. . . . . .

Section4.4CurveSketchingII

V63.0121, CalculusI

April1, 2009

Announcements

I

. . . . . .

GraphingChecklist

Tographafunction f, followthisplan:

0. Findwhen f ispositive, negative, zero, notdefined.

1. Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.

2. Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflection.

3. Puttogetherabigcharttoassemblemonotonicityandconcavitydata

4. Graph!

. . . . . .

Outline

MoreExamplesPointsofnondifferentiabilityHorizontalasymptotesVerticalasymptotesTrigonometricandpolynomialtogether

. . . . . .

ExampleGraph f(x) = x +

√|x|

I Thisfunctionlooksstrangebecauseoftheabsolutevalue.Butwheneverwebecomenervous, wecanjusttakecases.

I First, lookat f byitself. Wecantellthat f(0) = 0 andthatf(x) > 0 if x ispositive. Aretherenegativenumberswhicharezeroesfor f? Yes, if x = −1 thenx +

√|x| = −1 +

√1 = 0. Nootherzerosexist.

. . . . . .

ExampleGraph f(x) = x +

√|x|

I Thisfunctionlooksstrangebecauseoftheabsolutevalue.Butwheneverwebecomenervous, wecanjusttakecases.

I First, lookat f byitself. Wecantellthat f(0) = 0 andthatf(x) > 0 if x ispositive. Aretherenegativenumberswhicharezeroesfor f? Yes, if x = −1 thenx +

√|x| = −1 +

√1 = 0. Nootherzerosexist.

. . . . . .

Asymptoticbehavior

I Asymptotically, it’sclearthat limx→∞

f(x) = ∞, becauseboth

termstendto ∞.I Whatabout x → −∞? Thisisnowindeterminateoftheform

−∞ + ∞. Toresolveit, firstlet y = −x tomakeitlookmorefamiliar:

limx→−∞

(x +

√−x

)= lim

y→∞(−y +

√y) = lim

y→∞(√y− y)

Nowmultiplybytheconjugate:

limy→∞

(√y− y) ·

√y + y

√y + y

= limy→∞

y− y2√y + y

= −∞

. . . . . .

Thederivative

First, assume x > 0, so

f′(x) =ddx

(x +

√x)

= 1 +1

2√x

Thisisalwayspositive. Also, weseethat limx→0+

f′(x) = ∞ and

limx→∞

f′(x) = 1. If x isnegative, wehave

f′(x) =ddx

(x +

√−x

)= 1− 1

2√−x

Again, thislooksweirdbecause√−x appearstobeanegative

number. Butsince x < 0, −x > 0.

. . . . . .

Monotonicity

Weseethat f′(x) = 0 when

1 =1

2√−x

=⇒√−x =

12

=⇒ −x =14

=⇒ x = −14

Noticealsothat limx→0−

f′(x) = −∞ and limx→−∞

f′(x) = 1. Wecan’t

makeamulti-factorsignchartbecauseoftheabsolutevalue, buttheconclusionisthis:

..f′(x)

.f(x).

.−14

.0

. max

..0

.∓∞

. cusp

.+

.↗.−.↘

.+

.↗

. . . . . .

Concavity

If x > 0, then

f′′(x) =ddx

(1 +

12x−1/2

)= −1

4x−3/2

Thisisnegativewhenever x < 0. If x < 0, then

f′′(x) =ddx

(1− 1

2(−x)−1/2

)= −1

4(−x)−3/2

whichisalsoalwaysnegativefornegative x. Anotherwayto

writethisis f′′(x) = −14|x|−3/2. Hereisthesignchart:

..f′′(x)

.f(x)..0

.−∞.−−.⌢

.

..−−.⌢

. . . . . .

Synthesis

Nowwecanputthesethingstogether.

..f′(x)

.monotonicity.

.−14

.0 ..0

.∓∞.+

.↗.−.↘

.+

.↗.f′′(x)

.concavity..0

.−∞.−−.⌢

.−−.⌢

.−−.⌢

.f(x)

.shape.

.−1.0

. zero

..−1

4

.14

. max

..0.0

. cusp." . ."

. . . . . .

Graph

.

.f(x)

.shape.

.−1.0

. zero

..−1

4

.14

. max

..0.0

. cusp." . ."

.x

.f(x)

..(−1, 0) .

.(−14 ,

14)

..(0, 0)

. . . . . .

ExampleGraph f(x) = xe−x2

Beforetakingderivatives, wenoticethat f isodd, that f(0) = 0,and lim

x→∞f(x) = 0

. . . . . .

ExampleGraph f(x) = xe−x2

Beforetakingderivatives, wenoticethat f isodd, that f(0) = 0,and lim

x→∞f(x) = 0

. . . . . .

Monotonicity

Now

f′(x) = 1 · e−x2 + xe−x2(−2x) =(1− 2x2

)e−x2

=(1−

√2x

)(1 +

√2x

)e−x2

Thefactor e−x2 isalwayspositivesoitdoesn’tfigureintothesignof f′(x). Sooursignchartlookslikethis:

. .1−√2x.

.√

1/2

.0.+ .+ .−

.1 +√2x.

.−√

1/2

.0.− .+ .+

.f′(x)

.f(x).

.−√

1/2

.0

.min

..√

1/2

.0

. max

.−.↘

.+

.↗.−.↘

. . . . . .

Concavity

Nowwelookat f′′(x):

f′′(x) = (−4x)e−x2 + (1− 2x2)e−x2(−2x) =(4x3 − 6x

)e−x2

= 2x(2x2 − 3)e−x2

. .2x..0.0.− .− .+ .+

.√2x−

√3.

.√

3/2

.0.− .− .− .+

.√2x +

√3.

.−√

3/2

.0.− .+ .+ .+

.f′′(x)

.f(x).

.−√

3/2

.0

.IP

..0.0

.IP

..√

3/2

.0

.IP

.−−.⌢

.++.⌣

.−−.⌢

.++.⌣

. . . . . .

Synthesis

..f′(x)

.monotonicity.

.−√

1/2

.0 ..√

1/2

.0.−.↘

.−.↘

.+

.↗.+

.↗.−.↘

.−.↘

.f′′(x)

.concavity.

.−√

3/2

.0 ..0.0 .

.√

3/2

.0.−−.⌢

.++.⌣

.++.⌣

.−−.⌢

.−−.⌢

.++.⌣

.f(x)

.shape.

.−√

1/2

.− 1√2e

.min

..√

1/2

. 1√2e

. max

..−

√3/2

.−√

32e3

.IP

..0.0

.IP

..√

3/2

.√

32e3

.IP

. . . " ." . .

. . . . . .

Graph

.

.x

.f(x)

.

.(−

√1/2,− 1√

2e

)

..(√

1/2, 1√2e

)

.

.(−

√3/2,−

√32e3

)..(0, 0)

..(√

3/2,√

32e3

)

.f(x)

.shape.

.−√

1/2

.− 1√2e

.min

..√

1/2

. 1√2e

. max

..−

√3/2

.−√

32e3

.IP

..0.0

.IP

..√

3/2

.√

32e3

.IP

. . . " ." . .

. . . . . .

Example

Graph f(x) =1x

+1x2

. . . . . .

Step0Findwhen f ispositive, negative, zero, notdefined.

Weneedtofactor f:

f(x) =1x

+1x2

=x + 1x2

.

Thismeans f is 0 at −1 andhastroubleat 0. Infact,

limx→0

x + 1x2

= ∞,

so x = 0 isaverticalasymptoteofthegraph. Wecanmakeasignchartasfollows:

. .x + 1..0.−1

.− .+

.x2..0.0

.+ .+

.f(x)..∞.0

..0.−1

.− .+ .+

. . . . . .

Step0Findwhen f ispositive, negative, zero, notdefined. Weneedtofactor f:

f(x) =1x

+1x2

=x + 1x2

.

Thismeans f is 0 at −1 andhastroubleat 0. Infact,

limx→0

x + 1x2

= ∞,

so x = 0 isaverticalasymptoteofthegraph.

Wecanmakeasignchartasfollows:

. .x + 1..0.−1

.− .+

.x2..0.0

.+ .+

.f(x)..∞.0

..0.−1

.− .+ .+

. . . . . .

Step0Findwhen f ispositive, negative, zero, notdefined. Weneedtofactor f:

f(x) =1x

+1x2

=x + 1x2

.

Thismeans f is 0 at −1 andhastroubleat 0. Infact,

limx→0

x + 1x2

= ∞,

so x = 0 isaverticalasymptoteofthegraph. Wecanmakeasignchartasfollows:

. .x + 1..0.−1

.− .+

.x2..0.0

.+ .+

.f(x)..∞.0

..0.−1

.− .+ .+

. . . . . .

Forhorizontalasymptotes, noticethat

limx→∞

x + 1x2

= 0,

so y = 0 isahorizontalasymptoteofthegraph. Thesameistrueat −∞.

. . . . . .

Monotonicity

Wehavef′(x) = − 1

x2− 2

x3= −x + 2

x3.

Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:

. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min .VA

. . . . . .

Monotonicity

Wehavef′(x) = − 1

x2− 2

x3= −x + 2

x3.

Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:

. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min .VA

. . . . . .

Monotonicity

Wehavef′(x) = − 1

x2− 2

x3= −x + 2

x3.

Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:

. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−

.↘ .↗ .↘.min .VA

. . . . . .

Monotonicity

Wehavef′(x) = − 1

x2− 2

x3= −x + 2

x3.

Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:

. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘

.↗ .↘.min .VA

. . . . . .

Monotonicity

Wehavef′(x) = − 1

x2− 2

x3= −x + 2

x3.

Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:

. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗

.↘.min .VA

. . . . . .

Monotonicity

Wehavef′(x) = − 1

x2− 2

x3= −x + 2

x3.

Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:

. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min .VA

. . . . . .

Monotonicity

Wehavef′(x) = − 1

x2− 2

x3= −x + 2

x3.

Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:

. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min

.VA

. . . . . .

Monotonicity

Wehavef′(x) = − 1

x2− 2

x3= −x + 2

x3.

Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:

. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min .VA

. . . . . .

Concavity

Wehave

f′′(x) =2x3

+6x4

=2(x + 3)

x4.

Thecriticalpointsof f′ are −3 and 0. Signchart:

. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Concavity

Wehave

f′′(x) =2x3

+6x4

=2(x + 3)

x4.

Thecriticalpointsof f′ are −3 and 0. Signchart:

. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Concavity

Wehave

f′′(x) =2x3

+6x4

=2(x + 3)

x4.

Thecriticalpointsof f′ are −3 and 0. Signchart:

. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−−

.++ .++.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Concavity

Wehave

f′′(x) =2x3

+6x4

=2(x + 3)

x4.

Thecriticalpointsof f′ are −3 and 0. Signchart:

. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++

.++.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Concavity

Wehave

f′′(x) =2x3

+6x4

=2(x + 3)

x4.

Thecriticalpointsof f′ are −3 and 0. Signchart:

. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++

.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Concavity

Wehave

f′′(x) =2x3

+6x4

=2(x + 3)

x4.

Thecriticalpointsof f′ are −3 and 0. Signchart:

. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢

.⌣ .⌣

.IP .VA

. . . . . .

Concavity

Wehave

f′′(x) =2x3

+6x4

=2(x + 3)

x4.

Thecriticalpointsof f′ are −3 and 0. Signchart:

. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣

.⌣

.IP .VA

. . . . . .

Concavity

Wehave

f′′(x) =2x3

+6x4

=2(x + 3)

x4.

Thecriticalpointsof f′ are −3 and 0. Signchart:

. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Concavity

Wehave

f′′(x) =2x3

+6x4

=2(x + 3)

x4.

Thecriticalpointsof f′ are −3 and 0. Signchart:

. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣ .⌣

.IP

.VA

. . . . . .

Concavity

Wehave

f′′(x) =2x3

+6x4

=2(x + 3)

x4.

Thecriticalpointsof f′ are −3 and 0. Signchart:

. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+

.HA . .IP . .min . " .0 . " .VA . .HA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+.HA

. .IP . .min . " .0 . " .VA . .HA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+.HA .

.IP . .min . " .0 . " .VA . .HA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+.HA . .IP

. .min . " .0 . " .VA . .HA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+.HA . .IP .

.min . " .0 . " .VA . .HA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+.HA . .IP . .min

. " .0 . " .VA . .HA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+.HA . .IP . .min . "

.0 . " .VA . .HA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+.HA . .IP . .min . " .0

. " .VA . .HA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+.HA . .IP . .min . " .0 . "

.VA . .HA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+.HA . .IP . .min . " .0 . " .VA

. .HA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+.HA . .IP . .min . " .0 . " .VA .

.HA

. . . . . .

Synthesis

.

.f′

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.f′′

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.f

.shapeof f..∞.0

..0.−1

..−2.−1/4

..−3.−2/9

.−∞.0

.∞.0

.− .+ .+.HA . .IP . .min . " .0 . " .VA . .HA

. . . . . .

Graph

. .x

.y

..(−3,−2/9)

..(−2,−1/4)

. . . . . .

ProblemGraph f(x) = cos x− x

-5 5 10

-10

-5

5

. . . . . .

ProblemGraph f(x) = cos x− x

-5 5 10

-10

-5

5

. . . . . .

ProblemGraph f(x) = x ln x2

-3 -2 -1 1 2 3

-6

-4

-2

2

4

6

. . . . . .

ProblemGraph f(x) = x ln x2

-3 -2 -1 1 2 3

-6

-4

-2

2

4

6