn + yn = zn has no non-zero integer solutions for x y and z when n gt 2

Instantaneous Rate of Change

This is the Slope of the Tangent to the Curve given by the Limit as t 0 or

f(t+t) - f(t)

(t+t) - t

LimLimt t 0 0

dxdx

dtdt==

Also known as the first derivative of the function with respect to t

Or the rate of change of the function based on slight changes in t

This the instantaneous velocity v

= = vv

Rules of Differentiation

Constant Rule If f(x) = k then f ΄(x) = 0

eg Suppose f(x) = 3 What is f΄(x)

Power Function Rule If f(x) = cxn then f ΄(x) = cnxn-1

eg Suppose f(x) = 3x2 what is f΄(x)

Sum-Difference Rule

If f(x) = g(x) plusmn h(x) then f ΄(x) = g΄(x) plusmn h΄(x)

eg Suppose f(x) = x2 + 3x3 what is f΄(x)

eg Suppose f(x) = 17 ndash 4x what is f΄ (x)

More rules of Differentiation

Product Rule

If f(x) = g(x)h(x) then f΄(x) = g(x)hrsquo(x) + h(x)grsquo(x)eg Suppose f(x) = (4x3)(5-x2) What is frsquo(x)

Quotient Rule

If f(x) = g(x) h(x) then frsquo(x) = [h(x)grsquo(x) -g(x)hrsquo(x) ] h(x)2

eg Suppose f(x) = 2x2 (x-2) What is frsquo(x)

More rules of differentiation

Log Rule If f(x) = ln( g(x) ) then frsquo(x) = grsquo(x) g(x)eg Suppose f(x) = ln(x) What is frsquo(x)

Exponential-Function Rule

If f(x) = eg(x) then frsquo(x) = grsquo(x)eg(x)

eg Suppose f(x) = e3x what is frsquo(x)

Integration

The derivative stemmed from the need to compute the slope of a function f(x)

Integral calculus emerged from the need to identify the area between a function f(x) and the x-axis

For example suppose you wanted to know the area under the function f(x) = 2 on the range from 0 to 10 Then a numerical solution for this integral obviously exists equals to 20

x

f(x)

0 10

2

Integration cont

The integral of f(x) is defined as F(x) = f(x) dx

Frsquo(x) represents the ldquoanti-derivativerdquo of the function f(x) In other words ldquoF prime of x equals f of xrdquo

Frsquo(x) = f(x)

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668)

John Wallis James Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Advanced Topics in Physics - Calculus Application

The Rate of Change of a Function

Consider a function x = f(t)

t

x

A

bull ( t f(t) )

bull ( t + t f(t + t)) B

t t + + t

f(t + t)

f(t)

The Slope of the Secant Line AB

A

bull (t f(t) )

bull ( t + t f(t + t)) B

t t + + t

f(t + t)

ft)

y f(t+t) - f(t)m = =

x (t+ t) - t

Gives us the Gives us the average rate of average rate of change of position versus timechange of position versus time or the average velocity vor the average velocity vavgavg of of

an object an object

More useful is More useful is instantaneousinstantaneous rate of changerate of changePosition

x

tTime

1630rsquos Descartes and Fermat discover the general rule for the slope of tangent to a polynomial using the Limit as t 0

Reneacute Descartes

ldquoI think therefore I amrdquo

Pierre de Fermat

xn + yn = zn has no non-zero integer solutions for x y and z when n gt 2

Instantaneous Rate of Change

This is the Slope of the Tangent to the Curve given by the Limit as t 0 or

f(t+t) - f(t)

(t+t) - t

LimLimt t 0 0

dxdx

dtdt==

Also known as the first derivative of the function with respect to t

Or the rate of change of the function based on slight changes in t

This the instantaneous velocity v

= = vv

Rules of Differentiation

Constant Rule If f(x) = k then f ΄(x) = 0

eg Suppose f(x) = 3 What is f΄(x)

Power Function Rule If f(x) = cxn then f ΄(x) = cnxn-1

eg Suppose f(x) = 3x2 what is f΄(x)

Sum-Difference Rule

If f(x) = g(x) plusmn h(x) then f ΄(x) = g΄(x) plusmn h΄(x)

eg Suppose f(x) = x2 + 3x3 what is f΄(x)

eg Suppose f(x) = 17 ndash 4x what is f΄ (x)

More rules of Differentiation

Product Rule

If f(x) = g(x)h(x) then f΄(x) = g(x)hrsquo(x) + h(x)grsquo(x)eg Suppose f(x) = (4x3)(5-x2) What is frsquo(x)

Quotient Rule

If f(x) = g(x) h(x) then frsquo(x) = [h(x)grsquo(x) -g(x)hrsquo(x) ] h(x)2

eg Suppose f(x) = 2x2 (x-2) What is frsquo(x)

More rules of differentiation

Log Rule If f(x) = ln( g(x) ) then frsquo(x) = grsquo(x) g(x)eg Suppose f(x) = ln(x) What is frsquo(x)

Exponential-Function Rule

If f(x) = eg(x) then frsquo(x) = grsquo(x)eg(x)

eg Suppose f(x) = e3x what is frsquo(x)

Integration

The derivative stemmed from the need to compute the slope of a function f(x)

Integral calculus emerged from the need to identify the area between a function f(x) and the x-axis

For example suppose you wanted to know the area under the function f(x) = 2 on the range from 0 to 10 Then a numerical solution for this integral obviously exists equals to 20

x

f(x)

0 10

2

Integration cont

The integral of f(x) is defined as F(x) = f(x) dx

Frsquo(x) represents the ldquoanti-derivativerdquo of the function f(x) In other words ldquoF prime of x equals f of xrdquo

Frsquo(x) = f(x)

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668)

John Wallis James Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

The Slope of the Secant Line AB

A

bull (t f(t) )

bull ( t + t f(t + t)) B

t t + + t

f(t + t)

ft)

y f(t+t) - f(t)m = =

x (t+ t) - t

Gives us the Gives us the average rate of average rate of change of position versus timechange of position versus time or the average velocity vor the average velocity vavgavg of of

an object an object

More useful is More useful is instantaneousinstantaneous rate of changerate of changePosition

x

tTime

1630rsquos Descartes and Fermat discover the general rule for the slope of tangent to a polynomial using the Limit as t 0

Reneacute Descartes

ldquoI think therefore I amrdquo

Pierre de Fermat

xn + yn = zn has no non-zero integer solutions for x y and z when n gt 2

Instantaneous Rate of Change

This is the Slope of the Tangent to the Curve given by the Limit as t 0 or

f(t+t) - f(t)

(t+t) - t

LimLimt t 0 0

dxdx

dtdt==

Also known as the first derivative of the function with respect to t

Or the rate of change of the function based on slight changes in t

This the instantaneous velocity v

= = vv

Rules of Differentiation

Constant Rule If f(x) = k then f ΄(x) = 0

eg Suppose f(x) = 3 What is f΄(x)

Power Function Rule If f(x) = cxn then f ΄(x) = cnxn-1

eg Suppose f(x) = 3x2 what is f΄(x)

Sum-Difference Rule

If f(x) = g(x) plusmn h(x) then f ΄(x) = g΄(x) plusmn h΄(x)

eg Suppose f(x) = x2 + 3x3 what is f΄(x)

eg Suppose f(x) = 17 ndash 4x what is f΄ (x)

More rules of Differentiation

Product Rule

If f(x) = g(x)h(x) then f΄(x) = g(x)hrsquo(x) + h(x)grsquo(x)eg Suppose f(x) = (4x3)(5-x2) What is frsquo(x)

Quotient Rule

If f(x) = g(x) h(x) then frsquo(x) = [h(x)grsquo(x) -g(x)hrsquo(x) ] h(x)2

eg Suppose f(x) = 2x2 (x-2) What is frsquo(x)

More rules of differentiation

Log Rule If f(x) = ln( g(x) ) then frsquo(x) = grsquo(x) g(x)eg Suppose f(x) = ln(x) What is frsquo(x)

Exponential-Function Rule

If f(x) = eg(x) then frsquo(x) = grsquo(x)eg(x)

eg Suppose f(x) = e3x what is frsquo(x)

Integration

The derivative stemmed from the need to compute the slope of a function f(x)

Integral calculus emerged from the need to identify the area between a function f(x) and the x-axis

For example suppose you wanted to know the area under the function f(x) = 2 on the range from 0 to 10 Then a numerical solution for this integral obviously exists equals to 20

x

f(x)

0 10

2

Integration cont

The integral of f(x) is defined as F(x) = f(x) dx

Frsquo(x) represents the ldquoanti-derivativerdquo of the function f(x) In other words ldquoF prime of x equals f of xrdquo

Frsquo(x) = f(x)

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668)

John Wallis James Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

1630rsquos Descartes and Fermat discover the general rule for the slope of tangent to a polynomial using the Limit as t 0

Reneacute Descartes

ldquoI think therefore I amrdquo

Pierre de Fermat

xn + yn = zn has no non-zero integer solutions for x y and z when n gt 2

Instantaneous Rate of Change

This is the Slope of the Tangent to the Curve given by the Limit as t 0 or

f(t+t) - f(t)

(t+t) - t

LimLimt t 0 0

dxdx

dtdt==

Also known as the first derivative of the function with respect to t

Or the rate of change of the function based on slight changes in t

This the instantaneous velocity v

= = vv

Rules of Differentiation

Constant Rule If f(x) = k then f ΄(x) = 0

eg Suppose f(x) = 3 What is f΄(x)

Power Function Rule If f(x) = cxn then f ΄(x) = cnxn-1

eg Suppose f(x) = 3x2 what is f΄(x)

Sum-Difference Rule

If f(x) = g(x) plusmn h(x) then f ΄(x) = g΄(x) plusmn h΄(x)

eg Suppose f(x) = x2 + 3x3 what is f΄(x)

eg Suppose f(x) = 17 ndash 4x what is f΄ (x)

More rules of Differentiation

Product Rule

If f(x) = g(x)h(x) then f΄(x) = g(x)hrsquo(x) + h(x)grsquo(x)eg Suppose f(x) = (4x3)(5-x2) What is frsquo(x)

Quotient Rule

If f(x) = g(x) h(x) then frsquo(x) = [h(x)grsquo(x) -g(x)hrsquo(x) ] h(x)2

eg Suppose f(x) = 2x2 (x-2) What is frsquo(x)

More rules of differentiation

Log Rule If f(x) = ln( g(x) ) then frsquo(x) = grsquo(x) g(x)eg Suppose f(x) = ln(x) What is frsquo(x)

Exponential-Function Rule

If f(x) = eg(x) then frsquo(x) = grsquo(x)eg(x)

eg Suppose f(x) = e3x what is frsquo(x)

Integration

The derivative stemmed from the need to compute the slope of a function f(x)

Integral calculus emerged from the need to identify the area between a function f(x) and the x-axis

For example suppose you wanted to know the area under the function f(x) = 2 on the range from 0 to 10 Then a numerical solution for this integral obviously exists equals to 20

x

f(x)

0 10

2

Integration cont

The integral of f(x) is defined as F(x) = f(x) dx

Frsquo(x) represents the ldquoanti-derivativerdquo of the function f(x) In other words ldquoF prime of x equals f of xrdquo

Frsquo(x) = f(x)

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668)

John Wallis James Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Instantaneous Rate of Change

This is the Slope of the Tangent to the Curve given by the Limit as t 0 or

f(t+t) - f(t)

(t+t) - t

LimLimt t 0 0

dxdx

dtdt==

Also known as the first derivative of the function with respect to t

Or the rate of change of the function based on slight changes in t

This the instantaneous velocity v

= = vv

Rules of Differentiation

Constant Rule If f(x) = k then f ΄(x) = 0

eg Suppose f(x) = 3 What is f΄(x)

Power Function Rule If f(x) = cxn then f ΄(x) = cnxn-1

eg Suppose f(x) = 3x2 what is f΄(x)

Sum-Difference Rule

If f(x) = g(x) plusmn h(x) then f ΄(x) = g΄(x) plusmn h΄(x)

eg Suppose f(x) = x2 + 3x3 what is f΄(x)

eg Suppose f(x) = 17 ndash 4x what is f΄ (x)

More rules of Differentiation

Product Rule

If f(x) = g(x)h(x) then f΄(x) = g(x)hrsquo(x) + h(x)grsquo(x)eg Suppose f(x) = (4x3)(5-x2) What is frsquo(x)

Quotient Rule

If f(x) = g(x) h(x) then frsquo(x) = [h(x)grsquo(x) -g(x)hrsquo(x) ] h(x)2

eg Suppose f(x) = 2x2 (x-2) What is frsquo(x)

More rules of differentiation

Log Rule If f(x) = ln( g(x) ) then frsquo(x) = grsquo(x) g(x)eg Suppose f(x) = ln(x) What is frsquo(x)

Exponential-Function Rule

If f(x) = eg(x) then frsquo(x) = grsquo(x)eg(x)

eg Suppose f(x) = e3x what is frsquo(x)

Integration

The derivative stemmed from the need to compute the slope of a function f(x)

Integral calculus emerged from the need to identify the area between a function f(x) and the x-axis

For example suppose you wanted to know the area under the function f(x) = 2 on the range from 0 to 10 Then a numerical solution for this integral obviously exists equals to 20

x

f(x)

0 10

2

Integration cont

The integral of f(x) is defined as F(x) = f(x) dx

Frsquo(x) represents the ldquoanti-derivativerdquo of the function f(x) In other words ldquoF prime of x equals f of xrdquo

Frsquo(x) = f(x)

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668)

John Wallis James Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Rules of Differentiation

Constant Rule If f(x) = k then f ΄(x) = 0

eg Suppose f(x) = 3 What is f΄(x)

Power Function Rule If f(x) = cxn then f ΄(x) = cnxn-1

eg Suppose f(x) = 3x2 what is f΄(x)

Sum-Difference Rule

If f(x) = g(x) plusmn h(x) then f ΄(x) = g΄(x) plusmn h΄(x)

eg Suppose f(x) = x2 + 3x3 what is f΄(x)

eg Suppose f(x) = 17 ndash 4x what is f΄ (x)

More rules of Differentiation

Product Rule

If f(x) = g(x)h(x) then f΄(x) = g(x)hrsquo(x) + h(x)grsquo(x)eg Suppose f(x) = (4x3)(5-x2) What is frsquo(x)

Quotient Rule

If f(x) = g(x) h(x) then frsquo(x) = [h(x)grsquo(x) -g(x)hrsquo(x) ] h(x)2

eg Suppose f(x) = 2x2 (x-2) What is frsquo(x)

More rules of differentiation

Log Rule If f(x) = ln( g(x) ) then frsquo(x) = grsquo(x) g(x)eg Suppose f(x) = ln(x) What is frsquo(x)

Exponential-Function Rule

If f(x) = eg(x) then frsquo(x) = grsquo(x)eg(x)

eg Suppose f(x) = e3x what is frsquo(x)

Integration

The derivative stemmed from the need to compute the slope of a function f(x)

Integral calculus emerged from the need to identify the area between a function f(x) and the x-axis

For example suppose you wanted to know the area under the function f(x) = 2 on the range from 0 to 10 Then a numerical solution for this integral obviously exists equals to 20

x

f(x)

0 10

2

Integration cont

The integral of f(x) is defined as F(x) = f(x) dx

Frsquo(x) represents the ldquoanti-derivativerdquo of the function f(x) In other words ldquoF prime of x equals f of xrdquo

Frsquo(x) = f(x)

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668)

John Wallis James Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

More rules of Differentiation

Product Rule

If f(x) = g(x)h(x) then f΄(x) = g(x)hrsquo(x) + h(x)grsquo(x)eg Suppose f(x) = (4x3)(5-x2) What is frsquo(x)

Quotient Rule

If f(x) = g(x) h(x) then frsquo(x) = [h(x)grsquo(x) -g(x)hrsquo(x) ] h(x)2

eg Suppose f(x) = 2x2 (x-2) What is frsquo(x)

More rules of differentiation

Log Rule If f(x) = ln( g(x) ) then frsquo(x) = grsquo(x) g(x)eg Suppose f(x) = ln(x) What is frsquo(x)

Exponential-Function Rule

If f(x) = eg(x) then frsquo(x) = grsquo(x)eg(x)

eg Suppose f(x) = e3x what is frsquo(x)

Integration

The derivative stemmed from the need to compute the slope of a function f(x)

Integral calculus emerged from the need to identify the area between a function f(x) and the x-axis

For example suppose you wanted to know the area under the function f(x) = 2 on the range from 0 to 10 Then a numerical solution for this integral obviously exists equals to 20

x

f(x)

0 10

2

Integration cont

The integral of f(x) is defined as F(x) = f(x) dx

Frsquo(x) represents the ldquoanti-derivativerdquo of the function f(x) In other words ldquoF prime of x equals f of xrdquo

Frsquo(x) = f(x)

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668)

John Wallis James Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

More rules of differentiation

Log Rule If f(x) = ln( g(x) ) then frsquo(x) = grsquo(x) g(x)eg Suppose f(x) = ln(x) What is frsquo(x)

Exponential-Function Rule

If f(x) = eg(x) then frsquo(x) = grsquo(x)eg(x)

eg Suppose f(x) = e3x what is frsquo(x)

Integration

The derivative stemmed from the need to compute the slope of a function f(x)

Integral calculus emerged from the need to identify the area between a function f(x) and the x-axis

For example suppose you wanted to know the area under the function f(x) = 2 on the range from 0 to 10 Then a numerical solution for this integral obviously exists equals to 20

x

f(x)

0 10

2

Integration cont

The integral of f(x) is defined as F(x) = f(x) dx

Frsquo(x) represents the ldquoanti-derivativerdquo of the function f(x) In other words ldquoF prime of x equals f of xrdquo

Frsquo(x) = f(x)

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668)

John Wallis James Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Integration

The derivative stemmed from the need to compute the slope of a function f(x)

Integral calculus emerged from the need to identify the area between a function f(x) and the x-axis

For example suppose you wanted to know the area under the function f(x) = 2 on the range from 0 to 10 Then a numerical solution for this integral obviously exists equals to 20

x

f(x)

0 10

2

Integration cont

The integral of f(x) is defined as F(x) = f(x) dx

Frsquo(x) represents the ldquoanti-derivativerdquo of the function f(x) In other words ldquoF prime of x equals f of xrdquo

Frsquo(x) = f(x)

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668)

John Wallis James Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Integration cont

The integral of f(x) is defined as F(x) = f(x) dx

Frsquo(x) represents the ldquoanti-derivativerdquo of the function f(x) In other words ldquoF prime of x equals f of xrdquo

Frsquo(x) = f(x)

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668)

John Wallis James Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Hints of the reciprocity between the derivative and the integral resulted from studies of integration by Wallis (1658) and Gregory (1668)

John Wallis James Gregory

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

First published proof of the relationship between the Derivative and the Integral by Barrow (1670)

Isaac Barrow

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Derivative and the Integral first discovered by Newton (1666 unpublished) and later supported by Leibniz (1673)

Isaac Newton Gottfried Leibniz

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Joseph Fourier (1807) Put the emphasis on definite

integrals (he invented the notation ) and defined

them in terms of area between graph and x-axis

a

b

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

AL Cauchy First to define the integral as the limit of the summation

f xi 1 xi xi 1

Also the first (1823) to explicitly state and prove the second part of the FTC d

dxf t dt f x

a

x

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Bernhard Riemann (1822 - 1866) On the representation of a function as a limit of a series (Riemann Sum)

Defined as limit of f xi xi xi 1 f x

a

b

dx

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

S F LaCroix

(1765-1843)ldquoIntegral calculus is the inverse of differential calculus Its goal is to restore the functions from their differential coefficientsrdquo

ldquoWhen students find themselves stopped on a proof the professor should restrain from immediately pointing out the solution Let the students find it out for themselves and the error corrected may be more profitable than several theorems provedrdquo

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

The Fundamental Theorem of Calculus

If then f x a

b

dx F b F a F x f x

Vito Volterra in 1881 found a function f with an anti-derivative F so that F΄(x) = f(x) for all x but there is no interval over which the definite integral of f(x) exists

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Henri Lebesgue in 1901 came up with a totally different way of defining integrals using a ldquoStep Functionrdquo that is the same as the Riemann integral for well-behaved functions

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Rules of IntegrationRule 1) a dx = ax + c

eg What is 2 dx = ______________

Rule 2) xn dx = xn+1 (n+1) + ceg What is x3 dx ________________

Rule 3) a f(x) dx = a f(x) dx eg What is 17 x3 dx ________________

Rule 4) If u and v are functions of x (u+v) dx = u dx + v dxeg What is (5 x3 + 13x) dx ____________________________

Note that for each of these rules we must add a constant of integration

To find the area under a curve we use a Definite Integral

Find the area under the graph of

f(x) = 7 - x2

from x= -1 to x = 2

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Basic Properties of Integrals

Basic Properties of Integrals

Through this section we assume that all functions are continuous on a closed interval I = [ab] Below r is a real number f and g are functions

f x f fb c b

a a c

dx x dx x dx 3

f x 0c

c

dx 1 f x fb a

a b

dx x dx 2

f x g f gb b b

a a a

x dx x dx x dx 5

f x fb b

a a

r dx r x dx 4

These properties of integrals follow from the definition of integrals as limits of Riemann sums

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Evaluating the Definite Integral

Ex Calculate5

1

12 1x dx

x

55 2

1 1

12 1 lnx dx x x x

x

2 25 ln 5 5 1 ln1 1

28 ln 5 2639056

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Substitution for Definite Integrals

Ex Calculate 1 1 22

02 3x x dx

2let 3u x x

then 2

dudx

x

1 41 22 1 2

0 02 3x x x dx u du

43 2

0

2

3u

16

3

Notice limits change

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Computing Area

Ex Find the area enclosed by the x-axis the vertical lines x = 0 x = 2 and the graph of

23

02x dx

Gives the area since 2x3 is nonnegative on [0 2]

22

3 4

00

12

2x dx x 4 41 1

2 02 2

8

Antiderivative Fund Thm of Calculus

22 y x

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Derivatives of Functions of Higher Degree

Ex 1 Suppose f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) after expanding the binomial

Solution f(x) = (2x + 5)3 = (2x)3 + 3(2x)25 + 3(2x) 52 + 52

= 8x3 + 60x2 + 150x + 125

f rsquo(x) = 24x2 + 120x + 150 = 6(4x2 + 20x + 25) = 6(2x + 5)(2x + 5) = 6(2x + 5)2

f rsquorsquo(x) = 48x + 120 = 24(2x + 5)

Notice similarities between the solutions and the original function

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

For More Efficient Solutions of Problems of This Type

We can use the Chain Rule

If f (v) = vn and v is a function of x

then f (v) = nvn-1 dv

Now f(x) = (2x + 5)3 Find f rsquo(x) and f rsquorsquo(x) using the Chain Rule

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Lets Review the Following Before Going On

bull Basic Rules of Differentiation

bull The Product and Quotient Rules

bull The Chain Rule

bull Marginal Functions in Economics

bull Higher-Order Derivatives

bull Implicit Differentiation and Related Rates

bull Differentials

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Basic Differentiation Rules

1

Ex

2

0 is a constantd

c cdx

( ) 5

( ) 0

f x

f x

Ex

1 is a real numbern ndx nx n

dx

7

6

( )

( ) 7

f x x

f x x

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Basic Differentiation Rules

3

Ex

4

( ) ( ) is a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex

( ) ( ) d d d

f x g x f x g xdx dx dx

12( ) 7f x x

7 7( ) 3 8 24f x x x

11 11( ) 0 12 12f x x x

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

More Differentiation Rules

5

Ex 3 7 2( ) 2 5 3 8 1f x x x x x

( ) ( ) ( ) ( ) d d d

f x g x f x g x g x f xdx dx dx

Product Rule

2 7 2 3 6( ) 3 2 3 8 1 2 5 21 16f x x x x x x x x

9 7 6 4 2( ) 30 48 105 40 45 80 2f x x x x x x x

Derivative of the first function

Derivative of the second function

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

More Differentiation Rules

6

2

( ) ( ) ( ) ( )

( ) ( )

d dg x f x f x g xf xd dx dx

dx g x g x

Quotient Rule

lo hi hi lohi

lo lo lo

d dd

dx

Sometimes remembered as

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

More Differentiation Rules

6

Ex2

3 5( )

2

xf x

x

Quotient Rule (cont)

2

22

3 2 2 3 5( )

2

x x xf x

x

2

22

3 10 6

2

x x

x

Derivative of the numerator

Derivative of the denominator

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

More Differentiation Rules7 The Chain Rule

If ( ) ( ) thenh x g f x

( ) ( ) ( )h x g f x f x

Note h(x) is a composite function

If ( ) where ( ) theny h x g u u f x

dy dy du

dx du dx

Another Version

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

More Differentiation Rules

The General Power Rule

If ( ) ( ) real thenn

h x f x n

1( ) ( ) ( )

nh x n f x f x

Ex 1 22 2( ) 3 4 3 4f x x x x x

1 221

( ) 3 4 6 42

f x x x x

2

3 2

3 4

x

x x

The Chain Rule leads to

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Chain Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 13( ) 7

3 5 3 5 3 5

xxG x

x x x

Ex

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Chain Rule Example5 2 8 2 7 3y u u x x Ex

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Higher DerivativesThe second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative

Derivative Notations

nf

f

f

(4)f

Second

Third

Fourth

nth

2

2

d y

dx3

3

d y

dx4

4

d y

dxn

n

d y

dx

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Example of Higher Derivatives

5 3( ) 3 2 14f x x x Given find ( )f x

4 2( ) 15 6f x x x

3( ) 60 12f x x x

2( ) 180 12f x x

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Example of Higher Derivatives

Given2 1

( )3 2

xf x

x

find (2)f

2

2 2

2 3 2 3 2 1 7( ) 7 3 2

3 2 3 2

x xf x x

x x

3

3

42( ) 14 3 2 3

3 2f x x

x

3 3

42 42 21(2)

3243(2) 2f

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Implicit Differentiation

33 4 17y x x

y is expressed explicitly as a function of x

3 3 1y xy x

y is expressed implicitly as a function of x

To differentiate the implicit equation we write f (x) in place of y to get

3( ) ( ) 3 1f x x f x x

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Implicit Differentiation (cont)

Now differentiate

using the chain rule

3( ) ( ) 3 1f x x f x x

23 ( ) ( ) ( ) ( ) 3f x f x f x xf x

23 3y y y xy which can be written in the form

subbing in y

23 3y y x y

2

3

3

yy

y x

Solve for yrsquo

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Related Rates

Look at how the rate of change of one quantity is related to the rate of change of another quantity

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels west at 60 mihr How fast is the distance between them changing after 2 hours

Note The rate of change of the distance between them is related to the rate at which the cars are traveling

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Related Rates

Steps to solve a related rate problem1 Assign a variable to each quantity Draw a diagram if appropriate

2 Write down the known valuesrates

3 Relate variables with an equation

4 Differentiate the equation implicitly

5 Plug in values and solve

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Ex Two cars leave an intersection at the same time One car travels north at 35 mihr the other travels east at 60 mihr How fast is the distance between them changing after 2 hours

Distance = z

x

y35

dy

dt60

dx

dt

70y 120x

2 2 2x y z

2 2 2dx dy dz

x y zdt dt dt

2(120)(60) 2(70)(35) 2 10 193dz

dt

From original relationship

695 mihrdz

dt

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Increments

An increment in x represents a change from x1 to x2 and is defined by

2 1x x x Read ldquodelta xrdquo

An increment in y represents a change in y and is defined by

( ) ( )y f x x f x

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Differentials

Let y = f (x) be a differentiable function then the differential of x denoted dx is such that dx xThe differential of y denoted dy is

( ) ( )dy f x x f x dx

Note measures actual change in y y measures approximate change in dy y

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example

Example2( ) 3 findf x x x

2 and as changes from 3 to 302y dy x

Given

1 as changes from 3 to 302x x302 3 002x

(302) (3)y f f 243412 24 03412

( ) 6 1dy f x dx x dx 6(3) 1 (002) 034

• Slide 1
• Advanced Topics in Physics - Calculus Application The Rate of Change of a Function
• Slide 3
• Slide 4
• Instantaneous Rate of Change
• Rules of Differentiation
• More rules of Differentiation
• More rules of differentiation
• Integration
• Integration cont
• Slide 11
• Slide 12
• Slide 13
• Slide 14
• Slide 15
• Slide 16
• Slide 17
• Slide 18
• Slide 19
• Rules of Integration
• Basic Properties of Integrals
• Evaluating the Definite Integral
• Substitution for Definite Integrals
• Computing Area
• Derivatives of Functions of Higher Degree
• For More Efficient Solutions of Problems of This Type
• Slide 27
• Basic Differentiation Rules
• Slide 29
• More Differentiation Rules
• Slide 31
• Slide 32
• Slide 33
• Slide 34
• Slide 35
• Slide 36
• Higher Derivatives
• Example of Higher Derivatives
• Slide 39
• Implicit Differentiation
• Implicit Differentiation (cont)
• Related Rates
• Slide 43
• Slide 44
• Increments
• Differentials
• Example