Lesson 23: Antiderivatives
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. . . . . .
Section4.7Antiderivatives
V63.0121.034, CalculusI
November18, 2009
Announcements
I Wednesday, November25isaregularclassdayI nextandlastquizwillbetheweekafterThanksgiving(4.1–4.4, 4.7)
I FinalExam: Friday, December18, 2:00–3:50pm
..Imagecredit: IanHampton
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. . . . . .
WhytheMVT istheMITCMostImportantTheoremInCalculus!
TheoremLet f′ = 0 onaninterval (a,b). Then f isconstanton (a,b).
Proof.Pickanypoints x and y in (a,b) with x < y. Then f iscontinuouson [x, y] anddifferentiableon (x, y). ByMVT thereexistsapointz in (x, y) suchthat
f(y) − f(x)y− x
= f′(z) = 0.
So f(y) = f(x). Sincethisistrueforall x and y in (a,b), then f isconstant.
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. . . . . .
TheoremSuppose f and g aretwodifferentiablefunctionson (a,b) withf′ = g′. Then f and g differbyaconstant. Thatis, thereexistsaconstant C suchthat 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, aconstantI Thismeans f(x) − g(x) = C on (a,b)
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. . . . . .
Objectives
I Givenanexpressionforfunction f, findadifferentiablefunction Fsuchthat F′ = f (F iscalledan antiderivativefor f).
I Giventhegraphofafunction f, findadifferentiablefunction Fsuchthat F′ = f
I Useantiderivativestosolveproblemsinrectilinearmotion
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. . . . . .
Hardproblem, easycheck
ExampleFindanantiderivativefor f(x) = ln x.
Solution???
Exampleis F(x) = x ln x− x anantiderivativefor f(x) = ln x?
Solution
ddx
(x ln x− x) = 1 · ln x + x · 1x− 1
= ln x
Yes!
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. . . . . .
Hardproblem, easycheck
ExampleFindanantiderivativefor f(x) = ln x.
Solution???
Exampleis F(x) = x ln x− x anantiderivativefor f(x) = ln x?
Solution
ddx
(x ln x− x) = 1 · ln x + x · 1x− 1
= ln x
Yes!
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. . . . . .
Hardproblem, easycheck
ExampleFindanantiderivativefor f(x) = ln x.
Solution???
Exampleis F(x) = x ln x− x anantiderivativefor f(x) = ln x?
Solution
ddx
(x ln x− x) = 1 · ln x + x · 1x− 1
= ln x
Yes!
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. . . . . .
Hardproblem, easycheck
ExampleFindanantiderivativefor f(x) = ln x.
Solution???
Exampleis F(x) = x ln x− x anantiderivativefor f(x) = ln x?
Solution
ddx
(x ln x− x) = 1 · ln x + x · 1x− 1
= ln x
Yes!
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. . . . . .
Hardproblem, easycheck
ExampleFindanantiderivativefor f(x) = ln x.
Solution???
Exampleis F(x) = x ln x− x anantiderivativefor f(x) = ln x?
Solution
ddx
(x ln x− x) = 1 · ln x + x · 1x− 1
= ln x
Yes!
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. . . . . .
Outline
TabulatingAntiderivativesPowerfunctionsCombinationsExponentialfunctionsTrigonometricfunctions
FindingAntiderivativesGraphically
Rectilinearmotion
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. . . . . .
Antiderivativesofpowerfunctions
Recallthatthederivativeofapowerfunctionisapowerfunction.
FactThePowerRuleIf f(x) = xr, then f′(x) = rxr−1.
Soinlookingforantiderivativesofpowerfunctions, trypowerfunctions!
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. . . . . .
Antiderivativesofpowerfunctions
Recallthatthederivativeofapowerfunctionisapowerfunction.
FactThePowerRuleIf f(x) = xr, then f′(x) = rxr−1.
Soinlookingforantiderivativesofpowerfunctions, trypowerfunctions!
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. . . . . .
ExampleFindanantiderivativeforthefunction f(x) = x3.
Solution
I Tryapowerfunction F(x) = axr
I Then F′(x) = arxr−1, andwewantthistobeequalto x3.
I Apparently, r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 isanantiderivative.
I Check:ddx
(14x4
)= 4 · 1
4x4−1 = x3
I Anyothers? Yes, F(x) =14x4 + C isthemostgeneralform.
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. . . . . .
ExampleFindanantiderivativeforthefunction f(x) = x3.
Solution
I Tryapowerfunction F(x) = axr
I Then F′(x) = arxr−1, andwewantthistobeequalto x3.
I Apparently, r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 isanantiderivative.
I Check:ddx
(14x4
)= 4 · 1
4x4−1 = x3
I Anyothers? Yes, F(x) =14x4 + C isthemostgeneralform.
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. . . . . .
ExampleFindanantiderivativeforthefunction f(x) = x3.
Solution
I Tryapowerfunction F(x) = axr
I Then F′(x) = arxr−1, andwewantthistobeequalto x3.
I Apparently, r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 isanantiderivative.
I Check:ddx
(14x4
)= 4 · 1
4x4−1 = x3
I Anyothers? Yes, F(x) =14x4 + C isthemostgeneralform.
![Page 16: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/16.jpg)
. . . . . .
ExampleFindanantiderivativeforthefunction f(x) = x3.
Solution
I Tryapowerfunction F(x) = axr
I Then F′(x) = arxr−1, andwewantthistobeequalto x3.
I Apparently, r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 isanantiderivative.
I Check:ddx
(14x4
)= 4 · 1
4x4−1 = x3
I Anyothers? Yes, F(x) =14x4 + C isthemostgeneralform.
![Page 17: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/17.jpg)
. . . . . .
ExampleFindanantiderivativeforthefunction f(x) = x3.
Solution
I Tryapowerfunction F(x) = axr
I Then F′(x) = arxr−1, andwewantthistobeequalto x3.
I Apparently, r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 isanantiderivative.
I Check:ddx
(14x4
)= 4 · 1
4x4−1 = x3
I Anyothers? Yes, F(x) =14x4 + C isthemostgeneralform.
![Page 18: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/18.jpg)
. . . . . .
ExampleFindanantiderivativeforthefunction f(x) = x3.
Solution
I Tryapowerfunction F(x) = axr
I Then F′(x) = arxr−1, andwewantthistobeequalto x3.
I Apparently, r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 isanantiderivative.
I Check:ddx
(14x4
)= 4 · 1
4x4−1 = x3
I Anyothers? Yes, F(x) =14x4 + C isthemostgeneralform.
![Page 19: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/19.jpg)
. . . . . .
ExampleFindanantiderivativeforthefunction f(x) = x3.
Solution
I Tryapowerfunction F(x) = axr
I Then F′(x) = arxr−1, andwewantthistobeequalto x3.
I Apparently, r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 isanantiderivative.
I Check:ddx
(14x4
)= 4 · 1
4x4−1 = x3
I Anyothers?
Yes, F(x) =14x4 + C isthemostgeneralform.
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. . . . . .
ExampleFindanantiderivativeforthefunction f(x) = x3.
Solution
I Tryapowerfunction F(x) = axr
I Then F′(x) = arxr−1, andwewantthistobeequalto x3.
I Apparently, r− 1 = 3 =⇒ r = 4, and ar = 1 =⇒ a =14.
I So F(x) =14x4 isanantiderivative.
I Check:ddx
(14x4
)= 4 · 1
4x4−1 = x3
I Anyothers? Yes, F(x) =14x4 + C isthemostgeneralform.
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. . . . . .
Fact(ThePowerRuleforantiderivatives)If f(x) = xr, then
F(x) =1
r + 1xr+1
isanantiderivativefor f…
aslongas r ̸= −1.
FactIf f(x) = x−1 =
1x, then
F(x) = ln |x| + C
isanantiderivativefor f.
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. . . . . .
Fact(ThePowerRuleforantiderivatives)If f(x) = xr, then
F(x) =1
r + 1xr+1
isanantiderivativefor f aslongas r ̸= −1.
FactIf f(x) = x−1 =
1x, then
F(x) = ln |x| + C
isanantiderivativefor f.
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. . . . . .
Fact(ThePowerRuleforantiderivatives)If f(x) = xr, then
F(x) =1
r + 1xr+1
isanantiderivativefor f aslongas r ̸= −1.
FactIf f(x) = x−1 =
1x, then
F(x) = ln |x| + C
isanantiderivativefor f.
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. . . . . .
What’swiththeabsolutevalue?
I F(x) = ln |x| hasdomainallnonzeronumbers, while ln x isonlydefinedonpositivenumbers.
I Forpositivenumbers x,
ddx
ln |x| =ddx
ln x
(whichweknew)I Fornegativenumbers
ddx
ln |x| =ddx
ln(−x) =1−x
· (−1) =1x
I Weprefertheantiderivativewiththelargerdomain.
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. . . . . .
What’swiththeabsolutevalue?
I F(x) = ln |x| hasdomainallnonzeronumbers, while ln x isonlydefinedonpositivenumbers.
I Forpositivenumbers x,
ddx
ln |x| =ddx
ln x
(whichweknew)
I Fornegativenumbers
ddx
ln |x| =ddx
ln(−x) =1−x
· (−1) =1x
I Weprefertheantiderivativewiththelargerdomain.
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. . . . . .
What’swiththeabsolutevalue?
I F(x) = ln |x| hasdomainallnonzeronumbers, while ln x isonlydefinedonpositivenumbers.
I Forpositivenumbers x,
ddx
ln |x| =ddx
ln x
(whichweknew)I Fornegativenumbers
ddx
ln |x| =ddx
ln(−x) =1−x
· (−1) =1x
I Weprefertheantiderivativewiththelargerdomain.
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. . . . . .
What’swiththeabsolutevalue?
I F(x) = ln |x| hasdomainallnonzeronumbers, while ln x isonlydefinedonpositivenumbers.
I Forpositivenumbers x,
ddx
ln |x| =ddx
ln x
(whichweknew)I Fornegativenumbers
ddx
ln |x| =ddx
ln(−x) =1−x
· (−1) =1x
I Weprefertheantiderivativewiththelargerdomain.
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. . . . . .
Graphof ln |x|
. .x
.y
.f(x) = 1/x
.F(x) = ln |x|
.F(x) = ln |x|
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. . . . . .
Graphof ln |x|
. .x
.y
.f(x) = 1/x
.F(x) = ln |x|
.F(x) = ln |x|
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. . . . . .
Graphof ln |x|
. .x
.y
.f(x) = 1/x
.F(x) = ln |x|
.F(x) = ln |x|
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. . . . . .
Combinationsofantiderivatives
Fact(SumandConstantMultipleRuleforAntiderivatives)
I If F isanantiderivativeof f and G isanantiderivativeof g,then F + G isanantiderivativeof f + g.
I If F isanantiderivativeof f and c isaconstant, then cF isanantiderivativeof cf.
Proof.Thesefollowfromthesumandconstantmultipleruleforderivatives:
I If F′ = f and G′ = g, then
(F + G)′ = F′ + G′ = f + g
I Again, if F′ = f,(cF)′ = cF′ = cf
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. . . . . .
Combinationsofantiderivatives
Fact(SumandConstantMultipleRuleforAntiderivatives)
I If F isanantiderivativeof f and G isanantiderivativeof g,then F + G isanantiderivativeof f + g.
I If F isanantiderivativeof f and c isaconstant, then cF isanantiderivativeof cf.
Proof.Thesefollowfromthesumandconstantmultipleruleforderivatives:
I If F′ = f and G′ = g, then
(F + G)′ = F′ + G′ = f + g
I Again, if F′ = f,(cF)′ = cF′ = cf
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. . . . . .
ExampleFindanantiderivativefor f(x) = 16x + 5
SolutionTheexpression 8x2 isanantiderivativefor 16x, and 5x isanantiderivativefor 5. So
F(x) = 8x2 + 5x + C
istheantiderivativeof f.
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. . . . . .
ExampleFindanantiderivativefor f(x) = 16x + 5
SolutionTheexpression 8x2 isanantiderivativefor 16x, and 5x isanantiderivativefor 5. So
F(x) = 8x2 + 5x + C
istheantiderivativeof f.
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. . . . . .
ExponentialFunctions
FactIf f(x) = ax, f′(x) = (ln a)ax.
Accordingly,
FactIf f(x) = ax, then F(x) =
1ln a
ax + C istheantiderivativeof f.
Proof.Checkityourself.
Inparticular,
FactIf f(x) = ex, then F(x) = ex + C istheantiderivativeof F.
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. . . . . .
ExponentialFunctions
FactIf f(x) = ax, f′(x) = (ln a)ax.
Accordingly,
FactIf f(x) = ax, then F(x) =
1ln a
ax + C istheantiderivativeof f.
Proof.Checkityourself.
Inparticular,
FactIf f(x) = ex, then F(x) = ex + C istheantiderivativeof F.
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. . . . . .
ExponentialFunctions
FactIf f(x) = ax, f′(x) = (ln a)ax.
Accordingly,
FactIf f(x) = ax, then F(x) =
1ln a
ax + C istheantiderivativeof f.
Proof.Checkityourself.
Inparticular,
FactIf f(x) = ex, then F(x) = ex + C istheantiderivativeof F.
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. . . . . .
ExponentialFunctions
FactIf f(x) = ax, f′(x) = (ln a)ax.
Accordingly,
FactIf f(x) = ax, then F(x) =
1ln a
ax + C istheantiderivativeof f.
Proof.Checkityourself.
Inparticular,
FactIf f(x) = ex, then F(x) = ex + C istheantiderivativeof F.
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. . . . . .
Logarithmicfunctions?
I Rememberwefound
F(x) = x ln x− x
isanantiderivativeof f(x) = ln x.
I Thisisnotobvious. SeeCalcII forthefullstory.
I However, usingthefactthat loga x =ln xln a
, wegetthat
F(x) =1ln a
(x ln x− x) + C
istheantiderivativeof f(x) = loga(x).
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. . . . . .
Logarithmicfunctions?
I Rememberwefound
F(x) = x ln x− x
isanantiderivativeof f(x) = ln x.I Thisisnotobvious. SeeCalcII forthefullstory.
I However, usingthefactthat loga x =ln xln a
, wegetthat
F(x) =1ln a
(x ln x− x) + C
istheantiderivativeof f(x) = loga(x).
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. . . . . .
Logarithmicfunctions?
I Rememberwefound
F(x) = x ln x− x
isanantiderivativeof f(x) = ln x.I Thisisnotobvious. SeeCalcII forthefullstory.
I However, usingthefactthat loga x =ln xln a
, wegetthat
F(x) =1ln a
(x ln x− x) + C
istheantiderivativeof f(x) = loga(x).
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. . . . . .
Trigonometricfunctions
Fact
ddx
sin x = cos xddx
cos x = − sin x
Sototurnthesearound,
Fact
I Thefunction F(x) = − cos x + C istheantiderivativeoff(x) = sin x.
I Thefunction F(x) = sin x + C istheantiderivativeoff(x) = cos x.
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. . . . . .
Trigonometricfunctions
Fact
ddx
sin x = cos xddx
cos x = − sin x
Sototurnthesearound,
Fact
I Thefunction F(x) = − cos x + C istheantiderivativeoff(x) = sin x.
I Thefunction F(x) = sin x + C istheantiderivativeoff(x) = cos x.
![Page 44: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/44.jpg)
. . . . . .
Trigonometricfunctions
Fact
ddx
sin x = cos xddx
cos x = − sin x
Sototurnthesearound,
Fact
I Thefunction F(x) = − cos x + C istheantiderivativeoff(x) = sin x.
I Thefunction F(x) = sin x + C istheantiderivativeoff(x) = cos x.
![Page 45: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/45.jpg)
. . . . . .
Outline
TabulatingAntiderivativesPowerfunctionsCombinationsExponentialfunctionsTrigonometricfunctions
FindingAntiderivativesGraphically
Rectilinearmotion
![Page 46: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/46.jpg)
. . . . . .
ProblemBelowisthegraphofafunction f. Drawthegraphofanantiderivativefor F.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.
. .
. .y = f(x)
![Page 47: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/47.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 48: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/48.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 49: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/49.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 50: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/50.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 51: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/51.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 52: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/52.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 53: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/53.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 54: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/54.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 55: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/55.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 56: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/56.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 57: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/57.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 58: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/58.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 59: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/59.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 60: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/60.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 61: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/61.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 62: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/62.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 63: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/63.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 64: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/64.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 65: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/65.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 66: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/66.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 67: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/67.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 68: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/68.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 69: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/69.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 70: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/70.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 71: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/71.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 72: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/72.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 73: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/73.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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
. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 74: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/74.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. "
." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 75: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/75.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ."
. . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 76: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/76.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." .
. . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 77: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/77.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . .
. ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 78: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/78.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . "
.? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 79: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/79.jpg)
. . . . . .
Using f tomakeasignchartfor F
Assuming F′ = f, wecanmakeasignchartfor f and f′ tofindtheintervalsofmonotonicityandconcavityforfor 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. " ." . . . ".? .? .? .? .? .?
Theonlyquestionleftis: Whatarethefunctionvalues?
![Page 80: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/80.jpg)
. . . . . .
Couldyourepeatthequestion?
ProblemBelowisthegraphofafunction f. Drawthegraphof theantiderivativefor F with F(1) = 0.
Solution
I Westartwith F(1) = 0.I Usingthesignchart, we
drawarcswiththespecifiedmonotonicityandconcavity
I It’shardertotellif/whenF crossestheaxis; moreaboutthatlater.
..x
.y
..1
..2
..3
..4
..5
..6
.
.
.. .
. .f
.F
.shape..1
..2
..3
..4
..5
..6. " ." . . . "
.IP .max
.IP .min
.
.
.
.
..
![Page 81: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/81.jpg)
. . . . . .
Outline
TabulatingAntiderivativesPowerfunctionsCombinationsExponentialfunctionsTrigonometricfunctions
FindingAntiderivativesGraphically
Rectilinearmotion
![Page 82: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/82.jpg)
. . . . . .
Saywhat?
I “Rectlinearmotion”justmeansmotionalongaline.I Oftenwearegiveninformationaboutthevelocityoraccelerationofamovingparticleandwewanttoknowtheequationsofmotion.
![Page 83: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/83.jpg)
. . . . . .
Example: DeadReckoning
![Page 84: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/84.jpg)
. . . . . .
ProblemSupposeaparticleofmass m isacteduponbyaconstantforce F.Findthepositionfunction s(t), thevelocityfunction v(t), andtheaccelerationfunction a(t).
Solution
I ByNewton’sSecondLaw(F = ma)aconstantforceinduces
aconstantacceleration. So a(t) = a =Fm.
I Since v′(t) = a(t), v(t) mustbeanantiderivativeoftheconstantfunction a. So
v(t) = at + C = at + v0
where v0 istheinitialvelocity.I Since s′(t) = v(t), s(t) mustbeanantiderivativeof v(t),
meaning
s(t) =12at2 + v0t + C =
12at2 + v0t + s0
![Page 85: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/85.jpg)
. . . . . .
ProblemSupposeaparticleofmass m isacteduponbyaconstantforce F.Findthepositionfunction s(t), thevelocityfunction v(t), andtheaccelerationfunction a(t).
Solution
I ByNewton’sSecondLaw(F = ma)aconstantforceinduces
aconstantacceleration. So a(t) = a =Fm.
I Since v′(t) = a(t), v(t) mustbeanantiderivativeoftheconstantfunction a. So
v(t) = at + C = at + v0
where v0 istheinitialvelocity.I Since s′(t) = v(t), s(t) mustbeanantiderivativeof v(t),
meaning
s(t) =12at2 + v0t + C =
12at2 + v0t + s0
![Page 86: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/86.jpg)
. . . . . .
ProblemSupposeaparticleofmass m isacteduponbyaconstantforce F.Findthepositionfunction s(t), thevelocityfunction v(t), andtheaccelerationfunction a(t).
Solution
I ByNewton’sSecondLaw(F = ma)aconstantforceinduces
aconstantacceleration. So a(t) = a =Fm.
I Since v′(t) = a(t), v(t) mustbeanantiderivativeoftheconstantfunction a. So
v(t) = at + C = at + v0
where v0 istheinitialvelocity.
I Since s′(t) = v(t), s(t) mustbeanantiderivativeof v(t),meaning
s(t) =12at2 + v0t + C =
12at2 + v0t + s0
![Page 87: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/87.jpg)
. . . . . .
ProblemSupposeaparticleofmass m isacteduponbyaconstantforce F.Findthepositionfunction s(t), thevelocityfunction v(t), andtheaccelerationfunction a(t).
Solution
I ByNewton’sSecondLaw(F = ma)aconstantforceinduces
aconstantacceleration. So a(t) = a =Fm.
I Since v′(t) = a(t), v(t) mustbeanantiderivativeoftheconstantfunction a. So
v(t) = at + C = at + v0
where v0 istheinitialvelocity.I Since s′(t) = v(t), s(t) mustbeanantiderivativeof v(t),
meaning
s(t) =12at2 + v0t + C =
12at2 + v0t + s0
![Page 88: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/88.jpg)
. . . . . .
ExampleDropaballofftheroofoftheSilverCenter. Whatisitsvelocitywhenithitstheground?
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,
sothevelocityatimpactis v(2√5) = −20
√5m/s.
![Page 89: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/89.jpg)
. . . . . .
ExampleDropaballofftheroofoftheSilverCenter. Whatisitsvelocitywhenithitstheground?
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,
sothevelocityatimpactis v(2√5) = −20
√5m/s.
![Page 90: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/90.jpg)
. . . . . .
ExampleTheskidmarksmadebyanautomobileindicatethatitsbrakeswerefullyappliedforadistanceof160 ftbeforeitcametoastop. Supposethatthecarinquestionhasaconstantdecelerationof 20 ft/s2 undertheconditionsoftheskid. Howfastwasthecartravelingwhenitsbrakeswerefirstapplied?
Solution(Setup)
I Weknowthatthecarisdeceleratedby a(t) = −20I Weknowthatwhen s(t) = 160, v(t) = 0.I Wewanttoknow v(0) = v0.
![Page 91: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/91.jpg)
. . . . . .
ExampleTheskidmarksmadebyanautomobileindicatethatitsbrakeswerefullyappliedforadistanceof160 ftbeforeitcametoastop. Supposethatthecarinquestionhasaconstantdecelerationof 20 ft/s2 undertheconditionsoftheskid. Howfastwasthecartravelingwhenitsbrakeswerefirstapplied?
Solution(Setup)
I Weknowthatthecarisdeceleratedby a(t) = −20I Weknowthatwhen s(t) = 160, v(t) = 0.
I Wewanttoknow v(0) = v0.
![Page 92: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/92.jpg)
. . . . . .
ExampleTheskidmarksmadebyanautomobileindicatethatitsbrakeswerefullyappliedforadistanceof160 ftbeforeitcametoastop. Supposethatthecarinquestionhasaconstantdecelerationof 20 ft/s2 undertheconditionsoftheskid. Howfastwasthecartravelingwhenitsbrakeswerefirstapplied?
Solution(Setup)
I Weknowthatthecarisdeceleratedby a(t) = −20I Weknowthatwhen s(t) = 160, v(t) = 0.I Wewanttoknow v(0) = v0.
![Page 93: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/93.jpg)
. . . . . .
Solution(Implementation)
Ingeneral, s(t) = s0 + v0t +12at2, sowehave
s(t) = v0t− 10t2
v(t) = v0 − 20t
forall t.
If t1 isthetimeittookforthecartostop,
160 = v0t1 − 10t210 = v0 − 20t1
Weneedtosolvethesetwoequations.
![Page 94: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/94.jpg)
. . . . . .
Solution(Implementation)
Ingeneral, s(t) = s0 + v0t +12at2, sowehave
s(t) = v0t− 10t2
v(t) = v0 − 20t
forall t. If t1 isthetimeittookforthecartostop,
160 = v0t1 − 10t210 = v0 − 20t1
Weneedtosolvethesetwoequations.
![Page 95: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/95.jpg)
. . . . . .
Wehavev0t1 − 10t21 = 160 v0 − 20t1 = 0
I Thesecondgives t1 = v0/20, sosubstituteintothefirst:
v0 ·v020
− 10( v020
)2= 160
or
v2020
−10v20400
= 160
2v20 − v20 = 160 · 40 = 6400
I So v0 = 80 ft/s ≈ 55mi/hr
![Page 96: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/96.jpg)
. . . . . .
Wehavev0t1 − 10t21 = 160 v0 − 20t1 = 0
I Thesecondgives t1 = v0/20, sosubstituteintothefirst:
v0 ·v020
− 10( v020
)2= 160
or
v2020
−10v20400
= 160
2v20 − v20 = 160 · 40 = 6400
I So v0 = 80 ft/s ≈ 55mi/hr
![Page 97: Lesson 23: Antiderivatives](https://reader034.fdocuments.in/reader034/viewer/2022052523/5559efe1d8b42a39498b54a4/html5/thumbnails/97.jpg)
. . . . . .
Wehavev0t1 − 10t21 = 160 v0 − 20t1 = 0
I Thesecondgives t1 = v0/20, sosubstituteintothefirst:
v0 ·v020
− 10( v020
)2= 160
or
v2020
−10v20400
= 160
2v20 − v20 = 160 · 40 = 6400
I So v0 = 80 ft/s ≈ 55mi/hr