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. . . . . .
Section4.3Graphingfunctions
Math1aIntroductiontoCalculus
March17, 2008
Announcements
◮ Thankyoufortheevaluations!◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323
..Image: Flickruser Cobalt123
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. . . . . .
Announcements
◮ Thankyoufortheevaluations!◮ ProblemSessionsSunday, Thursday, 7pm, SC 310◮ OfficehoursTues, Weds, 2–4pmSC 323
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. . . . . .
Outline
Thechecklist
Bigexample
Yourturn
Page 4
. . . . . .
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!
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. . . . . .
Outline
Thechecklist
Bigexample
Yourturn
Page 6
. . . . . .
Bigexample
Let f(x) =1x
+1x2. Wewilldoacompletedissectionof f.
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. . . . . .
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
.− .+ .+
Page 8
. . . . . .
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
.− .+ .+
Page 9
. . . . . .
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
.− .+ .+
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. . . . . .
Forhorizontalasymptotes, noticethat
limx→∞
x + 1x2
= 0,
so y = 0 isahorizontalasymptoteofthegraph. Thesameistrueat −∞.
Page 11
. . . . . .
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
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. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.min .VA
Page 13
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−
.↘ .↗ .↘.min .VA
Page 14
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘
.↗ .↘.min .VA
Page 15
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘ .↗
.↘.min .VA
Page 16
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.min .VA
Page 17
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.min
.VA
Page 18
. . . . . .
Step1
Find f′ andformitssignchart. Concludeinformationaboutincreasing/decreasingandlocalmax/min.Wehave
f′(x) = − 1x2
− 2x3
= −x + 2x3
.
Thecriticalpointsare x = −2 and x = 0. Wehavethefollowingsignchart:
. .−(x + 2)..0.−2
.+ .−
.x3..0.0
.− .+
.f′(x)
.f(x)..∞.0
..0.−2
.− .+ .−.↘ .↗ .↘
.min .VA
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. . . . . .
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
Page 20
. . . . . .
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
Page 21
. . . . . .
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
Page 22
. . . . . .
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
Page 23
. . . . . .
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
Page 24
. . . . . .
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
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. . . . . .
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
Page 26
. . . . . .
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
Page 27
. . . . . .
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
Page 28
. . . . . .
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
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. . . . . .
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
Page 30
. . . . . .
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
Page 31
. . . . . .
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
Page 32
. . . . . .
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
Page 33
. . . . . .
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
Page 34
. . . . . .
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
Page 35
. . . . . .
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
Page 36
. . . . . .
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
Page 37
. . . . . .
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
Page 38
. . . . . .
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
Page 39
. . . . . .
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
Page 40
. . . . . .
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
Page 41
. . . . . .
Step4
-4 -3 -2 -1 1 2
H-3,-2����
9L H-2,-
1����
4L
H-1,0L
Page 42
. . . . . .
Outline
Thechecklist
Bigexample
Yourturn
Page 43
. . . . . .
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.
Page 44
. . . . . .
Graphof f(x) = x4 − 4x3 + 10
-2 -1 1 2 3 4 5
20
40
60
80
Page 45
. . . . . .
Graphof f(x) =34(x2 − 1)2/3
-2 -1 1 2
0.5
1.0
1.5
Page 46
. . . . . .
Graphof f(x) =x3
3x2 + 1
-2 -1 1 2
-0.6
-0.4
-0.2
0.2
0.4
0.6
Page 47
. . . . . .
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