This document contains notes from a Calculus I class lecture on the derivative. The lecture covered the definition of the derivative and examples of how it can be used to model rates of change in various contexts like velocity, population growth, and marginal costs. It also discussed properties of the derivative like how the derivative of a function relates to whether the function is increasing or decreasing over an interval.

1. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Notes
Sec on 2.1–2.2
The Deriva ve
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
.
February 14, 2011
. NYUMathematics
.
Notes
Announcements
Quiz this week on
Sec ons 1.1–1.4
No class Monday,
February 21
.
.
Objectives Notes
The Derivative
Understand and state the defini on of
the deriva ve of a func on at a point.
Given a func on and a point in its
domain, decide if the func on is
differen able at the point and find the
value of the deriva ve at that point.
Understand and give several examples
of deriva ves modeling rates of change
in science.
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. 1
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2. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Objectives Notes
The Derivative as a Function
Given a func on f, use the defini on of
the deriva ve to find the deriva ve
func on f’.
Given a func on, find its second
deriva ve.
Given the graph of a func on, sketch
the graph of its deriva ve.
.
.
Notes
Outline
Rates of Change
Tangent Lines
Velocity
Popula on growth
Marginal costs
The deriva ve, defined
Deriva ves of (some) power func ons
What does f tell you about f′ ?
How can a func on fail to be differen able?
Other nota ons
The second deriva ve
.
.
The tangent problem Notes
A geometric rate of change
Problem
Given a curve and a point on the curve, find the slope of the line
tangent to the curve at that point.
Solu on
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. 2
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3. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Notes
A tangent problem
Example
Find the slope of the line tangent to the curve y = x2 at the point
(2, 4).
.
.
Notes
Graphically and numerically
y x2 − 22
x m=
x−2
3 5
9 2.5 4.5
2.1 4.1
6.25 2.01 4.01
limit
4.41
4.0401
4
3.9601
3.61 1.99 3.99
2.25 1.9 3.9
1 1.5 3.5
. x 1 3
1 1.52.1 3
1.99
2.01
1.92.5
2
.
.
The velocity problem Notes
Kinematics—Physical rates of change
Problem
Given the posi on func on of a moving object, find the velocity of
the object at a certain instant in me.
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. 3
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4. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Notes
A velocity problem
Example Solu on
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 a er dropping
it and h is meters above the
ground. How fast is it falling one
second a er we drop it?
.
.
Notes
Numerical evidence
h(t) = 50 − 5t2
Fill in the table:
h(t) − h(1)
t vave =
t−1
2
1.5
1.1
1.01
1.001
.
.
Notes
Velocity in general
Upshot
y = h(t)
If the height func on is given h(t0 )
by h(t), the instantaneous ∆h
velocity at me 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
.
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. 4
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5. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Population growth Notes
Biological Rates of Change
Problem
Given the popula on func on of a group of organisms, find the rate
of growth of the popula on at a par cular instant.
Solu on
.
.
Notes
Population growth example
Example
Suppose the popula on of fish in the East River is given by the
func on
3et
P(t) =
1 + et
where t is in years since 2000 and P is in millions of fish. Is the fish
popula on growing fastest in 1990, 2000, or 2010? (Es mate
numerically)
Answer
.
.
Notes
Derivation
Solu on
Let ∆t be an increment in me and ∆P the corresponding change in
popula on:
∆P = P(t + ∆t) − P(t)
This depends on ∆t, so ideally we would want
( )
∆P 1 3et+∆t 3et
lim = lim −
∆t→0 ∆t ∆t→0 ∆t 1 + et+∆t 1 + et
But rather than compute a complicated limit analy cally, let us
approximate numerically. We will try a small ∆t, for instance 0.1.
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. 5
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6. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Notes
Numerical evidence
Solu on (Con nued)
To approximate the popula on change in year n, use the difference
P(t + ∆t) − P(t)
quo ent , where ∆t = 0.1 and t = n − 2000.
∆t
( )
P(−10 + 0.1) − P(−10) 1 3e−9.9 3e−10
r1990 ≈ = −
0.1 0.1 1 + e−9.9 1 + e−10
=
( )
P(0.1) − P(0) 1 3e0.1 3e0
r2000 ≈ = −
0.1 0.1 1 + e0.1 1 + e0
=
.
.
Notes
Solu on (Con nued)
( )
P(10 + 0.1) − P(10) 1 3e10.1 3e10
r2010 ≈ = 10.1
−
0.1 0.1 1+e 1 + e10
=
.
.
Marginal costs Notes
Rates of change in economics
Problem
Given the produc on cost of a good, find the marginal cost of
produc on a er having produced a certain quan ty.
Solu on
The marginal cost a er producing q is given by
C(q + ∆q) − C(q)
MC = lim
∆q→0 ∆q
.
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. 6
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7. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Notes
Marginal Cost Example
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
.
.
Notes
Comparisons
Solu on
C(q) = q3 − 12q2 + 60q
Fill in the table:
q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q)
4
5
6
.
.
Notes
Outline
Rates of Change
Tangent Lines
Velocity
Popula on growth
Marginal costs
The deriva ve, defined
Deriva ves of (some) power func ons
What does f tell you about f′ ?
How can a func on fail to be differen able?
Other nota ons
The second deriva ve
.
.
. 7
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8. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Notes
The definition
All of these rates of change are found the same way!
Defini on
Let f be a func on 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 func on is said to be differen able at a and f′ (a) is the
deriva ve of f at a.
.
.
Notes
Derivative of the squaring function
Example
Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a).
Solu on
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
.
.
Notes
Derivative of the reciprocal
Example
1
Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2).
x
Solu on
y
.
x
.
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. 8
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9. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Notes
What does f tell you about f′?
If f is a func on, we can compute the deriva ve f′ (x) at each
point x where f is differen able, and come up with another
func on, the deriva ve func on.
What can we say about this func on f′ ?
If f is decreasing on an interval, f′ is nega ve (technically, nonposi ve)
on that interval
If f is increasing on an interval, f′ is posi ve (technically, nonnega ve)
on that interval
.
.
Notes
What does f tell you about f′?
Fact
If f is decreasing on the open interval (a, b), then f′ ≤ 0 on (a, b).
Picture Proof.
If f is decreasing, then all secant lines
point downward, hence have y
nega ve slope. The deriva ve is a
limit of slopes of secant lines, which
are all nega ve, so the limit must be
≤ 0. .
x
.
.
Notes
What does f tell you about f′?
Fact
If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b).
The Real Proof.
If ∆x > 0, then
f(x + ∆x) − f(x)
f(x + ∆x) < f(x) =⇒ <0
∆x
If ∆x < 0, then x + ∆x < x, and
f(x + ∆x) − f(x)
f(x + ∆x) > f(x) =⇒ <0
. ∆x
.
. 9
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10. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Notes
What does f tell you about f′?
Fact
If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b).
The Real Proof.
f(x + ∆x) − f(x)
Either way, < 0, so
∆x
f(x + ∆x) − f(x)
f′ (x) = lim ≤0
∆x→0 ∆x
.
.
Notes
Going the Other Way?
Ques on
If a func on has a nega ve deriva ve on an interval, must it be
decreasing on that interval?
Answer
Maybe.
.
.
Notes
Outline
Rates of Change
Tangent Lines
Velocity
Popula on growth
Marginal costs
The deriva ve, defined
Deriva ves of (some) power func ons
What does f tell you about f′ ?
How can a func on fail to be differen able?
Other nota ons
The second deriva ve
.
.
. 10
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11. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Notes
Differentiability is super-continuity
Theorem
If f is differen able at a, then f is con nuous 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
.
.
Differentiability FAIL Notes
Kinks
Example
Let f have the graph on the le -hand side below. Sketch the graph of
the deriva ve f′ .
f(x) f′ (x)
. x . x
.
.
Differentiability FAIL Notes
Cusps
Example
Let f have the graph on the le -hand side below. Sketch the graph of
the deriva ve f′ .
f(x) f′ (x)
. x . x
.
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. 11
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12. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Differentiability FAIL Notes
Vertical Tangents
Example
Let f have the graph on the le -hand side below. Sketch the graph of
the deriva ve f′ .
f(x) f′ (x)
. x . x
.
.
Differentiability FAIL Notes
Weird, Wild, Stuff
Example
f(x) f′ (x)
. x . x
This func on is differen able But the deriva ve is not
at 0. con nuous at 0!
.
.
Notes
Outline
Rates of Change
Tangent Lines
Velocity
Popula on growth
Marginal costs
The deriva ve, defined
Deriva ves of (some) power func ons
What does f tell you about f′ ?
How can a func on fail to be differen able?
Other nota ons
The second deriva ve
.
.
. 12
.

13. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Notes
Notation
Newtonian nota on
f′ (x) y′ (x) y′
Leibnizian nota on
dy d df
f(x)
dx dx dx
These all mean the same thing.
.
.
Meet the Mathematician Notes
Isaac Newton
English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis
Principia Mathema ca
published 1687
.
.
Meet the Mathematician Notes
Gottfried Leibniz
German, 1646–1716
Eminent philosopher as
well as mathema cian
Contemporarily disgraced
by the calculus priority
dispute
.
.
. 13
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14. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Notes
Outline
Rates of Change
Tangent Lines
Velocity
Popula on growth
Marginal costs
The deriva ve, defined
Deriva ves of (some) power func ons
What does f tell you about f′ ?
How can a func on fail to be differen able?
Other nota ons
The second deriva ve
.
.
Notes
The second derivative
If f is a func on, so is f′ , and we can seek its deriva ve.
f′′ = (f′ )′
It measures the rate of change of the rate of change! Leibnizian
nota on:
d2 y d2 d2 f
2 2
f(x)
dx dx dx2
.
.
Notes
Function, derivative, second derivative
y
f(x) = x2
f′ (x) = 2x
f′′ (x) = 2
. x
.
.
. 14
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15. . V63.0121.001: Calculus I
. Sec on 2.1–2.2: The Deriva ve
. February 14, 2011
Summary Notes
What have we learned today?
The deriva ve measures instantaneous rate of change
The deriva ve has many interpreta ons: slope of the tangent
line, velocity, marginal quan es, etc.
The deriva ve reflects the monotonicity (increasing-ness or
decreasing-ness) of the graph
.
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Notes
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Notes
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. 15
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