lesson 7: the derivative (section 41 handout)

Section 2.1–2.2The Derivative and Rates of Change

The Derivative as a Function

V63.0121.041, Calculus I

New York University

September 26, 2010

Announcements

I Quiz this week in recitation on §§1.1–1.4

I Get-to-know-you/photo due Friday October 1

Announcements

I Quiz this week in recitationon §§1.1–1.4

I Get-to-know-you/photo dueFriday October 1

V63.0121.041, Calculus I (NYU) Section 2.1–2.2 The Derivative September 26, 2010 2 / 46

Format of written work

Please:

I Use scratch paper and copyyour final work onto freshpaper.

I Use loose-leaf paper (nottorn from a notebook).

I Write your name, lecturesection, assignment number,recitation, and date at thetop.

I Staple your homeworktogether.

See the website for more information.


Notes

Notes

Notes

1

Section 2.1–2.2 : The DerivativeV63.0121.041, Calculus I September 26, 2010

Objectives for Section 2.1

I Understand and state thedefinition of the derivative ofa function at a point.

I Given a function and a pointin its domain, decide if thefunction is differentiable atthe point and find the valueof the derivative at thatpoint.

I Understand and give severalexamples of derivativesmodeling rates of change inscience.


Objectives for Section 2.2

I Given a function f , use thedefinition of the derivativeto find the derivativefunction f’.

I Given a function, find itssecond derivative.

I Given the graph of afunction, sketch the graph ofits derivative.


Outline

Rates of ChangeTangent LinesVelocityPopulation growthMarginal costs

The derivative, definedDerivatives of (some) power functionsWhat does f tell you about f ′?

How can a function fail to be differentiable?

Other notations

The second derivative


Notes

Notes

Notes

2


The tangent problem

Problem

Given a curve and a point on the curve, find the slope of the line tangentto the curve at that point.

Example

Find the slope of the line tangent to the curve y = x2 at the point (2, 4).

Upshot

If the curve is given by y = f (x), and the point on the curve is (a, f (a)),then the slope of the tangent line is given by

mtangent = limx→a

f (x)− f (a)

x − a


Graphically and numerically

x

y

2

4

3

9

2.5

6.25

2.1

4.41

2.01

4.0401

1

1

1.5

2.25

1.9

3.61

1.99

3.9601

x m =x2 − 22

x − 23 5

2.5 4.5

2.1 4.1

2.01 4.01

limit

4

1.99 3.99

1.9 3.9

1.5 3.5

1 3


Velocity

Problem

Given the position function of a moving object, find the velocity of the object at acertain instant in time.

Example

Drop a ball off the roof of the Silver Center so that its height can be described by

h(t) = 50− 5t2

where t is seconds after dropping it and h is meters above the ground. How fast isit falling one second after we drop it?

Solution

The answer is

v = limt→1

(50− 5t2)− 45

t − 1= lim

t→1

5− 5t2

t − 1= lim

t→1

5(1− t)(1 + t)

t − 1

= (−5) limt→1

(1 + t) = −5 · 2 = −10


Notes

Notes

Notes

3


Numerical evidence

h(t) = 50− 5t2

Fill in the table:

t vave =h(t)− h(1)

t − 12

− 15

1.5

− 12.5

1.1

− 10.5

1.01

− 10.05

1.001

− 10.005


Velocity in general

Upshot

If the height function is given byh(t), the instantaneous velocityat time t0 is given by

v = limt→t0

h(t)− h(t0)

t − t0

= lim∆t→0

h(t0 + ∆t)− h(t0)

∆t

t

y = h(t)

t0 t

h(t0)

h(t0 + ∆t)

∆t

∆h


Population growth

Problem

Given the population function of a group of organisms, find the rate ofgrowth of the population at a particular instant.

Example

Suppose the population of fish in the East River is given by the function

P(t) =3et

1 + et

where t is in years since 2000 and P is in millions of fish. Is the fishpopulation growing fastest in 1990, 2000, or 2010? (Estimate numerically)

Solution

We estimate the rates of growth to be 0.000143229, 0.749376, and0.0001296. So the population is growing fastest in 2000.


Notes

Notes

Notes

4


Derivation

Let ∆t be an increment in time and ∆P the corresponding change inpopulation:

∆P = P(t + ∆t)− P(t)

This depends on ∆t, so ideally we would want

lim∆t→0

∆P

∆t= lim

∆t→0

1

∆t

(3et+∆t

1 + et+∆t− 3et

1 + et

)But rather than compute a complicated limit analytically, let usapproximate numerically. We will try a small ∆t, for instance 0.1.


Numerical evidence

To approximate the population change in year n, use the difference

quotientP(t + ∆t)− P(t)

∆t, where ∆t = 0.1 and t = n − 2000.

r1990 ≈P(−10 + 0.1)− P(−10)

0.1=

1

0.1

(3e−9.9

1 + e−9.9− 3e−10

1 + e−10

)=

0.000143229

r2000 ≈P(0.1)− P(0)

0.1=

1

0.1

(3e0.1

1 + e0.1− 3e0

1 + e0

)=

0.749376

r2010 ≈P(10 + 0.1)− P(10)

0.1=

1

0.1

(3e10.1

1 + e10.1− 3e10

1 + e10

)=

0.0001296


Population growth in general

Upshot

The instantaneous population growth is given by

lim∆t→0

P(t + ∆t)− P(t)

∆t


Notes

Notes

Notes

5


Marginal costs

Problem

Given the production cost of a good, find the marginal cost of productionafter having produced a certain quantity.

Example

Suppose the cost of producing q tons of rice on our paddy in a year is

C (q) = q3 − 12q2 + 60q

We are currently producing 5 tons a year. Should we change that?

Answer

If q = 5, then C = 125, ∆C = 19, while AC = 25. So we should producemore to lower average costs.


Comparisons

Solution

C (q) = q3 − 12q2 + 60q

Fill in the table:

q C (q) AC (q) = C (q)/q ∆C = C (q + 1)− C (q)

4

112 28 13

5

125 25 19

6

144 24 31


Marginal Cost in General

Upshot

I The incremental cost

∆C = C (q + 1)− C (q)

is useful, but is still only an average rate of change.

I The marginal cost after producing q given by

MC = lim∆q→0

C (q + ∆q)− C (q)

∆q

is more useful since it’s an instantaneous rate of change.


Notes

Notes

Notes

6


Outline




Other notations



The definition

All of these rates of change are found the same way!

Definition

Let f be a function and a a point in the domain of f . If the limit

f ′(a) = limh→0

f (a + h)− f (a)

h= lim

x→a

f (x)− f (a)

x − a

exists, the function is said to be differentiable at a and f ′(a) is thederivative of f at a.


Derivative of the squaring function

Example

Suppose f (x) = x2. Use the definition of derivative to find f ′(a).

Solution

f ′(a) = limh→0

f (a + h)− f (a)

h= lim

h→0

(a + h)2 − a2

h

= limh→0

(a2 + 2ah + h2)− a2

h= lim

h→0

2ah + h2

h

= limh→0

(2a + h) = 2a.


Notes

Notes

Notes

7


Derivative of the reciprocal function

Example

Suppose f (x) =1

x. Use the

definition of the derivative to findf ′(2).

Solution

f ′(2) = limx→2

1/x − 1/2

x − 2

= limx→2

2− x

2x(x − 2)

= limx→2

−1

2x= −1

4

x

x


What does f tell you about f ′?

I If f is a function, we can compute the derivative f ′(x) at each pointx where f is differentiable, and come up with another function, thederivative function.

I What can we say about this function f ′?I If f is decreasing on an interval, f ′ is negative (technically, nonpositive)

on that intervalI If f is increasing on an interval, f ′ is positive (technically, nonnegative)

on that interval


What does f tell you about f ′?

Fact

If f is decreasing on (a, b), then f ′ ≤ 0 on (a, b).

Proof.

If f is decreasing on (a, b), and ∆x > 0, then

f (x + ∆x) < f (x) =⇒ f (x + ∆x)− f (x)

∆x< 0

But if ∆x < 0, then x + ∆x < x , and

f (x + ∆x) > f (x) =⇒ f (x + ∆x)− f (x)

∆x< 0

still! Either way,f (x + ∆x)− f (x)

∆x< 0, so

f ′(x) = lim∆x→0

f (x + ∆x)− f (x)

∆x≤ 0


Notes

Notes

Notes

8


Outline




Other notations



Differentiability is super-continuity

Theorem

If f is differentiable at a, then f is continuous at a.

Proof.

We have

limx→a

(f (x)− f (a)) = limx→a

f (x)− f (a)

x − a· (x − a)

= limx→a

f (x)− f (a)

x − a· limx→a

(x − a)

= f ′(a) · 0 = 0

Note the proper use of the limit law: if the factors each have a limit at a,the limit of the product is the product of the limits.


Differentiability FAILKinks

x

f (x)

x

f ′(x)


Notes

Notes

Notes

9


Differentiability FAILCusps

x

f (x)

x

f ′(x)


Differentiability FAILVertical Tangents

x

f (x)

x

f ′(x)


Differentiability FAILWeird, Wild, Stuff

x

f (x)

This function is differentiable at0.

x

f ′(x)

But the derivative is notcontinuous at 0!


Notes

Notes

Notes

10


Outline




Other notations



Notation

I Newtonian notation

f ′(x) y ′(x) y ′

I Leibnizian notation

dy

dx

d

dxf (x)

df

dx

These all mean the same thing.


Meet the Mathematician: Isaac Newton

I English, 1643–1727

I Professor at Cambridge(England)

I Philosophiae NaturalisPrincipia Mathematicapublished 1687


Notes

Notes

Notes

11


Meet the Mathematician: Gottfried Leibniz

I German, 1646–1716

I Eminent philosopher as wellas mathematician

I Contemporarily disgraced bythe calculus priority dispute


Outline




Other notations




If f is a function, so is f ′, and we can seek its derivative.

f ′′ = (f ′)′

It measures the rate of change of the rate of change! Leibnizian notation:

d2y

dx2

d2

dx2f (x)

d2f

dx2


Notes

Notes

Notes

12


function, derivative, second derivative

x

y

f (x) = x2

f ′(x) = 2x

f ′′(x) = 2


What have we learned today?

I The derivative measures instantaneous rate of change

I The derivative has many interpretations: slope of the tangent line,velocity, marginal quantities, etc.

I The derivative reflects the monotonicity (increasing or decreasing) ofthe graph


Notes

Notes

Notes

13


lesson 7: the derivative (section 41 handout)

Documents

derivative v63

function v63

derivative function

nyu section

calculus isection

etwhere t

t ptquotient

t ptthis