lesson 26: the fundamental theorem of calculus (section 10 version)
DESCRIPTION
The First Fundamental Theorem of Calculus looks at the area function and its derivative. It so happens that the derivative of the area function is the original integrand.TRANSCRIPT
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Section5.4TheFundamentalTheoremofCalculus
V63.0121, CalculusI
April22, 2009
Announcements
I Quiz6nextweekon§§5.1–5.2
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. . . . . .
Thedefiniteintegralasalimit
DefinitionIf f isafunctiondefinedon [a,b], the definiteintegralof f from ato b isthenumber∫ b
af(x)dx = lim
∆x→0
n∑i=1
f(ci) ∆x
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. . . . . .
Theorem(TheSecondFundamentalTheoremofCalculus)Suppose f isintegrableon [a,b] and f = F′ foranotherfunction F,then ∫ b
af(x)dx = F(b) − F(a).
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. . . . . .
TheIntegralasTotalChange
Anotherwaytostatethistheoremis:∫ b
aF′(x)dx = F(b) − F(a),
or theintegralofaderivativealonganintervalisthetotalchangebetweenthesidesofthatinterval. Thishasmanyramifications:
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. . . . . .
TheIntegralasTotalChange
Anotherwaytostatethistheoremis:∫ b
aF′(x)dx = F(b) − F(a),
or theintegralofaderivativealonganintervalisthetotalchangebetweenthesidesofthatinterval. Thishasmanyramifications:
TheoremIf v(t) representsthevelocityofaparticlemovingrectilinearly,then ∫ t1
t0v(t)dt = s(t1) − s(t0).
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. . . . . .
TheIntegralasTotalChange
Anotherwaytostatethistheoremis:∫ b
aF′(x)dx = F(b) − F(a),
or theintegralofaderivativealonganintervalisthetotalchangebetweenthesidesofthatinterval. Thishasmanyramifications:
TheoremIf MC(x) representsthemarginalcostofmaking x unitsofaproduct, then
C(x) = C(0) +
∫ x
0MC(q)dq.
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. . . . . .
TheIntegralasTotalChange
Anotherwaytostatethistheoremis:∫ b
aF′(x)dx = F(b) − F(a),
or theintegralofaderivativealonganintervalisthetotalchangebetweenthesidesofthatinterval. Thishasmanyramifications:
TheoremIf ρ(x) representsthedensityofathinrodatadistanceof x fromitsend, thenthemassoftherodupto x is
m(x) =
∫ x
0ρ(s)ds.
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. . . . . .
Myfirsttableofintegrals∫[f(x) + g(x)] dx =
∫f(x)dx +
∫g(x)dx∫
xn dx =xn+1
n + 1+ C (n ̸= −1)∫
ex dx = ex + C∫sin x dx = − cos x + C∫cos x dx = sin x + C∫sec2 x dx = tan x + C∫
sec x tan x dx = sec x + C∫1
1 + x2dx = arctan x + C
∫cf(x)dx = c
∫f(x)dx∫
1xdx = ln |x| + C∫
ax dx =ax
ln a+ C∫
csc2 x dx = − cot x + C∫csc x cot x dx = − csc x + C∫
1√1− x2
dx = arcsin x + C
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. . . . . .
Outline
Myfirsttableofintegrals
TheFirstFundamentalTheoremofCalculusTheAreaFunctionStatementandproofof1FTCBiographies
Differentiationoffunctionsdefinedbyintegrals“Contrived”examplesErfOtherapplications
Factsabout g from fA problem
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. . . . . .
Anareafunction
Let f(t) = t3 anddefine g(x) =
∫ x
0f(t)dt. Canweevaluatethe
integralin g(x)?
..0 .x
Dividingtheinterval [0, x] into n pieces
gives ∆x =xnand xi = 0 + i∆x =
ixn.
So
Rn =xn· x
3
n3+
xn· (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 → ∞.
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. . . . . .
Anareafunction
Let f(t) = t3 anddefine g(x) =
∫ x
0f(t)dt. Canweevaluatethe
integralin g(x)?
..0 .x
Dividingtheinterval [0, x] into n pieces
gives ∆x =xnand xi = 0 + i∆x =
ixn.
So
Rn =xn· x
3
n3+
xn· (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 → ∞.
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. . . . . .
Anareafunction, continued
So
g(x) =x4
4.
Thismeansthatg′(x) = x3.
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. . . . . .
Anareafunction, continued
So
g(x) =x4
4.
Thismeansthatg′(x) = x3.
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. . . . . .
Theareafunction
Let f beafunctionwhichisintegrable(i.e., continuousorwithfinitelymanyjumpdiscontinuities)on [a,b]. Define
g(x) =
∫ x
af(t)dt.
I Whenis g increasing?
I Whenis g decreasing?I Overasmallinterval, what’stheaveragerateofchangeof g?
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. . . . . .
Theareafunction
Let f beafunctionwhichisintegrable(i.e., continuousorwithfinitelymanyjumpdiscontinuities)on [a,b]. Define
g(x) =
∫ x
af(t)dt.
I Whenis g increasing?I Whenis g decreasing?
I Overasmallinterval, what’stheaveragerateofchangeof g?
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. . . . . .
Theareafunction
Let f beafunctionwhichisintegrable(i.e., continuousorwithfinitelymanyjumpdiscontinuities)on [a,b]. Define
g(x) =
∫ x
af(t)dt.
I Whenis g increasing?I Whenis g decreasing?I Overasmallinterval, what’stheaveragerateofchangeof g?
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. . . . . .
Theorem(TheFirstFundamentalTheoremofCalculus)Let f beanintegrablefunctionon [a,b] anddefine
g(x) =
∫ x
af(t)dt.
If f iscontinuousat x in (a,b), then g isdifferentiableat x and
g′(x) = f(x).
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. . . . . .
Proof.Let h > 0 begivensothat x + h < b. Wehave
g(x + h) − g(x)h
=
1h
∫ x+h
xf(t)dt.
Let Mh bethemaximumvalueof f on [x, x + h], and mh theminimumvalueof f on [x, x + h]. From§5.2wehave
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 tendto f(x). Zappa-dappa.
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. . . . . .
Proof.Let h > 0 begivensothat x + h < b. Wehave
g(x + h) − g(x)h
=1h
∫ x+h
xf(t)dt.
Let Mh bethemaximumvalueof f on [x, x + h], and mh theminimumvalueof f on [x, x + h]. From§5.2wehave
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 tendto f(x). Zappa-dappa.
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. . . . . .
Proof.Let h > 0 begivensothat x + h < b. Wehave
g(x + h) − g(x)h
=1h
∫ x+h
xf(t)dt.
Let Mh bethemaximumvalueof f on [x, x + h], and mh theminimumvalueof f on [x, x + h]. From§5.2wehave
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 tendto f(x). Zappa-dappa.
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. . . . . .
Proof.Let h > 0 begivensothat x + h < b. Wehave
g(x + h) − g(x)h
=1h
∫ x+h
xf(t)dt.
Let Mh bethemaximumvalueof f on [x, x + h], and mh theminimumvalueof f on [x, x + h]. From§5.2wehave
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 tendto f(x). Zappa-dappa.
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. . . . . .
Proof.Let h > 0 begivensothat x + h < b. Wehave
g(x + h) − g(x)h
=1h
∫ x+h
xf(t)dt.
Let Mh bethemaximumvalueof f on [x, x + h], and mh theminimumvalueof f on [x, x + h]. From§5.2wehave
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 tendto f(x). Zappa-dappa.
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. . . . . .
Proof.Let h > 0 begivensothat x + h < b. Wehave
g(x + h) − g(x)h
=1h
∫ x+h
xf(t)dt.
Let Mh bethemaximumvalueof f on [x, x + h], and mh theminimumvalueof f on [x, x + h]. From§5.2wehave
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 tendto f(x). Zappa-dappa.
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. . . . . .
Proof.Let h > 0 begivensothat x + h < b. Wehave
g(x + h) − g(x)h
=1h
∫ x+h
xf(t)dt.
Let Mh bethemaximumvalueof f on [x, x + h], and mh theminimumvalueof f on [x, x + h]. From§5.2wehave
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 tendto f(x). Zappa-dappa.
![Page 25: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/25.jpg)
. . . . . .
MeettheMathematician: JamesGregory
I Scottish, 1638-1675I AstronomerandGeometer
I Conceivedtranscendentalnumbersandfoundevidencethatπ wastranscendental
I Provedageometricversionof1FTC asalemmabutdidn’ttakeitfurther
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. . . . . .
MeettheMathematician: IsaacBarrow
I English, 1630-1677I ProfessorofGreek,theology, andmathematicsatCambridge
I Hadafamousstudent
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. . . . . .
MeettheMathematician: IsaacNewton
I English, 1643–1727I ProfessoratCambridge(England)
I PhilosophiaeNaturalisPrincipiaMathematicapublished1687
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. . . . . .
MeettheMathematician: GottfriedLeibniz
I German, 1646–1716I Eminentphilosopheraswellasmathematician
I Contemporarilydisgracedbythecalculusprioritydispute
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. . . . . .
DifferentiationandIntegrationasreverseprocesses
Puttingtogether1FTC and2FTC,wegetabeautifulrelationshipbetweenthetwofundamentalconceptsincalculus.
Iddx
∫ x
af(t)dt = f(x)
I ∫ b
aF′(x)dx = F(b) − F(a).
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. . . . . .
DifferentiationandIntegrationasreverseprocesses
Puttingtogether1FTC and2FTC,wegetabeautifulrelationshipbetweenthetwofundamentalconceptsincalculus.
Iddx
∫ x
af(t)dt = f(x)
I ∫ b
aF′(x)dx = F(b) − F(a).
![Page 31: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/31.jpg)
. . . . . .
Outline
Myfirsttableofintegrals
TheFirstFundamentalTheoremofCalculusTheAreaFunctionStatementandproofof1FTCBiographies
Differentiationoffunctionsdefinedbyintegrals“Contrived”examplesErfOtherapplications
Factsabout g from fA problem
![Page 32: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/32.jpg)
. . . . . .
Differentiationofareafunctions
Example
Let g(x) =
∫ x
0t3 dt. Weknow g′(x) = x3. Whatifinsteadwehad
h(x) =
∫ 3x
0t3 dt.
Whatis h′(x)?
SolutionWecanthinkof h asthecomposition 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.
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. . . . . .
Differentiationofareafunctions
Example
Let g(x) =
∫ x
0t3 dt. Weknow g′(x) = x3. Whatifinsteadwehad
h(x) =
∫ 3x
0t3 dt.
Whatis h′(x)?
SolutionWecanthinkof h asthecomposition 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.
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. . . . . .
Example
Let h(x) =
∫ sin2 x
0(17t2 + 4t− 4)dt. Whatis h′(x)?
SolutionWehave
ddx
∫ sin2 x
0(17t2 + 4t− 4)dt
=(17(sin2 x)2 + 4(sin2 x) − 4
)· ddx
sin2 x
=(17 sin4 x + 4 sin2 x− 4
)· 2 sin x cos x
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. . . . . .
Example
Let h(x) =
∫ sin2 x
0(17t2 + 4t− 4)dt. Whatis h′(x)?
SolutionWehave
ddx
∫ sin2 x
0(17t2 + 4t− 4)dt
=(17(sin2 x)2 + 4(sin2 x) − 4
)· ddx
sin2 x
=(17 sin4 x + 4 sin2 x− 4
)· 2 sin x cos x
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. . . . . .
ErfHere’safunctionwithafunnynamebutanimportantrole:
erf(x) =2√π
∫ x
0e−t2 dt.
Itturnsout erf istheshapeofthebellcurve. Wecan’tfind erf(x),explicitly, butwedoknowitsderivative.
erf′(x) =2√πe−x2 .
Example
Findddx
erf(x2).
SolutionBythechainrulewehave
ddx
erf(x2) = erf′(x2)ddx
x2 =2√πe−(x2)22x =
4√πxe−x4 .
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. . . . . .
ErfHere’safunctionwithafunnynamebutanimportantrole:
erf(x) =2√π
∫ x
0e−t2 dt.
Itturnsout erf istheshapeofthebellcurve.
Wecan’tfind erf(x),explicitly, butwedoknowitsderivative.
erf′(x) =2√πe−x2 .
Example
Findddx
erf(x2).
SolutionBythechainrulewehave
ddx
erf(x2) = erf′(x2)ddx
x2 =2√πe−(x2)22x =
4√πxe−x4 .
![Page 38: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/38.jpg)
. . . . . .
ErfHere’safunctionwithafunnynamebutanimportantrole:
erf(x) =2√π
∫ x
0e−t2 dt.
Itturnsout erf istheshapeofthebellcurve. Wecan’tfind erf(x),explicitly, butwedoknowitsderivative.
erf′(x) =
2√πe−x2 .
Example
Findddx
erf(x2).
SolutionBythechainrulewehave
ddx
erf(x2) = erf′(x2)ddx
x2 =2√πe−(x2)22x =
4√πxe−x4 .
![Page 39: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/39.jpg)
. . . . . .
ErfHere’safunctionwithafunnynamebutanimportantrole:
erf(x) =2√π
∫ x
0e−t2 dt.
Itturnsout erf istheshapeofthebellcurve. Wecan’tfind erf(x),explicitly, butwedoknowitsderivative.
erf′(x) =2√πe−x2 .
Example
Findddx
erf(x2).
SolutionBythechainrulewehave
ddx
erf(x2) = erf′(x2)ddx
x2 =2√πe−(x2)22x =
4√πxe−x4 .
![Page 40: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/40.jpg)
. . . . . .
ErfHere’safunctionwithafunnynamebutanimportantrole:
erf(x) =2√π
∫ x
0e−t2 dt.
Itturnsout erf istheshapeofthebellcurve. Wecan’tfind erf(x),explicitly, butwedoknowitsderivative.
erf′(x) =2√πe−x2 .
Example
Findddx
erf(x2).
SolutionBythechainrulewehave
ddx
erf(x2) = erf′(x2)ddx
x2 =2√πe−(x2)22x =
4√πxe−x4 .
![Page 41: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/41.jpg)
. . . . . .
ErfHere’safunctionwithafunnynamebutanimportantrole:
erf(x) =2√π
∫ x
0e−t2 dt.
Itturnsout erf istheshapeofthebellcurve. Wecan’tfind erf(x),explicitly, butwedoknowitsderivative.
erf′(x) =2√πe−x2 .
Example
Findddx
erf(x2).
SolutionBythechainrulewehave
ddx
erf(x2) = erf′(x2)ddx
x2 =2√πe−(x2)22x =
4√πxe−x4 .
![Page 42: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/42.jpg)
. . . . . .
Otherfunctionsdefinedbyintegrals
I Thefuturevalueofanasset:
FV(t) =
∫ ∞
tπ(τ)e−rτ dτ
where π(τ) istheprofitabilityattime τ and r isthediscountrate.
I Theconsumersurplusofagood:
CS(q∗) =
∫ q∗
0(f(q) − p∗)dq
where f(q) isthedemandfunctionand p∗ and q∗ theequilibriumpriceandquantity.
![Page 43: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/43.jpg)
. . . . . .
Outline
Myfirsttableofintegrals
TheFirstFundamentalTheoremofCalculusTheAreaFunctionStatementandproofof1FTCBiographies
Differentiationoffunctionsdefinedbyintegrals“Contrived”examplesErfOtherapplications
Factsabout g from fA problem
![Page 44: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/44.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
![Page 45: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/45.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Whatistheparticle’svelocityattime t = 5?
![Page 46: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/46.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Whatistheparticle’svelocityattime t = 5?
SolutionRecallthatbytheFTC wehave
s′(t) = f(t).
So s′(5) = f(5) = 2.
![Page 47: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/47.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Istheaccelerationofthepar-ticleattime t = 5 positiveornegative?
![Page 48: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/48.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Istheaccelerationofthepar-ticleattime t = 5 positiveornegative?
SolutionWehave s′′(5) = f′(5), whichlooksnegativefromthegraph.
![Page 49: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/49.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Whatistheparticle’spositionattime t = 3?
![Page 50: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/50.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Whatistheparticle’spositionattime t = 3?
SolutionSinceon [0,3], f(x) = x, wehave
s(3) =
∫ 3
0x dx =
92.
![Page 51: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/51.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Atwhattimeduringthefirst9secondsdoes s haveitslargestvalue?
![Page 52: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/52.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Atwhattimeduringthefirst9secondsdoes s haveitslargestvalue?
Solution
![Page 53: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/53.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Atwhattimeduringthefirst9secondsdoes s haveitslargestvalue?
SolutionThecriticalpointsof s arethezerosof s′ = f.
![Page 54: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/54.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Atwhattimeduringthefirst9secondsdoes s haveitslargestvalue?
SolutionBylookingatthegraph, weseethat f ispositivefromt = 0 to t = 6, thennegativefrom t = 6 to t = 9.
![Page 55: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/55.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Atwhattimeduringthefirst9secondsdoes s haveitslargestvalue?
SolutionTherefore s isincreasingon[0, 6], thendecreasingon[6, 9]. Soitslargestvalueisatt = 6.
![Page 56: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/56.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Approximately when is theaccelerationzero?
![Page 57: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/57.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
Approximately when is theaccelerationzero?
Solutions′′ = 0 when f′ = 0, whichhappensat t = 4 and t = 7.5(approximately)
![Page 58: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/58.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
When is theparticlemovingtowardtheorigin? Awayfromtheorigin?
![Page 59: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/59.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
When is theparticlemovingtowardtheorigin? Awayfromtheorigin?
SolutionTheparticleismovingawayfromtheoriginwhen s > 0and s′ > 0.
![Page 60: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/60.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
When is theparticlemovingtowardtheorigin? Awayfromtheorigin?
SolutionSince s(0) = 0 and s′ > 0 on(0, 6), weknowtheparticleismovingawayfromtheoriginthen.
![Page 61: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/61.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..1
..2
..3
..4
..5
..6
..7
..8
..9
.1
.2
.3
.4
.• .(1,1)
.• .(2,2)
.• .(3,3).• .(5,2)
When is theparticlemovingtowardtheorigin? Awayfromtheorigin?
SolutionAfter t = 6, s′ < 0, sotheparticleismovingtowardtheorigin.
![Page 62: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/62.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..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 ornegative) of the origin doestheparticlelieattime t = 9?
![Page 63: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/63.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..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 ornegative) of the origin doestheparticlelieattime t = 9?
SolutionWehave s(9) =∫ 6
0f(x)dx +
∫ 9
6f(x)dx,
wheretheleftintegralispositiveandtherightintegralisnegative.
![Page 64: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/64.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..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 ornegative) of the origin doestheparticlelieattime t = 9?
SolutionInordertodecidewhethers(9) ispositiveornegative,weneedtodecideifthefirstareaismorepositivethanthesecondareaisnegative.
![Page 65: Lesson 26: The Fundamental Theorem of Calculus (Section 10 version)](https://reader034.vdocuments.mx/reader034/viewer/2022052619/5559c659d8b42a236c8b55b7/html5/thumbnails/65.jpg)
. . . . . .
Factsabout g from f
Let f bethefunctionwhosegraphisgivenbelow.Supposethethepositionattime t secondsofaparticlemoving
alongacoordinateaxisis s(t) =
∫ t
0f(x)dx meters. Usethegraph
toanswerthefollowingquestions.
. ..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 ornegative) of the origin doestheparticlelieattime t = 9?
SolutionThisappearstobethecase,so s(9) ispositive.