Lesson 23: Antiderivatives
-
Upload
mel-anthony-pepito -
Category
Technology
-
view
156 -
download
0
Transcript of Lesson 23: Antiderivatives
![Page 1: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/1.jpg)
Section 4.7Antiderivatives
V63.0121.002.2010Su, Calculus I
New York University
June 16, 2010
Announcements
I Quiz 4 Thursday on 4.1–4.4
. . . . . .
![Page 2: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/2.jpg)
. . . . . .
Announcements
I Quiz 4 Thursday on4.1–4.4
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 2 / 33
![Page 3: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/3.jpg)
. . . . . .
Objectives
I Given a ”simple“elementary function, find afunction whose derivativeis that function.
I Remember that a functionwhose derivative is zeroalong an interval must bezero along that interval.
I Solve problems involvingrectilinear motion.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 3 / 33
![Page 4: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/4.jpg)
. . . . . .
Outline
What is an antiderivative?
Tabulating AntiderivativesPower functionsCombinationsExponential functionsTrigonometric functions
Finding Antiderivatives Graphically
Rectilinear motion
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 4 / 33
![Page 5: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/5.jpg)
. . . . . .
What is an antiderivative?
DefinitionLet f be a function. An antiderivative for f is a function F such thatF′ = f.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 5 / 33
![Page 6: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/6.jpg)
. . . . . .
Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution???
Example
is F(x) = x ln x− x an antiderivative for f(x) = ln x?
Solution
ddx
(x ln x− x) = 1 · ln x+ x · 1x− 1 = ln x"
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 6 / 33
![Page 7: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/7.jpg)
. . . . . .
Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution???
Example
is F(x) = x ln x− x an antiderivative for f(x) = ln x?
Solution
ddx
(x ln x− x) = 1 · ln x+ x · 1x− 1 = ln x"
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 6 / 33
![Page 8: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/8.jpg)
. . . . . .
Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution???
Example
is F(x) = x ln x− x an antiderivative for f(x) = ln x?
Solution
ddx
(x ln x− x) = 1 · ln x+ x · 1x− 1 = ln x"
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 6 / 33
![Page 9: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/9.jpg)
. . . . . .
Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution???
Example
is F(x) = x ln x− x an antiderivative for f(x) = ln x?
Solution
ddx
(x ln x− x)
= 1 · ln x+ x · 1x− 1 = ln x"
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 6 / 33
![Page 10: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/10.jpg)
. . . . . .
Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution???
Example
is F(x) = x ln x− x an antiderivative for f(x) = ln x?
Solution
ddx
(x ln x− x) = 1 · ln x+ x · 1x− 1
= ln x"
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 6 / 33
![Page 11: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/11.jpg)
. . . . . .
Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution???
Example
is F(x) = x ln x− x an antiderivative for f(x) = ln x?
Solution
ddx
(x ln x− x) = 1 · ln x+ x · 1x− 1 = ln x
"
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 6 / 33
![Page 12: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/12.jpg)
. . . . . .
Hard problem, easy check
Example
Find an antiderivative for f(x) = ln x.
Solution???
Example
is F(x) = x ln x− x an antiderivative for f(x) = ln x?
Solution
ddx
(x ln x− x) = 1 · ln x+ x · 1x− 1 = ln x"
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 6 / 33
![Page 13: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/13.jpg)
. . . . . .
Why the MVT is the MITCMost Important Theorem In Calculus!
TheoremLet 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. Then f is continuous on[x, y] and differentiable on (x, y). By MVT there exists a point z in (x, y)such that
f(y)− f(x)y− x
= f′(z) =⇒ 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.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 7 / 33
![Page 14: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/14.jpg)
. . . . . .
When two functions have the same derivative
TheoremSuppose f and g are two differentiable functions on (a,b) with f′ = g′.Then f and g differ by a constant. That is, there exists a constant Csuch that f(x) = g(x) + C.
Proof.
I Let h(x) = f(x)− g(x)I Then h′(x) = f′(x)− g′(x) = 0 on (a,b)I So h(x) = C, a constantI This means f(x)− g(x) = C on (a,b)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 8 / 33
![Page 15: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/15.jpg)
. . . . . .
Outline
What is an antiderivative?
Tabulating AntiderivativesPower functionsCombinationsExponential functionsTrigonometric functions
Finding Antiderivatives Graphically
Rectilinear motion
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 9 / 33
![Page 16: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/16.jpg)
. . . . . .
Antiderivatives of power functions
Recall that the derivative of apower function is a powerfunction.
Fact (The Power Rule)
If f(x) = xr, then f′(x) = rxr−1.
So in looking for antiderivativesof power functions, try powerfunctions!
..x
.y.f(x) = x2
.f′(x) = 2x
.F(x) = ?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 10 / 33
![Page 17: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/17.jpg)
. . . . . .
Antiderivatives of power functions
Recall that the derivative of apower function is a powerfunction.
Fact (The Power Rule)
If f(x) = xr, then f′(x) = rxr−1.
So in looking for antiderivativesof power functions, try powerfunctions!
..x
.y.f(x) = x2
.f′(x) = 2x
.F(x) = ?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 10 / 33
![Page 18: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/18.jpg)
. . . . . .
Antiderivatives of power functions
Recall that the derivative of apower function is a powerfunction.
Fact (The Power Rule)
If f(x) = xr, then f′(x) = rxr−1.
So in looking for antiderivativesof power functions, try powerfunctions!
..x
.y.f(x) = x2
.f′(x) = 2x
.F(x) = ?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 10 / 33
![Page 19: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/19.jpg)
. . . . . .
Antiderivatives of power functions
Recall that the derivative of apower function is a powerfunction.
Fact (The Power Rule)
If f(x) = xr, then f′(x) = rxr−1.
So in looking for antiderivativesof power functions, try powerfunctions!
..x
.y.f(x) = x2
.f′(x) = 2x
.F(x) = ?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 10 / 33
![Page 20: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/20.jpg)
. . . . . .
Example
Find an antiderivative for the function f(x) = x3.
Solution
I Try a power function F(x) = axr
I Then F′(x) = arxr−1, so we want arxr−1 = x3.
I r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 is an antiderivative.
I Check:ddx
(14x4)
= 4 · 14x4−1 = x3 "
I Any others? Yes, F(x) =14x4 + C is the most general form.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 11 / 33
![Page 21: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/21.jpg)
. . . . . .
Example
Find an antiderivative for the function f(x) = x3.
Solution
I Try a power function F(x) = axr
I Then F′(x) = arxr−1, so we want arxr−1 = x3.
I r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 is an antiderivative.
I Check:ddx
(14x4)
= 4 · 14x4−1 = x3 "
I Any others? Yes, F(x) =14x4 + C is the most general form.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 11 / 33
![Page 22: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/22.jpg)
. . . . . .
Example
Find an antiderivative for the function f(x) = x3.
Solution
I Try a power function F(x) = axr
I Then F′(x) = arxr−1, so we want arxr−1 = x3.
I r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 is an antiderivative.
I Check:ddx
(14x4)
= 4 · 14x4−1 = x3 "
I Any others? Yes, F(x) =14x4 + C is the most general form.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 11 / 33
![Page 23: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/23.jpg)
. . . . . .
Example
Find an antiderivative for the function f(x) = x3.
Solution
I Try a power function F(x) = axr
I Then F′(x) = arxr−1, so we want arxr−1 = x3.
I r− 1 = 3 =⇒ r = 4
, and ar = 1 =⇒ a =14.
I So F(x) =14x4 is an antiderivative.
I Check:ddx
(14x4)
= 4 · 14x4−1 = x3 "
I Any others? Yes, F(x) =14x4 + C is the most general form.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 11 / 33
![Page 24: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/24.jpg)
. . . . . .
Example
Find an antiderivative for the function f(x) = x3.
Solution
I Try a power function F(x) = axr
I Then F′(x) = arxr−1, so we want arxr−1 = x3.
I r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 is an antiderivative.
I Check:ddx
(14x4)
= 4 · 14x4−1 = x3 "
I Any others? Yes, F(x) =14x4 + C is the most general form.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 11 / 33
![Page 25: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/25.jpg)
. . . . . .
Example
Find an antiderivative for the function f(x) = x3.
Solution
I Try a power function F(x) = axr
I Then F′(x) = arxr−1, so we want arxr−1 = x3.
I r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 is an antiderivative.
I Check:ddx
(14x4)
= 4 · 14x4−1 = x3 "
I Any others? Yes, F(x) =14x4 + C is the most general form.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 11 / 33
![Page 26: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/26.jpg)
. . . . . .
Example
Find an antiderivative for the function f(x) = x3.
Solution
I Try a power function F(x) = axr
I Then F′(x) = arxr−1, so we want arxr−1 = x3.
I r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 is an antiderivative.
I Check:ddx
(14x4)
= 4 · 14x4−1 = x3
"
I Any others? Yes, F(x) =14x4 + C is the most general form.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 11 / 33
![Page 27: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/27.jpg)
. . . . . .
Example
Find an antiderivative for the function f(x) = x3.
Solution
I Try a power function F(x) = axr
I Then F′(x) = arxr−1, so we want arxr−1 = x3.
I r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 is an antiderivative.
I Check:ddx
(14x4)
= 4 · 14x4−1 = x3 "
I Any others? Yes, F(x) =14x4 + C is the most general form.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 11 / 33
![Page 28: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/28.jpg)
. . . . . .
Example
Find an antiderivative for the function f(x) = x3.
Solution
I Try a power function F(x) = axr
I Then F′(x) = arxr−1, so we want arxr−1 = x3.
I r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 is an antiderivative.
I Check:ddx
(14x4)
= 4 · 14x4−1 = x3 "
I Any others?
Yes, F(x) =14x4 + C is the most general form.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 11 / 33
![Page 29: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/29.jpg)
. . . . . .
Example
Find an antiderivative for the function f(x) = x3.
Solution
I Try a power function F(x) = axr
I Then F′(x) = arxr−1, so we want arxr−1 = x3.
I r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 is an antiderivative.
I Check:ddx
(14x4)
= 4 · 14x4−1 = x3 "
I Any others? Yes, F(x) =14x4 + C is the most general form.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 11 / 33
![Page 30: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/30.jpg)
. . . . . .
Fact (The Power Rule for antiderivatives)
If f(x) = xr, then
F(x) =1
r+ 1xr+1
is an antiderivative for f…
as long as r ̸= −1.
Fact
If f(x) = x−1 =1x, then
F(x) = ln |x|+ C
is an antiderivative for f.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 12 / 33
![Page 31: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/31.jpg)
. . . . . .
Fact (The Power Rule for antiderivatives)
If f(x) = xr, then
F(x) =1
r+ 1xr+1
is an antiderivative for f as long as r ̸= −1.
Fact
If f(x) = x−1 =1x, then
F(x) = ln |x|+ C
is an antiderivative for f.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 12 / 33
![Page 32: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/32.jpg)
. . . . . .
Fact (The Power Rule for antiderivatives)
If f(x) = xr, then
F(x) =1
r+ 1xr+1
is an antiderivative for f as long as r ̸= −1.
Fact
If f(x) = x−1 =1x, then
F(x) = ln |x|+ C
is an antiderivative for f.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 12 / 33
![Page 33: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/33.jpg)
. . . . . .
What's with the absolute value?
F(x) = ln |x| =
{ln(x) if x > 0;ln(−x) if x < 0.
I The domain of F is all nonzero numbers, while ln x is only definedon positive numbers.
I If x > 0,ddx
ln |x| = ddx
ln(x) =1x"
I If x < 0,
ddx
ln |x| = ddx
ln(−x) =1−x
· (−1) =1x"
I We prefer the antiderivative with the larger domain.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 13 / 33
![Page 34: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/34.jpg)
. . . . . .
What's with the absolute value?
F(x) = ln |x| =
{ln(x) if x > 0;ln(−x) if x < 0.
I The domain of F is all nonzero numbers, while ln x is only definedon positive numbers.
I If x > 0,ddx
ln |x|
=ddx
ln(x) =1x"
I If x < 0,
ddx
ln |x| = ddx
ln(−x) =1−x
· (−1) =1x"
I We prefer the antiderivative with the larger domain.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 13 / 33
![Page 35: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/35.jpg)
. . . . . .
What's with the absolute value?
F(x) = ln |x| =
{ln(x) if x > 0;ln(−x) if x < 0.
I The domain of F is all nonzero numbers, while ln x is only definedon positive numbers.
I If x > 0,ddx
ln |x| = ddx
ln(x)
=1x"
I If x < 0,
ddx
ln |x| = ddx
ln(−x) =1−x
· (−1) =1x"
I We prefer the antiderivative with the larger domain.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 13 / 33
![Page 36: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/36.jpg)
. . . . . .
What's with the absolute value?
F(x) = ln |x| =
{ln(x) if x > 0;ln(−x) if x < 0.
I The domain of F is all nonzero numbers, while ln x is only definedon positive numbers.
I If x > 0,ddx
ln |x| = ddx
ln(x) =1x
"
I If x < 0,
ddx
ln |x| = ddx
ln(−x) =1−x
· (−1) =1x"
I We prefer the antiderivative with the larger domain.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 13 / 33
![Page 37: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/37.jpg)
. . . . . .
What's with the absolute value?
F(x) = ln |x| =
{ln(x) if x > 0;ln(−x) if x < 0.
I The domain of F is all nonzero numbers, while ln x is only definedon positive numbers.
I If x > 0,ddx
ln |x| = ddx
ln(x) =1x"
I If x < 0,
ddx
ln |x| = ddx
ln(−x) =1−x
· (−1) =1x"
I We prefer the antiderivative with the larger domain.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 13 / 33
![Page 38: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/38.jpg)
. . . . . .
What's with the absolute value?
F(x) = ln |x| =
{ln(x) if x > 0;ln(−x) if x < 0.
I The domain of F is all nonzero numbers, while ln x is only definedon positive numbers.
I If x > 0,ddx
ln |x| = ddx
ln(x) =1x"
I If x < 0,
ddx
ln |x|
=ddx
ln(−x) =1−x
· (−1) =1x"
I We prefer the antiderivative with the larger domain.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 13 / 33
![Page 39: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/39.jpg)
. . . . . .
What's with the absolute value?
F(x) = ln |x| =
{ln(x) if x > 0;ln(−x) if x < 0.
I The domain of F is all nonzero numbers, while ln x is only definedon positive numbers.
I If x > 0,ddx
ln |x| = ddx
ln(x) =1x"
I If x < 0,
ddx
ln |x| = ddx
ln(−x)
=1−x
· (−1) =1x"
I We prefer the antiderivative with the larger domain.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 13 / 33
![Page 40: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/40.jpg)
. . . . . .
What's with the absolute value?
F(x) = ln |x| =
{ln(x) if x > 0;ln(−x) if x < 0.
I The domain of F is all nonzero numbers, while ln x is only definedon positive numbers.
I If x > 0,ddx
ln |x| = ddx
ln(x) =1x"
I If x < 0,
ddx
ln |x| = ddx
ln(−x) =1−x
· (−1)
=1x"
I We prefer the antiderivative with the larger domain.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 13 / 33
![Page 41: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/41.jpg)
. . . . . .
What's with the absolute value?
F(x) = ln |x| =
{ln(x) if x > 0;ln(−x) if x < 0.
I The domain of F is all nonzero numbers, while ln x is only definedon positive numbers.
I If x > 0,ddx
ln |x| = ddx
ln(x) =1x"
I If x < 0,
ddx
ln |x| = ddx
ln(−x) =1−x
· (−1) =1x
"
I We prefer the antiderivative with the larger domain.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 13 / 33
![Page 42: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/42.jpg)
. . . . . .
What's with the absolute value?
F(x) = ln |x| =
{ln(x) if x > 0;ln(−x) if x < 0.
I The domain of F is all nonzero numbers, while ln x is only definedon positive numbers.
I If x > 0,ddx
ln |x| = ddx
ln(x) =1x"
I If x < 0,
ddx
ln |x| = ddx
ln(−x) =1−x
· (−1) =1x"
I We prefer the antiderivative with the larger domain.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 13 / 33
![Page 43: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/43.jpg)
. . . . . .
What's with the absolute value?
F(x) = ln |x| =
{ln(x) if x > 0;ln(−x) if x < 0.
I The domain of F is all nonzero numbers, while ln x is only definedon positive numbers.
I If x > 0,ddx
ln |x| = ddx
ln(x) =1x"
I If x < 0,
ddx
ln |x| = ddx
ln(−x) =1−x
· (−1) =1x"
I We prefer the antiderivative with the larger domain.V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 13 / 33
![Page 44: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/44.jpg)
. . . . . .
Graph of ln |x|
. .x
.y
.f(x) = 1/x
.F(x) =
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 14 / 33
![Page 45: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/45.jpg)
. . . . . .
Graph of ln |x|
. .x
.y
.f(x) = 1/x
.F(x) = ln(x)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 14 / 33
![Page 46: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/46.jpg)
. . . . . .
Graph of ln |x|
. .x
.y
.f(x) = 1/x
.F(x) = ln |x|
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 14 / 33
![Page 47: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/47.jpg)
. . . . . .
Combinations of antiderivatives
Fact (Sum and Constant Multiple Rule for Antiderivatives)
I If F is an antiderivative of f and G is an antiderivative of g, thenF+G is an antiderivative of f+ g.
I If F is an antiderivative of f and c is a constant, then cF is anantiderivative of cf.
Proof.These follow from the sum and constant multiple rule for derivatives:
I If F′ = f and G′ = g, then
(F+G)′ = F′ +G′ = f+ g
I Or, if F′ = f,(cF)′ = cF′ = cf
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 15 / 33
![Page 48: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/48.jpg)
. . . . . .
Combinations of antiderivatives
Fact (Sum and Constant Multiple Rule for Antiderivatives)
I If F is an antiderivative of f and G is an antiderivative of g, thenF+G is an antiderivative of f+ g.
I If F is an antiderivative of f and c is a constant, then cF is anantiderivative of cf.
Proof.These follow from the sum and constant multiple rule for derivatives:
I If F′ = f and G′ = g, then
(F+G)′ = F′ +G′ = f+ g
I Or, if F′ = f,(cF)′ = cF′ = cf
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 15 / 33
![Page 49: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/49.jpg)
. . . . . .
Antiderivatives of Polynomials
Example
Find an antiderivative for f(x) = 16x+ 5.
Solution
The expression12x2 is an antiderivative for x, and x is an antiderivative
for 1. So
F(x) = 16 ·(12x2)+ 5 · x+ C = 8x2 + 5x+ C
is the antiderivative of f.
QuestionWhy do we not need two C’s?
AnswerA combination of two arbitrary constants is still an arbitrary constant.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 16 / 33
![Page 50: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/50.jpg)
. . . . . .
Antiderivatives of Polynomials
Example
Find an antiderivative for f(x) = 16x+ 5.
Solution
The expression12x2 is an antiderivative for x, and x is an antiderivative
for 1. So
F(x) = 16 ·(12x2)+ 5 · x+ C = 8x2 + 5x+ C
is the antiderivative of f.
QuestionWhy do we not need two C’s?
AnswerA combination of two arbitrary constants is still an arbitrary constant.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 16 / 33
![Page 51: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/51.jpg)
. . . . . .
Antiderivatives of Polynomials
Example
Find an antiderivative for f(x) = 16x+ 5.
Solution
The expression12x2 is an antiderivative for x, and x is an antiderivative
for 1. So
F(x) = 16 ·(12x2)+ 5 · x+ C = 8x2 + 5x+ C
is the antiderivative of f.
QuestionWhy do we not need two C’s?
AnswerA combination of two arbitrary constants is still an arbitrary constant.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 16 / 33
![Page 52: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/52.jpg)
. . . . . .
Antiderivatives of Polynomials
Example
Find an antiderivative for f(x) = 16x+ 5.
Solution
The expression12x2 is an antiderivative for x, and x is an antiderivative
for 1. So
F(x) = 16 ·(12x2)+ 5 · x+ C = 8x2 + 5x+ C
is the antiderivative of f.
QuestionWhy do we not need two C’s?
AnswerA combination of two arbitrary constants is still an arbitrary constant.V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 16 / 33
![Page 53: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/53.jpg)
. . . . . .
Exponential Functions
FactIf f(x) = ax, f′(x) = (ln a)ax.
Accordingly,
Fact
If f(x) = ax, then F(x) =1ln a
ax + C is the antiderivative of f.
Proof.Check it yourself.
In particular,
FactIf f(x) = ex, then F(x) = ex + C is the antiderivative of f.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 17 / 33
![Page 54: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/54.jpg)
. . . . . .
Exponential Functions
FactIf f(x) = ax, f′(x) = (ln a)ax.
Accordingly,
Fact
If f(x) = ax, then F(x) =1ln a
ax + C is the antiderivative of f.
Proof.Check it yourself.
In particular,
FactIf f(x) = ex, then F(x) = ex + C is the antiderivative of f.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 17 / 33
![Page 55: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/55.jpg)
. . . . . .
Exponential Functions
FactIf f(x) = ax, f′(x) = (ln a)ax.
Accordingly,
Fact
If f(x) = ax, then F(x) =1ln a
ax + C is the antiderivative of f.
Proof.Check it yourself.
In particular,
FactIf f(x) = ex, then F(x) = ex + C is the antiderivative of f.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 17 / 33
![Page 56: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/56.jpg)
. . . . . .
Exponential Functions
FactIf f(x) = ax, f′(x) = (ln a)ax.
Accordingly,
Fact
If f(x) = ax, then F(x) =1ln a
ax + C is the antiderivative of f.
Proof.Check it yourself.
In particular,
FactIf f(x) = ex, then F(x) = ex + C is the antiderivative of f.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 17 / 33
![Page 57: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/57.jpg)
. . . . . .
Logarithmic functions?
I Remember we found
F(x) = x ln x− x
is an antiderivative of f(x) = ln x.
I This is not obvious. See Calc II for the full story.
I However, using the fact that loga x =ln xln a
, we get:
FactIf f(x) = loga(x)
F(x) =1ln a
(x ln x− x) + C = x loga x−1ln a
x+ C
is the antiderivative of f(x).
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 18 / 33
![Page 58: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/58.jpg)
. . . . . .
Logarithmic functions?
I Remember we found
F(x) = x ln x− x
is an antiderivative of f(x) = ln x.I This is not obvious. See Calc II for the full story.
I However, using the fact that loga x =ln xln a
, we get:
FactIf f(x) = loga(x)
F(x) =1ln a
(x ln x− x) + C = x loga x−1ln a
x+ C
is the antiderivative of f(x).
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 18 / 33
![Page 59: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/59.jpg)
. . . . . .
Logarithmic functions?
I Remember we found
F(x) = x ln x− x
is an antiderivative of f(x) = ln x.I This is not obvious. See Calc II for the full story.
I However, using the fact that loga x =ln xln a
, we get:
FactIf f(x) = loga(x)
F(x) =1ln a
(x ln x− x) + C = x loga x−1ln a
x+ C
is the antiderivative of f(x).
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 18 / 33
![Page 60: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/60.jpg)
. . . . . .
Trigonometric functions
Fact
ddx
sin x = cos xddx
cos x = − sin x
So to turn these around,
Fact
I The function F(x) = − cos x+C is the antiderivative of f(x) = sin x.I The function F(x) = sin x+ C is the antiderivative of f(x) = cos x.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 19 / 33
![Page 61: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/61.jpg)
. . . . . .
Trigonometric functions
Fact
ddx
sin x = cos xddx
cos x = − sin x
So to turn these around,
Fact
I The function F(x) = − cos x+C is the antiderivative of f(x) = sin x.
I The function F(x) = sin x+ C is the antiderivative of f(x) = cos x.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 19 / 33
![Page 62: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/62.jpg)
. . . . . .
Trigonometric functions
Fact
ddx
sin x = cos xddx
cos x = − sin x
So to turn these around,
Fact
I The function F(x) = − cos x+C is the antiderivative of f(x) = sin x.I The function F(x) = sin x+ C is the antiderivative of f(x) = cos x.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 19 / 33
![Page 63: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/63.jpg)
. . . . . .
More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution???
AnswerF(x) = ln(sec x).
Check
ddx
=1
sec x· ddx
sec x =1
sec x· sec x tan x = tan x"
More about this later.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 20 / 33
![Page 64: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/64.jpg)
. . . . . .
More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution???
AnswerF(x) = ln(sec x).
Check
ddx
=1
sec x· ddx
sec x =1
sec x· sec x tan x = tan x"
More about this later.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 20 / 33
![Page 65: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/65.jpg)
. . . . . .
More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution???
AnswerF(x) = ln(sec x).
Check
ddx
=1
sec x· ddx
sec x =1
sec x· sec x tan x = tan x"
More about this later.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 20 / 33
![Page 66: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/66.jpg)
. . . . . .
More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution???
AnswerF(x) = ln(sec x).
Check
ddx
=1
sec x· ddx
sec x
=1
sec x· sec x tan x = tan x"
More about this later.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 20 / 33
![Page 67: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/67.jpg)
. . . . . .
More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution???
AnswerF(x) = ln(sec x).
Check
ddx
=1
sec x· ddx
sec x =1
sec x· sec x tan x
= tan x"
More about this later.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 20 / 33
![Page 68: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/68.jpg)
. . . . . .
More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution???
AnswerF(x) = ln(sec x).
Check
ddx
=1
sec x· ddx
sec x =1
sec x· sec x tan x = tan x
"
More about this later.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 20 / 33
![Page 69: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/69.jpg)
. . . . . .
More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution???
AnswerF(x) = ln(sec x).
Check
ddx
=1
sec x· ddx
sec x =1
sec x· sec x tan x = tan x"
More about this later.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 20 / 33
![Page 70: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/70.jpg)
. . . . . .
More Trig
Example
Find an antiderivative of f(x) = tan x.
Solution???
AnswerF(x) = ln(sec x).
Check
ddx
=1
sec x· ddx
sec x =1
sec x· sec x tan x = tan x"
More about this later.V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 20 / 33
![Page 71: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/71.jpg)
. . . . . .
Outline
What is an antiderivative?
Tabulating AntiderivativesPower functionsCombinationsExponential functionsTrigonometric functions
Finding Antiderivatives Graphically
Rectilinear motion
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 21 / 33
![Page 72: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/72.jpg)
. . . . . .
Finding Antiderivatives Graphically
ProblemBelow is the graph of a function f. Draw the graph of an antiderivativefor f.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.
. .
. .y = f(x)
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 22 / 33
![Page 73: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/73.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 74: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/74.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+
.+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 75: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/75.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+
.− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 76: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/76.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .−
.− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 77: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/77.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .−
.+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 78: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/78.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+
.↗ .↗ .↘ .↘ .↗. max .min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 79: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/79.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗
.↗ .↘ .↘ .↗. max .min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 80: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/80.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗
.↘ .↘ .↗. max .min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 81: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/81.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘
.↘ .↗. max .min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 82: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/82.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘
.↗. max .min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 83: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/83.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗
. max .min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 84: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/84.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max
.min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 85: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/85.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 86: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/86.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++
.−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 87: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/87.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−−
.−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 88: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/88.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−−
.++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 89: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/89.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++
.++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 90: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/90.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++
.⌣ .⌢ .⌢ .⌣ .⌣.
IP.
IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 91: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/91.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣
.⌢ .⌢ .⌣ .⌣.
IP.
IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 92: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/92.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢
.⌢ .⌣ .⌣.
IP.
IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 93: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/93.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢
.⌣ .⌣.
IP.
IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 94: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/94.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣
.⌣.
IP.
IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 95: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/95.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 96: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/96.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 97: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/97.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6
. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 98: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/98.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6. "
." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 99: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/99.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6. " ."
. . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 100: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/100.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6. " ." .
. . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 101: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/101.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6. " ." . .
. ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 102: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/102.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 103: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/103.jpg)
. . . . . .
Using f to make a sign chart for F
Assuming F′ = f, we can make a sign chart for f and f′ to find theintervals of monotonicity and concavity for F:
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
.
. .f = F′
.F..1
..2
..3
..4
..5
..6
.+ .+ .− .− .+.↗ .↗ .↘ .↘ .↗. max .
min
.f′ = F′′
.F..1
..2
..3
..4
..5
..6
.++ .−− .−− .++ .++.⌣ .⌢ .⌢ .⌣ .⌣
.IP
.IP
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . ".? .? .? .? .? .?
The only question left is: What are the function values?
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 23 / 33
![Page 104: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/104.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.I Using the sign chart, we
draw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
..
.
..
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 105: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/105.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.
I Using the sign chart, wedraw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
..
.
..
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 106: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/106.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.I Using the sign chart, we
draw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
..
.
..
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 107: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/107.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.I Using the sign chart, we
draw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
.
.
.
..
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 108: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/108.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.I Using the sign chart, we
draw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
.
.
.
..
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 109: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/109.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.I Using the sign chart, we
draw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
..
.
..
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 110: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/110.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.I Using the sign chart, we
draw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
..
.
..
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 111: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/111.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.I Using the sign chart, we
draw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
..
.
..
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 112: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/112.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.I Using the sign chart, we
draw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
..
.
..
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 113: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/113.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.I Using the sign chart, we
draw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
..
.
.
.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 114: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/114.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.I Using the sign chart, we
draw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
..
.
..
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 115: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/115.jpg)
. . . . . .
Could you repeat the question?
ProblemBelow is the graph of a function f. Draw the graph of the antiderivativefor f with F(1) = 0.
Solution
I We start with F(1) = 0.I Using the sign chart, we
draw arcs with thespecified monotonicity andconcavity
I It’s harder to tell if/when Fcrosses the axis; moreabout that later.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
..
.
..
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 24 / 33
![Page 116: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/116.jpg)
. . . . . .
Outline
What is an antiderivative?
Tabulating AntiderivativesPower functionsCombinationsExponential functionsTrigonometric functions
Finding Antiderivatives Graphically
Rectilinear motion
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 25 / 33
![Page 117: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/117.jpg)
. . . . . .
Say what?
I “Rectilinear motion” just means motion along a line.I Often we are given information about the velocity or acceleration
of a moving particle and we want to know the equations of motion.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 26 / 33
![Page 118: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/118.jpg)
. . . . . .
Application: Dead Reckoning
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 27 / 33
![Page 119: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/119.jpg)
. . . . . .
Application: Dead Reckoning
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 27 / 33
![Page 120: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/120.jpg)
. . . . . .
ProblemSuppose a particle of mass m is acted upon by a constant force F.Find the position function s(t), the velocity function v(t), and theacceleration function a(t).
Solution
I By Newton’s Second Law (F = ma) a constant force induces a
constant acceleration. So a(t) = a =Fm.
I Since v′(t) = a(t), v(t) must be an antiderivative of the constantfunction a. So
v(t) = at+ C = at+ v0
where v0 is the initial velocity.I Since s′(t) = v(t), s(t) must be an antiderivative of v(t), meaning
s(t) =12at2 + v0t+ C =
12at2 + v0t+ s0
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 28 / 33
![Page 121: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/121.jpg)
. . . . . .
ProblemSuppose a particle of mass m is acted upon by a constant force F.Find the position function s(t), the velocity function v(t), and theacceleration function a(t).
Solution
I By Newton’s Second Law (F = ma) a constant force induces a
constant acceleration. So a(t) = a =Fm.
I Since v′(t) = a(t), v(t) must be an antiderivative of the constantfunction a. So
v(t) = at+ C = at+ v0
where v0 is the initial velocity.I Since s′(t) = v(t), s(t) must be an antiderivative of v(t), meaning
s(t) =12at2 + v0t+ C =
12at2 + v0t+ s0
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 28 / 33
![Page 122: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/122.jpg)
. . . . . .
ProblemSuppose a particle of mass m is acted upon by a constant force F.Find the position function s(t), the velocity function v(t), and theacceleration function a(t).
Solution
I By Newton’s Second Law (F = ma) a constant force induces a
constant acceleration. So a(t) = a =Fm.
I Since v′(t) = a(t), v(t) must be an antiderivative of the constantfunction a. So
v(t) = at+ C = at+ v0
where v0 is the initial velocity.
I Since s′(t) = v(t), s(t) must be an antiderivative of v(t), meaning
s(t) =12at2 + v0t+ C =
12at2 + v0t+ s0
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 28 / 33
![Page 123: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/123.jpg)
. . . . . .
ProblemSuppose a particle of mass m is acted upon by a constant force F.Find the position function s(t), the velocity function v(t), and theacceleration function a(t).
Solution
I By Newton’s Second Law (F = ma) a constant force induces a
constant acceleration. So a(t) = a =Fm.
I Since v′(t) = a(t), v(t) must be an antiderivative of the constantfunction a. So
v(t) = at+ C = at+ v0
where v0 is the initial velocity.I Since s′(t) = v(t), s(t) must be an antiderivative of v(t), meaning
s(t) =12at2 + v0t+ C =
12at2 + v0t+ s0
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 28 / 33
![Page 124: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/124.jpg)
. . . . . .
An earlier Hatsumon
Example
Drop a ball off the roof of the Silver Center. What is its velocity when ithits the ground?
SolutionAssume s0 = 100m, 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
√5m/s.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 29 / 33
![Page 125: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/125.jpg)
. . . . . .
An earlier Hatsumon
Example
Drop a ball off the roof of the Silver Center. What is its velocity when ithits the ground?
SolutionAssume s0 = 100m, 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
√5m/s.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 29 / 33
![Page 126: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/126.jpg)
. . . . . .
Finding initial velocity from stopping distance
Example
The skid marks made by an automobile indicate that its brakes werefully applied for a distance of 160 ft before it came to a stop. Supposethat the car in question has a constant deceleration of 20 ft/s2 under theconditions of the skid. How fast was the car traveling when its brakeswere first applied?
Solution (Setup)
I While breaking, the car has acceleration a(t) = −20I Measure time 0 and position 0 when the car starts braking. So
s(0) = 0.I The car stops at time some t1, when v(t1) = 0.I We know that when s(t1) = 160.I We want to know v(0), or v0.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 30 / 33
![Page 127: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/127.jpg)
. . . . . .
Finding initial velocity from stopping distance
Example
The skid marks made by an automobile indicate that its brakes werefully applied for a distance of 160 ft before it came to a stop. Supposethat the car in question has a constant deceleration of 20 ft/s2 under theconditions of the skid. How fast was the car traveling when its brakeswere first applied?
Solution (Setup)
I While breaking, the car has acceleration a(t) = −20
I Measure time 0 and position 0 when the car starts braking. Sos(0) = 0.
I The car stops at time some t1, when v(t1) = 0.I We know that when s(t1) = 160.I We want to know v(0), or v0.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 30 / 33
![Page 128: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/128.jpg)
. . . . . .
Finding initial velocity from stopping distance
Example
The skid marks made by an automobile indicate that its brakes werefully applied for a distance of 160 ft before it came to a stop. Supposethat the car in question has a constant deceleration of 20 ft/s2 under theconditions of the skid. How fast was the car traveling when its brakeswere first applied?
Solution (Setup)
I While breaking, the car has acceleration a(t) = −20I Measure time 0 and position 0 when the car starts braking. So
s(0) = 0.I The car stops at time some t1, when v(t1) = 0.
I We know that when s(t1) = 160.I We want to know v(0), or v0.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 30 / 33
![Page 129: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/129.jpg)
. . . . . .
Finding initial velocity from stopping distance
Example
The skid marks made by an automobile indicate that its brakes werefully applied for a distance of 160 ft before it came to a stop. Supposethat the car in question has a constant deceleration of 20 ft/s2 under theconditions of the skid. How fast was the car traveling when its brakeswere first applied?
Solution (Setup)
I While breaking, the car has acceleration a(t) = −20I Measure time 0 and position 0 when the car starts braking. So
s(0) = 0.I The car stops at time some t1, when v(t1) = 0.I We know that when s(t1) = 160.I We want to know v(0), or v0.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 30 / 33
![Page 130: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/130.jpg)
. . . . . .
Implementing the Solution
In general,
s(t) = s0 + v0t+12at2
Since s0 = 0 and a = −20, we have
s(t) = v0t− 10t2
v(t) = v0 − 20t
for all t.
Plugging in t = t1,
160 = v0t1 − 10t210 = v0 − 20t1
We need to solve these two equations.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 31 / 33
![Page 131: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/131.jpg)
. . . . . .
Implementing the Solution
In general,
s(t) = s0 + v0t+12at2
Since s0 = 0 and a = −20, we have
s(t) = v0t− 10t2
v(t) = v0 − 20t
for all t. Plugging in t = t1,
160 = v0t1 − 10t210 = v0 − 20t1
We need to solve these two equations.
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 31 / 33
![Page 132: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/132.jpg)
. . . . . .
Solving
We havev0t1 − 10t21 = 160 v0 − 20t1 = 0
I The second gives t1 = v0/20, so substitute into the first:
v0 ·v020
− 10( v020
)2= 160
or
v2020
−10v20400
= 160
2v20 − v20 = 160 · 40 = 6400
I So v0 = 80 ft/s ≈ 55mi/hr
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 32 / 33
![Page 133: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/133.jpg)
. . . . . .
Solving
We havev0t1 − 10t21 = 160 v0 − 20t1 = 0
I The second gives t1 = v0/20, so substitute into the first:
v0 ·v020
− 10( v020
)2= 160
or
v2020
−10v20400
= 160
2v20 − v20 = 160 · 40 = 6400
I So v0 = 80 ft/s ≈ 55mi/hr
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 32 / 33
![Page 134: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/134.jpg)
. . . . . .
Solving
We havev0t1 − 10t21 = 160 v0 − 20t1 = 0
I The second gives t1 = v0/20, so substitute into the first:
v0 ·v020
− 10( v020
)2= 160
or
v2020
−10v20400
= 160
2v20 − v20 = 160 · 40 = 6400
I So v0 = 80 ft/s ≈ 55mi/hr
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 32 / 33
![Page 135: Lesson 23: Antiderivatives](https://reader036.fdocuments.in/reader036/viewer/2022081400/55508e9eb4c905a85c8b4f8f/html5/thumbnails/135.jpg)
. . . . . .
Summary
I Antiderivatives are a usefulconcept, especially inmotion
I We can graph anantiderivative from thegraph of a function
I We can computeantiderivatives, but notalways
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f.
..
.
...F
f(x) = e−x2
f′(x) = ???
V63.0121.002.2010Su, Calculus I (NYU) Section 4.7 Antiderivatives June 16, 2010 33 / 33