Differentiation Recap

xinchange

yinchange

x

y

The first derivative gives the ratio for which f(x) changes w.r.t. change in the x value.

F(x) – Non linearxinchange

yinchange

x

y

For a Non linear function, we can not take just ANY two points

If we wish to find the gradient at x=1 we must move the other point at (x=3) closer and closer to the point at x=1

The closer the points are together the more accurate the approximation of the gradient

x=3 moved to x=2

The approximation is move accurate then before

The blue line is the approximation to the tangent

x=2 moved to x=1.5

Again the approximation is move accurate then before

x=1.5 moved to x=1.01

The approximation is now very accurate as the points are virtually coincidentThis red line through the

point is called the TANGENT of f(x) at the point x=1

The tangent has the same gradient as the curve f(x) at the point in question and touches f(x) at x=1

This red dashed line through the point is called the Normal of f(x) at the point x=1

It’s Perpendicular to the tangent

The tangent has the same gradient as the curve f(x) at the point in question and touches f(x) at x=1

Definition of derivative

y

x

0

xx

yLimit We call this limit

dx

dy

0

xx

yLimit

Some centuries ago Leibnitz and Isaac Newton Both independently applied this limit to many different functions

And noticed a general pattern.

This provided the basic rule of differentiation. And made the process very easy for polynomial functions

The general rule is

1

n

n

nkxxfdx

d

kxxf

1

n

n

nkxdx

dy

kxy

Otherwise written as

Increasing functions

This function is increasing because

0dx

dy Always +ve

Decreasing functions

This function is increasing because

0dx

dy always-ve

Where the gradient is Zero

In the above graphs the gradient passes through zero at x=0 we can write

00

xdx

dyThese points of zero gradient are very important in mathematics applications as at these points the rate of change of the function is zero

Consider this function

The gradient is zero at these two points

The tangents are horizontal

The function does not change value when the gradient is zero

And therefore with we can find local

Maximum and Minimum values of functions

0dx

dy

0dx

dy

Importance

0dx

dy Will tell us when a function is at a local maximum or minimum

Can be used to find the maximum or minimum values in various questions involving rates of change

But since BOTH Max and Min have gradient = 0 we need a way of distinguishing between the two

Function Max/MinWe could plot the function and look at the graph

But an easier way is to consider the 2nd derivative

2

2

dx

yd

dx

dy

dx

d

Consider the following function f(x) = 2x3-4x-4

f’(x) = 6x2-4 f’’(x)=12x

f’(x)=12xf’’(x)=6x2-4

f(x)=2x3-4x-4

f’(x)=0

f(x) decreasingas f’(x)<0

f(x) increasingas f’(x)>0

f ’’(x)<0Concave up

f ’’(x)>0Concave down

First derivative:

y is positive Curve is rising.

y is negative Curve is falling.

y is zero Possible local maximum or minimum.

Second derivative:

y is positive Curve is concave up. (MIN)

y is negative Curve is concave down. (MAX)

Multiple choice Test

QuestionShow that the function is a decreasing function xxxxf 222

3

1)( 23

ANSWER

)224()(' 2 xxxf

)224)2(()(' 2 xxf

)18)2(()(' 2 xxf

)()(' hereinsidexf

)224()(' 2 xxxf

x

y

O B D

R

C

A (1 , 5 )

x

2

.

The diagram above shows part of the curve C with equation y = 9 - 2x -

,

(a) Verify that b = 4. (1) (1)The tangent to C at the point A cuts the x-axis at the point D, as shown in thediagram above.

(b) Show that an equation of the tangent to C at A is y + x = 6. (4)(c) Find the coordinates of the point D.(d) Find the Area of the ABD, assume it is a triangle

The point A(1, 5) lies on C and the curve crosses the x-axis at B(b, 0), where b is a constant and b > 0.

Answer(a) y = 9 – 2b - = 0 => b = 4

b

2

(c) Let y = 0 and x = 6 so D is (6, 0)

(d) Area of shaded triangle is 4.5

Rates of ChangeThe following function f(s) = 3t is shown below

What does the gradient of the line represent ?

Rates of ChangeThe following function f(s) = 3t is shown below

What does the gradient of the line represent ?

dt

dSVelocity

The steeper the line the faster the velocityThe Black line is fastest as it arrives at B the quickest

The velocity is the Gradient

What about this graph?

What does the gradient represent

timeinChange

velocityinChange

t

V

What about this graph?

What does the gradient represent

timeinChange

velocityinChange

t

V

t

Vonaccelerati

decelerating

accelerating

0t

V ConstantVelocity

Summary

2

2

dt

sd

dt

ds

dt

dAccor

dt

dVAcc

dt

dsVel

)(tfs We will need these formula for some rates of change Questions