Lesson 32: The Fundamental Theorem Of Calculus
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Transcript of Lesson 32: The Fundamental Theorem Of Calculus
![Page 1: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/1.jpg)
Section 5.4The Fundamental Theorem of Calculus
Math 1a
December 12, 2007
Announcements
I my next office hours: Today 1–3 (SC 323)
I MT II is graded. Come to OH to talk about it
I Final seview sessions: Wed 1/9 and Thu 1/10 in Hall D, Sun1/13 in Hall C, all 7–8:30pm
I Final tentatively scheduled for January 17, 9:15am
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Outline
The Area Function
FTC1StatementProofBiographies
Differentiation of functions defined by integrals“Contrived” examplesErfOther applications
FTC2
Facts about g from fA problem
![Page 3: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/3.jpg)
An area function
Let f (t) = t2 and define g(x) =
∫ x
0t3 dt. Can we evaluate the
integral in g(x)?
0 x
Dividing the interval [0, x ] into n pieces
gives ∆x =x
nand xi = 0 + i∆x =
ix
n.
So
Rn =x
n· x3
n3+
x
n· (2x)3
n3+ · · ·+ x
n· (nx)3
n3
=x4
n4
(13 + 23 + 33 + · · ·+ n3
)=
x4
n4
[12n(n + 1)
]2=
x4n2(n + 1)2
4n4→ x4
4
as n→∞.
![Page 4: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/4.jpg)
An area function
Let f (t) = t2 and define g(x) =
∫ x
0t3 dt. Can we evaluate the
integral in g(x)?
0 x
Dividing the interval [0, x ] into n pieces
gives ∆x =x
nand xi = 0 + i∆x =
ix
n.
So
Rn =x
n· x3
n3+
x
n· (2x)3
n3+ · · ·+ x
n· (nx)3
n3
=x4
n4
(13 + 23 + 33 + · · ·+ n3
)=
x4
n4
[12n(n + 1)
]2=
x4n2(n + 1)2
4n4→ x4
4
as n→∞.
![Page 5: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/5.jpg)
An area function, continued
So
g(x) =x4
4.
This means thatg ′(x) = x3.
![Page 6: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/6.jpg)
An area function, continued
So
g(x) =x4
4.
This means thatg ′(x) = x3.
![Page 7: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/7.jpg)
The area function
Let f be a function which is integrable (i.e., continuous or withfinitely many jump discontinuities) on [a, b]. Define
g(x) =
∫ t
af (t) dt.
I When is g increasing?
I When is g decreasing?
I Over a small interval, what’s the average rate of change of g?
![Page 8: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/8.jpg)
The area function
Let f be a function which is integrable (i.e., continuous or withfinitely many jump discontinuities) on [a, b]. Define
g(x) =
∫ t
af (t) dt.
I When is g increasing?
I When is g decreasing?
I Over a small interval, what’s the average rate of change of g?
![Page 9: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/9.jpg)
The area function
Let f be a function which is integrable (i.e., continuous or withfinitely many jump discontinuities) on [a, b]. Define
g(x) =
∫ t
af (t) dt.
I When is g increasing?
I When is g decreasing?
I Over a small interval, what’s the average rate of change of g?
![Page 10: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/10.jpg)
Outline
The Area Function
FTC1StatementProofBiographies
Differentiation of functions defined by integrals“Contrived” examplesErfOther applications
FTC2
Facts about g from fA problem
![Page 11: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/11.jpg)
Theorem (The First Fundamental Theorem of Calculus)
Let f be an integrable function on [a, b] and define
g(x) =
∫ x
af (t) dt.
If f is continuous at x in (a, b), then g is differentiable at x and
g ′(x) = f (x).
![Page 12: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/12.jpg)
Proof.Let h > 0 be given so that x + h < b. We have
g(x + h)− g(x)
h=
1
h
∫ x+h
xf (t) dt.
Let Mh be the maximum value of f on [x , x + h], and mh theminimum value of f on [x , x + h]. From §5.2 we have
mh · h ≤
∫ x+h
xf (t) dt
≤ Mh · h
So
mh ≤g(x + h)− g(x)
h≤ Mh.
As h→ 0, both mh and Mh tend to f (x). Zappa-dappa.
![Page 13: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/13.jpg)
Proof.Let h > 0 be given so that x + h < b. We have
g(x + h)− g(x)
h=
1
h
∫ x+h
xf (t) dt.
Let Mh be the maximum value of f on [x , x + h], and mh theminimum value of f on [x , x + h]. From §5.2 we have
mh · h ≤
∫ x+h
xf (t) dt
≤ Mh · h
So
mh ≤g(x + h)− g(x)
h≤ Mh.
As h→ 0, both mh and Mh tend to f (x). Zappa-dappa.
![Page 14: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/14.jpg)
Proof.Let h > 0 be given so that x + h < b. We have
g(x + h)− g(x)
h=
1
h
∫ x+h
xf (t) dt.
Let Mh be the maximum value of f on [x , x + h], and mh theminimum value of f on [x , x + h]. From §5.2 we have
mh · h ≤
∫ x+h
xf (t) dt
≤ Mh · h
So
mh ≤g(x + h)− g(x)
h≤ Mh.
As h→ 0, both mh and Mh tend to f (x). Zappa-dappa.
![Page 15: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/15.jpg)
Proof.Let h > 0 be given so that x + h < b. We have
g(x + h)− g(x)
h=
1
h
∫ x+h
xf (t) dt.
Let Mh be the maximum value of f on [x , x + h], and mh theminimum value of f on [x , x + h]. From §5.2 we have
mh · h ≤
∫ x+h
xf (t) dt ≤ Mh · h
So
mh ≤g(x + h)− g(x)
h≤ Mh.
As h→ 0, both mh and Mh tend to f (x). Zappa-dappa.
![Page 16: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/16.jpg)
Proof.Let h > 0 be given so that x + h < b. We have
g(x + h)− g(x)
h=
1
h
∫ x+h
xf (t) dt.
Let Mh be the maximum value of f on [x , x + h], and mh theminimum value of f on [x , x + h]. From §5.2 we have
mh · h ≤∫ x+h
xf (t) dt ≤ Mh · h
So
mh ≤g(x + h)− g(x)
h≤ Mh.
As h→ 0, both mh and Mh tend to f (x). Zappa-dappa.
![Page 17: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/17.jpg)
Proof.Let h > 0 be given so that x + h < b. We have
g(x + h)− g(x)
h=
1
h
∫ x+h
xf (t) dt.
Let Mh be the maximum value of f on [x , x + h], and mh theminimum value of f on [x , x + h]. From §5.2 we have
mh · h ≤∫ x+h
xf (t) dt ≤ Mh · h
So
mh ≤g(x + h)− g(x)
h≤ Mh.
As h→ 0, both mh and Mh tend to f (x). Zappa-dappa.
![Page 18: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/18.jpg)
Proof.Let h > 0 be given so that x + h < b. We have
g(x + h)− g(x)
h=
1
h
∫ x+h
xf (t) dt.
Let Mh be the maximum value of f on [x , x + h], and mh theminimum value of f on [x , x + h]. From §5.2 we have
mh · h ≤∫ x+h
xf (t) dt ≤ Mh · h
So
mh ≤g(x + h)− g(x)
h≤ Mh.
As h→ 0, both mh and Mh tend to f (x). Zappa-dappa.
![Page 19: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/19.jpg)
Meet the Mathematician: Isaac Barrow
I English, 1630-1677
I Professor of Greek,theology, andmathematics atCambridge
I Had a famous student
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Meet the Mathematician: Isaac Newton
I English, 1643–1727
I Professor at Cambridge(England)
I Philosophiae NaturalisPrincipia Mathematicapublished 1687
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Meet the Mathematician: Gottfried Leibniz
I German, 1646–1716
I Eminent philosopher aswell as mathematician
I Contemporarily disgracedby the calculus prioritydispute
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Outline
The Area Function
FTC1StatementProofBiographies
Differentiation of functions defined by integrals“Contrived” examplesErfOther applications
FTC2
Facts about g from fA problem
![Page 23: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/23.jpg)
Differentiation of area functions
Example
Let g(x) =
∫ x
0t3 dt. We know g ′(x) = x3. What if instead we
had
h(x) =
∫ 3x
0t3 dt.
What is h′(x)?
SolutionWe can think of h as the composition g ◦ k, where
g(u) =
∫ u
0t3 dt and k(x) = 3x. Then
h′(x) = g ′(k(x))k ′(x) = 3(k(x))3 = 3(3x)3 = 81x3.
![Page 24: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/24.jpg)
Differentiation of area functions
Example
Let g(x) =
∫ x
0t3 dt. We know g ′(x) = x3. What if instead we
had
h(x) =
∫ 3x
0t3 dt.
What is h′(x)?
SolutionWe can think of h as the composition g ◦ k, where
g(u) =
∫ u
0t3 dt and k(x) = 3x. Then
h′(x) = g ′(k(x))k ′(x) = 3(k(x))3 = 3(3x)3 = 81x3.
![Page 25: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/25.jpg)
Example
Let h(x) =
∫ sin2 x
0(17t2 + 4t − 4) dt. What is h′(x)?
SolutionWe have
d
dx
∫ sin2 x
0(17t2 + 4t − 4) dt
=(17(sin2 x)2 + 4(sin2 x)− 4
)· d
dxsin2 x
=(17 sin4 x + 4 sin2 x − 4
)· 2 sin x cos x
![Page 26: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/26.jpg)
Example
Let h(x) =
∫ sin2 x
0(17t2 + 4t − 4) dt. What is h′(x)?
SolutionWe have
d
dx
∫ sin2 x
0(17t2 + 4t − 4) dt
=(17(sin2 x)2 + 4(sin2 x)− 4
)· d
dxsin2 x
=(17 sin4 x + 4 sin2 x − 4
)· 2 sin x cos x
![Page 27: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/27.jpg)
ErfHere’s a function with a funny name but an important role:
erf(x) =2√π
∫ x
0e−t2
dt.
It turns out erf is the shape of the bell curve. We can’t find erf(x),explicitly, but we do know its derivative.
erf ′(x) =2√π
e−x2.
Example
Findd
dxerf(x2).
SolutionBy the chain rule we have
d
dxerf(x2) = erf ′(x2)
d
dxx2 =
2√2π
e−(x2)22x =4√π
xe−x4.
![Page 28: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/28.jpg)
ErfHere’s a function with a funny name but an important role:
erf(x) =2√π
∫ x
0e−t2
dt.
It turns out erf is the shape of the bell curve.
We can’t find erf(x),explicitly, but we do know its derivative.
erf ′(x) =2√π
e−x2.
Example
Findd
dxerf(x2).
SolutionBy the chain rule we have
d
dxerf(x2) = erf ′(x2)
d
dxx2 =
2√2π
e−(x2)22x =4√π
xe−x4.
![Page 29: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/29.jpg)
ErfHere’s a function with a funny name but an important role:
erf(x) =2√π
∫ x
0e−t2
dt.
It turns out erf is the shape of the bell curve. We can’t find erf(x),explicitly, but we do know its derivative.
erf ′(x) =
2√π
e−x2.
Example
Findd
dxerf(x2).
SolutionBy the chain rule we have
d
dxerf(x2) = erf ′(x2)
d
dxx2 =
2√2π
e−(x2)22x =4√π
xe−x4.
![Page 30: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/30.jpg)
ErfHere’s a function with a funny name but an important role:
erf(x) =2√π
∫ x
0e−t2
dt.
It turns out erf is the shape of the bell curve. We can’t find erf(x),explicitly, but we do know its derivative.
erf ′(x) =2√π
e−x2.
Example
Findd
dxerf(x2).
SolutionBy the chain rule we have
d
dxerf(x2) = erf ′(x2)
d
dxx2 =
2√2π
e−(x2)22x =4√π
xe−x4.
![Page 31: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/31.jpg)
ErfHere’s a function with a funny name but an important role:
erf(x) =2√π
∫ x
0e−t2
dt.
It turns out erf is the shape of the bell curve. We can’t find erf(x),explicitly, but we do know its derivative.
erf ′(x) =2√π
e−x2.
Example
Findd
dxerf(x2).
SolutionBy the chain rule we have
d
dxerf(x2) = erf ′(x2)
d
dxx2 =
2√2π
e−(x2)22x =4√π
xe−x4.
![Page 32: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/32.jpg)
ErfHere’s a function with a funny name but an important role:
erf(x) =2√π
∫ x
0e−t2
dt.
It turns out erf is the shape of the bell curve. We can’t find erf(x),explicitly, but we do know its derivative.
erf ′(x) =2√π
e−x2.
Example
Findd
dxerf(x2).
SolutionBy the chain rule we have
d
dxerf(x2) = erf ′(x2)
d
dxx2 =
2√2π
e−(x2)22x =4√π
xe−x4.
![Page 33: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/33.jpg)
Other functions defined by integrals
I The future value of an asset:
FV (t) =
∫ ∞t
π(τ)e−rτ dτ
where π(τ) is the profitability at time τ and r is the discountrate.
I The consumer surplus of a good:
CS(p∗) =
∫ p∗
0f (p) dp
where f (p) is the demand function and p∗ is the equilibriumprice (depends on supply)
![Page 34: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/34.jpg)
Outline
The Area Function
FTC1StatementProofBiographies
Differentiation of functions defined by integrals“Contrived” examplesErfOther applications
FTC2
Facts about g from fA problem
![Page 35: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/35.jpg)
Theorem (The Second Fundamental Theorem of Calculus,Weak Form)
If f is continuous on [a, b] and f = F ′ for another function F , then∫ b
af (t) dt = F (b)− F (a).
Proof.Let g be the area function. Since f is continuous on [a, b], g isdifferentiable on (a, b), and g ′ = f = F ′ on (a, b). Henceg(x) = F (x) + C for all x in [a, b] (remember this requires theMean Value Theorem!). Since g(a) = 0, we have C = −F (a).Therefore
g(b) = F (b)− F (a).
![Page 36: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/36.jpg)
Theorem (The Second Fundamental Theorem of Calculus,Weak Form)
If f is continuous on [a, b] and f = F ′ for another function F , then∫ b
af (t) dt = F (b)− F (a).
Proof.Let g be the area function. Since f is continuous on [a, b], g isdifferentiable on (a, b), and g ′ = f = F ′ on (a, b). Henceg(x) = F (x) + C for all x in [a, b] (remember this requires theMean Value Theorem!). Since g(a) = 0, we have C = −F (a).Therefore
g(b) = F (b)− F (a).
![Page 37: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/37.jpg)
Outline
The Area Function
FTC1StatementProofBiographies
Differentiation of functions defined by integrals“Contrived” examplesErfOther applications
FTC2
Facts about g from fA problem
![Page 38: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/38.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
![Page 39: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/39.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
What is the particle’s velocityat time t = 5?
![Page 40: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/40.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
What is the particle’s velocityat time t = 5?
SolutionRecall that by the FTC wehave
s ′(t) = f (t).
So s ′(5) = f (5) = 2.
![Page 41: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/41.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
Is the acceleration of the par-ticle at time t = 5 positive ornegative?
![Page 42: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/42.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
Is the acceleration of the par-ticle at time t = 5 positive ornegative?
SolutionWe have s ′′(5) = f ′(5), whichlooks negative from thegraph.
![Page 43: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/43.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
What is the particle’s positionat time t = 3?
![Page 44: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/44.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
What is the particle’s positionat time t = 3?
SolutionSince on [0, 3], f (x) = x, wehave
s(3) =
∫ 3
0x dx =
9
2.
![Page 45: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/45.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
At what time during the first 9seconds does s have its largestvalue?
![Page 46: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/46.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
At what time during the first 9seconds does s have its largestvalue?
Solution
![Page 47: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/47.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
At what time during the first 9seconds does s have its largestvalue?
SolutionThe critical points of s arethe zeros of s ′ = f .
![Page 48: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/48.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
At what time during the first 9seconds does s have its largestvalue?
SolutionBy looking at the graph, wesee that f is positive fromt = 0 to t = 6, then negativefrom t = 6 to t = 9.
![Page 49: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/49.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
At what time during the first 9seconds does s have its largestvalue?
SolutionTherefore s is increasing on[0, 6], then decreasing on[6, 9]. So its largest value isat t = 6.
![Page 50: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/50.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
Approximately when is the ac-celeration zero?
![Page 51: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/51.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
Approximately when is the ac-celeration zero?
Solutions ′′ = 0 when f ′ = 0, whichhappens at t = 4 and t = 7.5(approximately)
![Page 52: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/52.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
When is the particle movingtoward the origin? Away fromthe origin?
![Page 53: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/53.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
When is the particle movingtoward the origin? Away fromthe origin?
SolutionThe particle is moving awayfrom the origin when s > 0and s ′ > 0.
![Page 54: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/54.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
When is the particle movingtoward the origin? Away fromthe origin?
SolutionSince s(0) = 0 and s ′ > 0 on(0, 6), we know the particle ismoving away from the originthen.
![Page 55: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/55.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
When is the particle movingtoward the origin? Away fromthe origin?
SolutionAfter t = 6, s ′ < 0, so theparticle is moving toward theorigin.
![Page 56: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/56.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
On which side (positive or neg-ative) of the origin does theparticle lie at time t = 9?
![Page 57: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/57.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
On which side (positive or neg-ative) of the origin does theparticle lie at time t = 9?
SolutionWe have s(9) =∫ 6
0f (x) dx +
∫ 9
6f (x) dx,
where the left integral ispositive and the right integralis negative.
![Page 58: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/58.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
On which side (positive or neg-ative) of the origin does theparticle lie at time t = 9?
SolutionIn order to decide whethers(9) is positive or negative,we need to decide if the firstarea is more positive than thesecond area is negative.
![Page 59: Lesson 32: The Fundamental Theorem Of Calculus](https://reader034.fdocuments.net/reader034/viewer/2022042607/5559c65cd8b42a236c8b55b9/html5/thumbnails/59.jpg)
Facts about g from f
Let f be the function whose graph is given below.Suppose the the position at time t seconds of a particle moving
along a coordinate axis is s(t) =
∫ t
0f (x) dx meters. Use the
graph to answer the following questions.
1 2 3 4 5 6 7 8 9
1
2
3
4
• (1,1)
• (2,2)
• (3,3)• (5,2)
On which side (positive or neg-ative) of the origin does theparticle lie at time t = 9?
SolutionThis appears to be the case,so s(9) is positive.