Lesson 23: Antiderivatives

. . . . . .

Section4.7Antiderivatives

V63.0121.027, CalculusI

November19, 2009

Announcements

I NextwrittenassignmentwillbedueWednesday, Nov25I nextandlastquizwillbetheweekafterThanksgiving(4.1–4.4, 4.7)

I FinalExam: Friday, December18, 2:00–3:50pm

..Imagecredit: IanHampton

. . . . . .

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.

. . . . . .

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)

. . . . . .

Objectives

I Givenanexpressionforfunction f, findadifferentiablefunction Fsuchthat F′ = f (F iscalledan antiderivativefor f).

I Giventhegraphofafunction f, findadifferentiablefunction Fsuchthat F′ = f

I Useantiderivativestosolveproblemsinrectilinearmotion

. . . . . .

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!

. . . . . .

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!

. . . . . .

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!

. . . . . .

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!

. . . . . .

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!

. . . . . .

Outline

TabulatingAntiderivativesPowerfunctionsCombinationsExponentialfunctionsTrigonometricfunctions

FindingAntiderivativesGraphically

Rectilinearmotion

. . . . . .

Antiderivativesofpowerfunctions

Recallthatthederivativeofapowerfunctionisapowerfunction.

FactThePowerRuleIf f(x) = xr, then f′(x) = rxr−1.

Soinlookingforantiderivativesofpowerfunctions, trypowerfunctions!

. . . . . .

Antiderivativesofpowerfunctions

Recallthatthederivativeofapowerfunctionisapowerfunction.

FactThePowerRuleIf f(x) = xr, then f′(x) = rxr−1.

Soinlookingforantiderivativesofpowerfunctions, trypowerfunctions!

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

Graphof ln |x|

. .x

.y

.f(x) = 1/x

.F(x) = ln |x|

.F(x) = ln |x|

. . . . . .

Graphof ln |x|

. .x

.y

.f(x) = 1/x

.F(x) = ln |x|

.F(x) = ln |x|

. . . . . .

Graphof ln |x|

. .x

.y

.f(x) = 1/x

.F(x) = ln |x|

.F(x) = ln |x|

. . . . . .

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

. . . . . .

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

. . . . . .

ExampleFindanantiderivativefor f(x) = 16x + 5

SolutionTheexpression 8x2 isanantiderivativefor 16x, and 5x isanantiderivativefor 5. So

F(x) = 8x2 + 5x + C

istheantiderivativeof f.

. . . . . .

ExampleFindanantiderivativefor f(x) = 16x + 5

SolutionTheexpression 8x2 isanantiderivativefor 16x, and 5x isanantiderivativefor 5. So

F(x) = 8x2 + 5x + C

istheantiderivativeof f.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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).

. . . . . .

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).

. . . . . .

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).

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

Outline

TabulatingAntiderivativesPowerfunctionsCombinationsExponentialfunctionsTrigonometricfunctions

FindingAntiderivativesGraphically

Rectilinearmotion

. . . . . .

ProblemBelowisthegraphofafunction f. Drawthegraphofanantiderivativefor F.

..x

.y

..1

..2

..3

..4

..5

..6

.

.

.

. .

. .y = f(x)

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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?

. . . . . .

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

.

.

.

.

..

. . . . . .

Outline

TabulatingAntiderivativesPowerfunctionsCombinationsExponentialfunctionsTrigonometricfunctions

FindingAntiderivativesGraphically

Rectilinearmotion

. . . . . .

Saywhat?

I “Rectlinearmotion”justmeansmotionalongaline.I Oftenwearegiveninformationaboutthevelocityoraccelerationofamovingparticleandwewanttoknowtheequationsofmotion.

. . . . . .

Example: DeadReckoning

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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.

. . . . . .

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

. . . . . .

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

. . . . . .

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