At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.

1. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Sec on 4.7
An deriva ves
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
April 19, 2011
.
.
Notes
Announcements
Quiz 5 on Sec ons
4.1–4.4 April 28/29
Final Exam Thursday May
12, 2:00–3:50pm
I am teaching Calc II MW
2:00pm and Calc III TR
2:00pm both Fall ’11 and
Spring ’12
.
.
Notes
Objectives
Given a ”simple“ elementary
func on, find a func on whose
deriva ve is that func on.
Remember that a func on
whose deriva ve is zero along
an interval must be zero along
that interval.
Solve problems involving
rec linear mo on.
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. 1
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2. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Outline
What is an an deriva ve?
Tabula ng An deriva ves
Power func ons
Combina ons
Exponen al func ons
Trigonometric func ons
An deriva ves of piecewise func ons
Finding An deriva ves Graphically
Rec linear mo on
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.
Notes
What is an antiderivative?
Defini on
Let f be a func on. An an deriva ve for f is a func on F such that
F′ = f.
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.
Notes
Who cares?
Ques on
Why would we want the an deriva ve of a func on?
Answers
For the challenge of it
For applica ons when the deriva ve of a func on is known but
the original func on is not
Biggest applica on will be a er the Fundamental Theorem of
Calculus (Chapter 5)
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. 2
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3. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Hard problem, easy check
Example
Find an an deriva ve for f(x) = ln x.
Solu on
???
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.
Notes
Hard problem, easy check
Example
is F(x) = x ln x − x an an deriva ve for f(x) = ln x?
Solu on
d
dx
1
(x ln x − x) = 1 · ln x + x · − 1 = ln x
x
.
.
Why the MVT is the MITC Notes
Most Important Theorem In Calculus!
Theorem
Let f′ = 0 on an interval (a, b). Then f is constant on (a, b).
Proof.
Pick any points x and y in (a, b) with x y. By MVT there exists a
point z in (x, y) such that
f(y) = f(x) + f′ (z)(y − x)
But f′ (z) = 0, so f(y) = f(x). Since this is true for all x and y in (a, b),
then f is constant.
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. 3
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4. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Functions with the same derivative
Theorem
Suppose f and g are two differen able func ons on (a, b) with
f′ = g′ . Then f and g differ by a constant. That is, there exists a
constant C such that f(x) = g(x) + C.
Proof.
Let h(x) = f(x) − g(x)
Then h′ (x) = f′ (x) − g′ (x) = 0 on (a, b)
So h(x) = C, a constant
This means f(x) − g(x) = C on (a, b)
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.
Notes
Outline
What is an an deriva ve?
Tabula ng An deriva ves
Power func ons
Combina ons
Exponen al func ons
Trigonometric func ons
An deriva ves of piecewise func ons
Finding An deriva ves Graphically
Rec linear mo on
.
.
Notes
Antiderivatives of power functions
′
Recall that the deriva ve of a yf (x) = 2x
power func on is a power f(x) = x2
func on.
Fact (The Power Rule) F(x) = ?
If f(x) = xr , then f′ (x) = rxr−1 .
So in looking for
an deriva ves of power .
x
func ons, try power
func ons!
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. 4
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5. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Antiderivatives of power functions
Example
Find an an deriva ve for the func on f(x) = x3 .
Solu on
Try a power func on F(x) = axr
Then F′ (x) = arxr−1 , so we want arxr−1 = x3 .
1
r − 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a = .
4
1
So F(x) = x4 is an an deriva ve.
4
.
.
Notes
Antiderivatives of power functions
Example
Find an an deriva ve for the func on f(x) = x3 .
Solu on
1
So F(x) = x4 is an an deriva ve.
4
( )
Check:
d 1 4
dx 4
1
x = 4 · x4−1 = x3
4
1
Any others? Yes, F(x) = x4 + C is the most general form.
4
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Notes
General power functions
Fact (The Power Rule for an deriva ves)
If f(x) = xr , then
1 r+1
F(x) = x
r+1
is an an deriva ve for f… as long as r ̸= −1.
Fact
1
If f(x) = x−1 = , then F(x) = ln |x| + C is an an deriva ve for f.
x
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. 5
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6. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
What’s with the absolute value?
{
ln(x) if x 0;
F(x) = ln |x| =
ln(−x) if x 0.
The domain of F is all nonzero numbers, while ln x is only
defined on posi ve numbers.
If x 0,
d
dx
ln |x| =
d
dx
ln(x) =
1
x
If x 0,
d
dx
ln |x| =
d
dx
ln(−x) =
1
−x
· (−1) =
1
x
We prefer the an deriva ve with the larger domain.
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.
Notes
Graph of ln |x|
y
F(x) = ln |x|
. x = 1/x
f(x)
.
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Notes
Combinations of antiderivatives
Fact (Sum and Constant Mul ple Rule for An deriva ves)
If F is an an deriva ve of f and G is an an deriva ve of g, then
F + G is an an deriva ve of f + g.
If F is an an deriva ve of f and c is a constant, then cF is an
an deriva ve of cf.
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. 6
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7. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Combinations of antiderivatives
Proof.
These follow from the sum and constant mul ple rule for
deriva ves:
If F′ = f and G′ = g, then
(F + G)′ = F′ + G′ = f + g
Or, if F′ = f,
(cF)′ = cF′ = cf
.
.
Notes
Antiderivatives of Polynomials
Example
Find an an deriva ve for f(x) = 16x + 5.
Solu on
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Notes
Antiderivatives of Polynomials
Ques on
Do we need two C’s or just one?
Answer
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. 7
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8. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Exponential Functions
Fact
If f(x) = ax , f′ (x) = (ln a)ax .
Accordingly,
Fact
1 x
If f(x) = ax , then F(x) = a + C is the an deriva ve of f.
ln a
Proof.
Check it yourself.
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Notes
Exponential Functions
In par cular,
Fact
If f(x) = ex , then F(x) = ex + C is the an deriva ve of f.
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Notes
Logarithmic functions?
Remember we found F(x) = x ln x − x is an an deriva ve of
f(x) = ln x.
This is not obvious. See Calc II for the full story.
ln x
However, using the fact that loga x = , we get:
ln a
Fact
If f(x) = loga (x)
1 1
F(x) = (x ln x − x) + C = x loga x − x+C
ln a ln a
is the an deriva ve of f(x).
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. 8
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9. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Trigonometric functions
Fact
d d
sin x = cos x cos x = − sin x
dx dx
So to turn these around,
Fact
The func on F(x) = − cos x + C is the an deriva ve of
f(x) = sin x.
The func on F(x) = sin x + C is the an deriva ve of
f(x) = cos x.
.
.
Notes
More Trig
Example
Find an an deriva ve of f(x) = tan x.
Answer
F(x) = ln | sec x|.
Check
d
=
1
·
d
dx sec x dx
sec x =
1
sec x
· sec x tan x = tan x
. More about this later.
.
Notes
Antiderivatives of piecewise functions
Example
Let {
x if 0 ≤ x ≤ 1;
f(x) =
1 − x2 if 1 x.
Find the an deriva ve of f with F(0) = 1.
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. 9
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10. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Antiderivatives of piecewise functions
Solu on
.
.
Notes
.
.
Notes
Outline
What is an an deriva ve?
Tabula ng An deriva ves
Power func ons
Combina ons
Exponen al func ons
Trigonometric func ons
An deriva ves of piecewise func ons
Finding An deriva ves Graphically
Rec linear mo on
.
.
. 10
.

11. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Finding Antiderivatives Graphically
y
Problem
Pictured is the graph of a y = f(x)
func on f. Draw the graph of .
an an deriva ve for f. x
1 2 3 4 5 6
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.
Notes
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find the
intervals of monotonicity and concavity for F:
′
y . + + − − + f=F
1↗2↗3↘4↘5↗6 F
max min
′ ′′
++ −−−− ++ ++ f = F
. ⌣ ⌢ ⌢ ⌣ ⌣ F
x 1 2 3 4 5 6
1 2 3 4 5 6
IP IP
? ? ? ? ? ?F
1 2 3 4 5 6 shape
The only ques on le is: What are the func on values?
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Notes
Could you repeat the question?
Problem
Below is the graph of a func on f. Draw the graph of the
an deriva ve for f with F(1) = 0.
Solu on y
We start with F(1) = 0. f
.
Using the sign chart, we draw arcs x
1 2 3 4 5 6
with the specified monotonicity and
concavity F
It’s harder to tell if/when F crosses
1 2 3 4 5 6 shape
IP
max
IP
min
the axis; more about that later.
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. 11
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12. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Outline
What is an an deriva ve?
Tabula ng An deriva ves
Power func ons
Combina ons
Exponen al func ons
Trigonometric func ons
An deriva ves of piecewise func ons
Finding An deriva ves Graphically
Rec linear mo on
.
.
Notes
Say what?
“Rec linear mo on” just means mo on along a line.
O en we are given informa on about the velocity or
accelera on of a moving par cle and we want to know the
equa ons of mo on.
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Notes
Application: Dead Reckoning
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. 12
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13. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Problem Notes
Suppose a par cle of mass m is acted upon by a constant force F.
Find the posi on func on s(t), the velocity func on v(t), and the
accelera on func on a(t).
Solu on
By Newton’s Second Law (F = ma) a constant force induces a
F
constant accelera on. So a(t) = a = .
m
Since v′ (t) = a(t), v(t) must be an an deriva ve of the
constant func on a. So
v(t) = at + C = at + v0
. where v0 is the ini al velocity.
Since s′ (t) = v(t), s(t) must be an an deriva ve of v(t), .
meaning
1 1
s(t) = at2 + v0 t + C = at2 + v0 t + s0
2 2
Notes
An earlier Hatsumon
Example
Drop a ball off the roof of the Silver Center. What is its velocity when
it hits the ground?
Solu on
Assume s0 = 100 m, and v0 = 0. Approximate a = g ≈ −10. Then
s(t) = 100 − 5t2
√ √
So s(t) = 0 when t = 20 = 2 5. Then
v(t) = −10t,
. √ √
so the velocity at impact is v(2 5) = −20 5 m/s. .
Finding initial velocity from Notes
stopping distance
Example
The skid marks made by an automobile indicate that its brakes were
fully applied for a distance of 160 before it came to a stop.
Suppose that the car in ques on has a constant decelera on of
20 ft/s2 under the condi ons of the skid. How fast was the car
traveling when its brakes were first applied?
Solu on (Setup)
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. 13
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14. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Implementing the Solution
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.
Notes
Solving
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Notes
Summary of Antiderivatives so far
f(x) F(x)
1 r+1
x , r ̸= 1
r
x +C
r+1
1 −1
=x ln |x| + C
x x
e ex + C
x 1 x
a a +C
ln a
ln x x ln x − x + C
x ln x − x
loga x +C
ln a
sin x − cos x + C
. cos x sin x + C
tan x ln | tan x| + C
.
. 14
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15. . V63.0121.001: Calculus I
. Sec on 4.7: An deriva ves
. April 19, 2011
Notes
Final Thoughts
An deriva ves are a
useful concept, especially
in mo on y
We can graph an f
an deriva ve from the .
xF
graph of a func on 123456
We can compute
an deriva ves, but not
f(x) = e−x
2
always
. f′ (x) = ???
.
Notes
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Notes
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. 15
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