Download pdf - Lesson 18: Graphing

. . . . . .

Section4.3Graphingfunctions

Math1aIntroductiontoCalculus

March17, 2008

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. . . . . .

Announcements

◮ Thankyoufortheevaluations!◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323

. . . . . .

Outline

Thechecklist

Bigexample

Yourturn

. . . . . .

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. Puttogetherabigchart

4. Graph!

. . . . . .

Outline

Thechecklist

Bigexample

Yourturn

. . . . . .

Bigexample

Let f(x) =1x

+1x2. Wewilldoacompletedissectionof f.

. . . . . .

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

.


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

.


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 −∞.

. . . . . .

Step1

Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.

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

. . . . . .

Step1

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

f′(x) = − 1x2

− 2x3

= −x + 2x3

.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min .VA

. . . . . .

Step1


f′(x) = − 1x2

− 2x3

= −x + 2x3

.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−

.↘ .↗ .↘.min .VA

. . . . . .

Step1


f′(x) = − 1x2

− 2x3

= −x + 2x3

.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘

.↗ .↘.min .VA

. . . . . .

Step1


f′(x) = − 1x2

− 2x3

= −x + 2x3

.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗

.↘.min .VA

. . . . . .

Step1


f′(x) = − 1x2

− 2x3

= −x + 2x3

.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min .VA

. . . . . .

Step1


f′(x) = − 1x2

− 2x3

= −x + 2x3

.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min

.VA

. . . . . .

Step1


f′(x) = − 1x2

− 2x3

= −x + 2x3

.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min .VA

. . . . . .

Step2

Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.

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

. . . . . .

Step2

Find f′′ andformitssignchart. Concludeconcaveup/concavedownandinflectionpoints.Wehave

f′′(x) =2x3

+6x4

=2(x + 3)

x4.


. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Step2


f′′(x) =2x3

+6x4

=2(x + 3)

x4.


. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−−

.++ .++.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Step2


f′′(x) =2x3

+6x4

=2(x + 3)

x4.


. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++

.++.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Step2


f′′(x) =2x3

+6x4

=2(x + 3)

x4.


. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++

.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Step2


f′′(x) =2x3

+6x4

=2(x + 3)

x4.


. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢

.⌣ .⌣

.IP .VA

. . . . . .

Step2


f′′(x) =2x3

+6x4

=2(x + 3)

x4.


. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣

.⌣

.IP .VA

. . . . . .

Step2


f′′(x) =2x3

+6x4

=2(x + 3)

x4.


. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Step2


f′′(x) =2x3

+6x4

=2(x + 3)

x4.


. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣ .⌣

.IP

.VA

. . . . . .

Step2


f′′(x) =2x3

+6x4

=2(x + 3)

x4.


. .(x + 3)..0.−3

.− .+

.x4..0.0

.+ .+

.f′(x)

.f(x)..∞.0

..0.−3

.−− .++ .++.⌢ .⌣ .⌣

.IP .VA

. . . . . .

Step3

Puttogetherabigchart!

. .sign/value..∞.0

..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+

.monotonicity..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA . ✟ .IP . ✡.min . ✠.0 . ✠.VA . ✡ .HA

. . . . . .

Step3



..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+


..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA

. ✟ .IP . ✡.min . ✠.0 . ✠.VA . ✡ .HA

. . . . . .

Step3



..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+


..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA . ✟

.IP . ✡.min . ✠.0 . ✠.VA . ✡ .HA

. . . . . .

Step3



..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+


..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA . ✟ .IP

. ✡.min . ✠.0 . ✠.VA . ✡ .HA

. . . . . .

Step3



..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+


..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA . ✟ .IP . ✡

.min . ✠.0 . ✠.VA . ✡ .HA

. . . . . .

Step3



..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+


..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA . ✟ .IP . ✡.min

. ✠.0 . ✠.VA . ✡ .HA

. . . . . .

Step3



..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+


..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA . ✟ .IP . ✡.min . ✠

.0 . ✠.VA . ✡ .HA

. . . . . .

Step3



..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+


..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA . ✟ .IP . ✡.min . ✠.0

. ✠.VA . ✡ .HA

. . . . . .

Step3



..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+


..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA . ✟ .IP . ✡.min . ✠.0 . ✠

.VA . ✡ .HA

. . . . . .

Step3



..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+


..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA . ✟ .IP . ✡.min . ✠.0 . ✠.VA

. ✡ .HA

. . . . . .

Step3



..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+


..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA . ✟ .IP . ✡.min . ✠.0 . ✠.VA . ✡

.HA

. . . . . .

Step3



..0.−1

.−1/4.−2/9

.−∞.0

.∞.0

.− .+ .+


..0.−2

.− .+ .−.↘ .↗ .↘

.concavity..∞.0

..0.−3

.−− .++ .−−.⌢ .⌣ .⌣

.HA . ✟ .IP . ✡.min . ✠.0 . ✠.VA . ✡ .HA

. . . . . .

Step4

-4 -3 -2 -1 1 2

H-3,-2��

9L H-2,-

1��

4L

H-1,0L

. . . . . .

Outline

Thechecklist

Bigexample

Yourturn

. . . . . .

Yourturn

Plotthesefunctions(Groupwork):

1. f(x) = x4 − 4x3 + 10

2. f(x) =34(x2 − 1)2/3

3. f(x) =x3

3x2 + 14. f(x) = (2− x2)3/2.

. . . . . .

Graphof f(x) = x4 − 4x3 + 10

-2 -1 1 2 3 4 5

20

40

60

80

. . . . . .

Graphof f(x) =34(x2 − 1)2/3

-2 -1 1 2

0.5

1.0

1.5

. . . . . .

Graphof f(x) =x3

3x2 + 1

-2 -1 1 2

-0.6

-0.4

-0.2

0.2

0.4

0.6

. . . . . .

Graphof f(x) = (2− x2)3/2

-1.0 -0.5 0.5 1.0

0.5

1.0

1.5

2.0

2.5