Lesson 28: The Fundamental Theorem of Calculus
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Transcript of Lesson 28: The Fundamental Theorem of Calculus
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. . . . . .
Section5.4TheFundamentalTheoremofCalculus
Math1aIntroductiontoCalculus
April16, 2008
Announcements
◮ Midtermisfinished: x̄ ≈ 43, σ ≈ 6.◮ MidtermIII isWednesday4/30inclass◮ Friday5/2isMovieDay!◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323
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. . . . . .
Announcements
◮ Midtermisfinished: x̄ ≈ 43, σ ≈ 6.◮ MidtermIII isWednesday4/30inclass◮ Friday5/2isMovieDay!◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323
![Page 3: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/3.jpg)
. . . . . .
Outline
Lasttime: TheSecondFundamentalTheoremofCalculusMyfirsttableofintegrals
TheFirstFundamentalTheoremofCalculusTheAreaFunctionStatementandproofof1FTCBiographies
Differentiationoffunctionsdefinedbyintegrals“Contrived”examplesErfOtherapplications
Factsabout g from fA problem
![Page 4: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/4.jpg)
. . . . . .
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
![Page 5: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/5.jpg)
. . . . . .
Theorem(TheSecondFundamentalTheoremofCalculus)Suppose f isintegrableon [a,b] and f = F′ foranotherfunction F,then ∫ b
af(x)dx = F(b) − F(a).
![Page 6: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/6.jpg)
. . . . . .
TheIntegralasTotalChange
Anotherwaytostatethistheoremis:∫ b
aF′(x)dx = F(b) − F(a),
or theintegralofaderivativealonganintervalisthetotalchangebetweenthesidesofthatinterval. Thishasmanyramifications:
![Page 7: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/7.jpg)
. . . . . .
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).
![Page 8: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/8.jpg)
. . . . . .
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.
![Page 9: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/9.jpg)
. . . . . .
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.
![Page 10: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/10.jpg)
. . . . . .
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
![Page 11: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/11.jpg)
. . . . . .
Outline
Lasttime: TheSecondFundamentalTheoremofCalculusMyfirsttableofintegrals
TheFirstFundamentalTheoremofCalculusTheAreaFunctionStatementandproofof1FTCBiographies
Differentiationoffunctionsdefinedbyintegrals“Contrived”examplesErfOtherapplications
Factsabout g from fA problem
![Page 12: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/12.jpg)
. . . . . .
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 → ∞.
![Page 13: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/13.jpg)
. . . . . .
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 → ∞.
![Page 14: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/14.jpg)
. . . . . .
Anareafunction, continued
So
g(x) =x4
4.
Thismeansthatg′(x) = x3.
![Page 15: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/15.jpg)
. . . . . .
Anareafunction, continued
So
g(x) =x4
4.
Thismeansthatg′(x) = x3.
![Page 16: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/16.jpg)
. . . . . .
Theareafunction
Let f beafunctionwhichisintegrable(i.e., continuousorwithfinitelymanyjumpdiscontinuities)on [a,b]. Define
g(x) =
∫ x
af(t)dt.
◮ Whenis g increasing?
◮ Whenis g decreasing?◮ 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.
◮ Whenis g increasing?◮ Whenis g decreasing?
◮ Overasmallinterval, what’stheaveragerateofchangeof g?
![Page 18: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/18.jpg)
. . . . . .
Theareafunction
Let f beafunctionwhichisintegrable(i.e., continuousorwithfinitelymanyjumpdiscontinuities)on [a,b]. Define
g(x) =
∫ x
af(t)dt.
◮ Whenis g increasing?◮ Whenis g decreasing?◮ Overasmallinterval, what’stheaveragerateofchangeof g?
![Page 19: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/19.jpg)
. . . . . .
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).
![Page 20: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/20.jpg)
. . . . . .
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 21: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/21.jpg)
. . . . . .
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 22: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/22.jpg)
. . . . . .
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 23: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/23.jpg)
. . . . . .
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 24: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/24.jpg)
. . . . . .
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 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/25.jpg)
. . . . . .
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 26: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/26.jpg)
. . . . . .
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 27: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/27.jpg)
. . . . . .
MeettheMathematician: JamesGregory
◮ Scottish, 1638-1675◮ AstronomerandGeometer
◮ Conceivedtranscendentalnumbersandfoundevidencethatπ wastranscendental
◮ Provedageometricversionof1FTC asalemmabutdidn’ttakeitfurther
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. . . . . .
MeettheMathematician: IsaacBarrow
◮ English, 1630-1677◮ ProfessorofGreek,theology, andmathematicsatCambridge
◮ Hadafamousstudent
![Page 29: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/29.jpg)
. . . . . .
MeettheMathematician: IsaacNewton
◮ English, 1643–1727◮ ProfessoratCambridge(England)
◮ PhilosophiaeNaturalisPrincipiaMathematicapublished1687
![Page 30: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/30.jpg)
. . . . . .
MeettheMathematician: GottfriedLeibniz
◮ German, 1646–1716◮ Eminentphilosopheraswellasmathematician
◮ Contemporarilydisgracedbythecalculusprioritydispute
![Page 31: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/31.jpg)
. . . . . .
DifferentiationandIntegrationasreverseprocesses
Puttingtogether1FTC and2FTC,wegetabeautifulrelationshipbetweenthetwofundamentalconceptsincalculus.
◮ddx
∫ x
af(t)dt = f(x)
◮ ∫ b
aF′(x)dx = F(b) − F(a).
![Page 32: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/32.jpg)
. . . . . .
DifferentiationandIntegrationasreverseprocesses
Puttingtogether1FTC and2FTC,wegetabeautifulrelationshipbetweenthetwofundamentalconceptsincalculus.
◮ddx
∫ x
af(t)dt = f(x)
◮ ∫ b
aF′(x)dx = F(b) − F(a).
![Page 33: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/33.jpg)
. . . . . .
Outline
Lasttime: TheSecondFundamentalTheoremofCalculusMyfirsttableofintegrals
TheFirstFundamentalTheoremofCalculusTheAreaFunctionStatementandproofof1FTCBiographies
Differentiationoffunctionsdefinedbyintegrals“Contrived”examplesErfOtherapplications
Factsabout g from fA problem
![Page 34: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/34.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.
![Page 35: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/35.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.
![Page 36: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/36.jpg)
. . . . . .
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
![Page 37: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/37.jpg)
. . . . . .
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
![Page 38: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/42.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 43: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/43.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 44: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/44.jpg)
. . . . . .
Otherfunctionsdefinedbyintegrals
◮ Thefuturevalueofanasset:
FV(t) =
∫ ∞
tπ(τ)e−rτ dτ
where π(τ) istheprofitabilityattime τ and r isthediscountrate.
◮ Theconsumersurplusofagood:
CS(p∗) =
∫ p∗
0f(p)dp
where f(p) isthedemandfunctionand p∗ istheequilibriumprice(dependsonsupply)
![Page 45: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/45.jpg)
. . . . . .
Outline
Lasttime: TheSecondFundamentalTheoremofCalculusMyfirsttableofintegrals
TheFirstFundamentalTheoremofCalculusTheAreaFunctionStatementandproofof1FTCBiographies
Differentiationoffunctionsdefinedbyintegrals“Contrived”examplesErfOtherapplications
Factsabout g from fA problem
![Page 46: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
![Page 47: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Whatistheparticle’svelocityattime t = 5?
![Page 48: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Whatistheparticle’svelocityattime t = 5?
SolutionRecallthatbytheFTC wehave
s′(t) = f(t).
So s′(5) = f(5) = 2.
![Page 49: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Istheaccelerationofthepar-ticleattime t = 5 positiveornegative?
![Page 50: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Istheaccelerationofthepar-ticleattime t = 5 positiveornegative?
SolutionWehave s′′(5) = f′(5), whichlooksnegativefromthegraph.
![Page 51: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Whatistheparticle’spositionattime t = 3?
![Page 52: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Whatistheparticle’spositionattime t = 3?
SolutionSinceon [0,3], f(x) = x, wehave
s(3) =
∫ 3
0x dx =
92.
![Page 53: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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?
![Page 54: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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?
Solution
![Page 55: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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?
SolutionThecriticalpointsof s arethezerosof s′ = f.
![Page 56: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Atwhattimeduringthefirst9secondsdoes s haveitslargestvalue?
SolutionBylookingatthegraph, weseethat f ispositivefromt = 0 to t = 6, thennegativefrom t = 6 to t = 9.
![Page 57: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Atwhattimeduringthefirst9secondsdoes s haveitslargestvalue?
SolutionTherefore s isincreasingon[0, 6], thendecreasingon[6, 9]. Soitslargestvalueisatt = 6.
![Page 58: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Approximately when is theaccelerationzero?
![Page 59: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Approximately when is theaccelerationzero?
Solutions′′ = 0 when f′ = 0, whichhappensat t = 4 and t = 7.5(approximately)
![Page 60: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Whenistheparticlemovingtowardtheorigin? Awayfromtheorigin?
![Page 61: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Whenistheparticlemovingtowardtheorigin? Awayfromtheorigin?
SolutionTheparticleismovingawayfromtheoriginwhen s > 0and s′ > 0.
![Page 62: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Whenistheparticlemovingtowardtheorigin? Awayfromtheorigin?
SolutionSince s(0) = 0 and s′ > 0 on(0, 6), weknowtheparticleismovingawayfromtheoriginthen.
![Page 63: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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)
Whenistheparticlemovingtowardtheorigin? Awayfromtheorigin?
SolutionAfter t = 6, s′ < 0, sotheparticleismovingtowardtheorigin.
![Page 64: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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?
![Page 65: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/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?
SolutionWehave s(9) =∫ 6
0f(x)dx +
∫ 9
6f(x)dx,
wheretheleftintegralispositiveandtherightintegralisnegative.
![Page 66: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/66.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 67: Lesson 28: The Fundamental Theorem of Calculus](https://reader034.fdocuments.us/reader034/viewer/2022051313/549ec36fb37959af618b4814/html5/thumbnails/67.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.