Lesson 7: The Derivative (Section 41 handout)

V63.0121.041, Calculus I Section 2.1–2.2 : The Derivative September 26, 2010

Section 2.1–2.2 Notes

The Derivative and Rates of Change
The Derivative as a Function

V63.0121.041, Calculus I

New York University

September 26, 2010

Announcements

Quiz this week in recitation on §§1.1–1.4
Get-to-know-you/photo due Friday October 1

Announcements
Notes

Quiz this week in recitation
on §§1.1–1.4
Get-to-know-you/photo due
Friday October 1

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

Format of written work
Notes

Please:
Use scratch paper and copy
your ﬁnal work onto fresh
paper.
Use loose-leaf paper (not
torn from a notebook).
Write your name, lecture
section, assignment number,
recitation, and date at the
top.
Staple your homework
together.
See the website for more information.


1


Objectives for Section 2.1
Notes

Understand and state the
definition of the derivative of
a function at a point.
Given a function and a point
in its domain, decide if the
function is differentiable at
the point and find the value
of the derivative at that
point.
Understand and give several
examples of derivatives
modeling rates of change in
science.


Objectives for Section 2.2
Notes

Given a function f , use the
definition of the derivative
to find the derivative
function f’.
Given a function, find its
second derivative.
Given the graph of a
function, sketch the graph of
its derivative.


Outline
Notes

Rates of Change
Tangent Lines
Velocity
Population growth
Marginal costs

The derivative, defined
Derivatives of (some) power functions
What does f tell you about f ?

How can a function fail to be differentiable?

Other notations

The second derivative


2


The tangent problem
Notes
Problem
Given a curve and a point on the curve, find the slope of the line tangent
to the curve at that point.

Example
Find the slope of the line tangent to the curve y = x 2 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

f (x) − f (a)
mtangent = lim
x→a x −a


Graphically and numerically
Notes
y
x 2 − 22
x m=
x −2
3 5
9 2.5 4.5
2.1 4.1
2.01 4.01
6.25
limit
4.41 1.99 3.99
4.0401
4
3.9601 1.9 3.9
3.61
1.5 3.5
2.25
1 3
1
x
1 1.5 2.1 3
1.99
1.9 2.5
2.01
2

Velocity
Notes
Problem
Given the position function of a moving object, find the velocity of the object at a
certain 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 − 5t 2

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

Solution


3


Numerical evidence
Notes

h(t) = 50 − 5t 2
Fill in the table:
h(t) − h(1)
t vave =
t −1
2
1.5
1.1
1.01
1.001


Velocity in general
Notes

y = h(t)
Upshot
h(t0 )
If the height function is given by
h(t), the instantaneous velocity ∆h
at time t0 is given by
h(t0 + ∆t)
h(t) − h(t0 )
v = lim
t→t0 t − t0
h(t0 + ∆t) − h(t0 )
= lim
∆t→0 ∆t
∆t
t
t0 t


Population growth
Notes
Problem
Given the population function of a group of organisms, find the rate of
growth of the population at a particular instant.

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

3e t
P(t) =
1 + et
where t is in years since 2000 and P is in millions of fish. Is the fish
population growing fastest in 1990, 2000, or 2010? (Estimate numerically)

Solution


4


Derivation
Notes

Let ∆t be an increment in time and ∆P the corresponding change in
population:
∆P = P(t + ∆t) − P(t)
This depends on ∆t, so ideally we would want

∆P 1 3e t+∆t 3e t
lim = lim −
∆t→0 ∆t ∆t→0 ∆t 1 + e t+∆t 1 + et

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


Numerical evidence
Notes
To approximate the population change in year n, use the diﬀerence
P(t + ∆t) − P(t)
quotient , where ∆t = 0.1 and t = n − 2000.
∆t

P(−10 + 0.1) − P(−10) 1 3e −9.9 3e −10
r1990 ≈ = −9.9
−
0.1 0.1 1+e 1 + e −10
=
P(0.1) − P(0) 1 3e 0.1 3e 0
r2000 ≈ = −
0.1 0.1 1 + e 0.1 1 + e 0
=
P(10 + 0.1) − P(10) 1 3e 10.1 3e 10
r2010 ≈ = 10.1
−
0.1 0.1 1+e 1 + e 10
=


Population growth in general
Notes

Upshot
The instantaneous population growth is given by

P(t + ∆t) − P(t)
lim
∆t→0 ∆t


5


Marginal costs
Notes
Problem
Given the production cost of a good, ﬁnd the marginal cost of production
after having produced a certain quantity.

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

C (q) = q 3 − 12q 2 + 60q

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

Answer


Comparisons
Notes

Solution

C (q) = q 3 − 12q 2 + 60q
Fill in the table:

q C (q) AC (q) = C (q)/q ∆C = C (q + 1) − C (q)
4
5
6


Marginal Cost in General
Notes

Upshot

The incremental cost

∆C = C (q + 1) − C (q)

is useful, but is still only an average rate of change.
The marginal cost after producing q given by

C (q + ∆q) − C (q)
MC = lim
∆q→0 ∆q

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


6


Outline
Notes

Rates of Change
Tangent Lines
Velocity
Population growth
Marginal costs



Other notations



The definition
Notes

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 + h) − f (a) f (x) − f (a)
f (a) = lim = lim
h→0 h x→a x −a

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


Derivative of the squaring function
Notes

Example
Suppose f (x) = x 2 . Use the definition of derivative to find f (a).

Solution

f (a + h) − f (a) (a + h)2 − a2
f (a) = lim = lim
h→0 h h→0 h
(a2 + 2ah + h2 ) − a2 2ah + h2
= lim = lim
h→0 h h→0 h
= lim (2a + h) = 2a.
h→0


7


Derivative of the reciprocal function
Notes
Example
1
Suppose f (x) = . Use the
x
definition of the derivative to find
f (2). x

Solution

1/x − 1/2
f (2) = lim
x→2 x −2
2−x x
= lim
x→2 2x(x − 2)
−1 1
= lim =−
x→2 2x 4


Notes

If f is a function, we can compute the derivative f (x) at each point
x where f is differentiable, and come up with another function, the
derivative function.
What can we say about this function f ?
If f is decreasing on an interval, f is negative (technically, nonpositive)
on that interval
If f is increasing on an interval, f is positive (technically, nonnegative)
on that interval


Notes
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) =⇒ <0
∆x
But if ∆x < 0, then x + ∆x < x, and
f (x + ∆x) − f (x)
f (x + ∆x) > f (x) =⇒ <0
∆x
f (x + ∆x) − f (x)
still! Either way, < 0, so
∆x
f (x + ∆x) − f (x)
f (x) = lim ≤0
∆x→0 ∆x


8


Outline
Notes

Rates of Change
Tangent Lines
Velocity
Population growth
Marginal costs



Other notations



Differentiability is super-continuity
Notes
Theorem
If f is differentiable at a, then f is continuous at a.

Proof.
We have
f (x) − f (a)
lim (f (x) − f (a)) = lim · (x − a)
x→a x→a x −a
f (x) − f (a)
= lim · lim (x − a)
x→a x −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 FAIL
Kinks Notes

f (x) f (x)

x x


9


Cusps Notes

f (x) f (x)

x x


Vertical Tangents Notes

f (x) f (x)

x x


Weird, Wild, Stuﬀ Notes

f (x) f (x)

x x

This function is diﬀerentiable at But the derivative is not
0. continuous at 0!


10


Outline
Notes

Rates of Change
Tangent Lines
Velocity
Population growth
Marginal costs



Other notations



Notation
Notes

Newtonian notation

f (x) y (x) y

Leibnizian notation
dy d df
f (x)
dx dx dx
These all mean the same thing.


Meet the Mathematician: Isaac Newton
Notes

English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis
Principia Mathematica
published 1687


11


Meet the Mathematician: Gottfried Leibniz
Notes

German, 1646–1716
Eminent philosopher as well
as mathematician
Contemporarily disgraced by
the calculus priority dispute


Outline
Notes

Rates of Change
Tangent Lines
Velocity
Population growth
Marginal costs



Other notations



Notes

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:

d 2y d2 d 2f
f (x)
dx 2 dx 2 dx 2


12


function, derivative, second derivative
Notes
y
f (x) = x 2

f (x) = 2x

f (x) = 2
x


What have we learned today?
Notes

The derivative measures instantaneous rate of change
The derivative has many interpretations: slope of the tangent line,
velocity, marginal quantities, etc.
The derivative reﬂects the monotonicity (increasing or decreasing) of
the graph


Notes

13

Lesson 7: The Derivative (Section 41 handout)

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