Lesson 3.6 (Continued) Graphing Exponential Functions 1 3.4.2: Graphing Exponential Functions.
Lesson 22: Graphing
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Transcript of Lesson 22: Graphing
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. . . . . .
Section4.4CurveSketching
V63.0121.027, CalculusI
November17, 2009
Announcements
I NextwrittenassignmentwillbedueWednesday, Nov25I nextandlastquizwillbetheweekafterThanksgivingI FinalExam: Friday, December18, 2:00–3:50pm
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. . . . . .
Outline
TheProcedure
SimpleexamplesA cubicfunctionA quarticfunction
MoreExamplesPointsofnondifferentiabilityHorizontalasymptotesVerticalasymptotesTrigonometricandpolynomialtogetherLogarithmic
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. . . . . .
Objective
Givenafunction, graphitcompletely, indicating
I zeroesI asymptotesifapplicableI criticalpointsI local/globalmax/minI inflectionpoints
.
.Imagecredit: ImageOfSurgery
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. . . . . .
TheIncreasing/DecreasingTest
Theorem(TheIncreasing/DecreasingTest)If f′ > 0 on (a,b), then f isincreasingon (a,b). If f′ < 0 on (a,b),then f isdecreasingon (a,b).
Proof.Picktwopoints x and y in (a,b) with x < y. Wemustshowf(x) < f(y). ByMVT thereexistsapoint c in (x, y) suchthat
f(y) − f(x)y− x
= f′(c) > 0.
Sof(y) − f(x) = f′(c)(y− x) > 0.
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. . . . . .
Theorem(ConcavityTest)
I If f′′(x) > 0 forall x in I, thenthegraphof f isconcaveupwardon I
I If f′′(x) < 0 forall x in I, thenthegraphof f isconcavedownwardon I
Proof.Suppose f′′(x) > 0 on I. Thismeans f′ isincreasingon I. Let a andx bein I. Thetangentlinethrough (a, f(a)) isthegraphof
L(x) = f(a) + f′(a)(x− a)
ByMVT,thereexistsa b between a and x withf(x) − f(a)
x− a= f′(b).
So
f(x) = f(a) + f′(b)(x− a) ≥ f(a) + f′(a)(x− a) = L(x)
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. . . . . .
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!
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. . . . . .
Outline
TheProcedure
SimpleexamplesA cubicfunctionA quarticfunction
MoreExamplesPointsofnondifferentiabilityHorizontalasymptotesVerticalasymptotesTrigonometricandpolynomialtogetherLogarithmic
![Page 8: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/8.jpg)
. . . . . .
Graphingacubic
ExampleGraph f(x) = 2x3 − 3x2 − 12x.
(Step0)First, let’sfindthezeros. Wecanatleastfactoroutonepowerof x:
f(x) = x(2x2 − 3x− 12)
so f(0) = 0. Theotherfactorisaquadratic, sowetheothertworootsare
x =3±
√32 − 4(2)(−12)
4=
3±√105
4
It’sOK toskipthisstepfornowsincetherootsaresocomplicated.
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. . . . . .
Graphingacubic
ExampleGraph f(x) = 2x3 − 3x2 − 12x.
(Step0)First, let’sfindthezeros. Wecanatleastfactoroutonepowerof x:
f(x) = x(2x2 − 3x− 12)
so f(0) = 0. Theotherfactorisaquadratic, sowetheothertworootsare
x =3±
√32 − 4(2)(−12)
4=
3±√105
4
It’sOK toskipthisstepfornowsincetherootsaresocomplicated.
![Page 10: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/10.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
.
.x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+ .− .+
.↗ .↘ .↗.max .min
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. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
. .x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+ .− .+
.↗ .↘ .↗.max .min
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. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
. .x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+ .− .+
.↗ .↘ .↗.max .min
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. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
. .x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+ .− .+
.↗ .↘ .↗.max .min
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. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
. .x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+
.− .+
.↗ .↘ .↗.max .min
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. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
. .x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+ .−
.+
.↗ .↘ .↗.max .min
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. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
. .x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+ .− .+
.↗ .↘ .↗.max .min
![Page 17: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/17.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
. .x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+ .− .+
.↗
.↘ .↗.max .min
![Page 18: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/18.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
. .x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+ .− .+
.↗ .↘
.↗.max .min
![Page 19: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/19.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
. .x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+ .− .+
.↗ .↘ .↗
.max .min
![Page 20: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/20.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
. .x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+ .− .+
.↗ .↘ .↗.max
.min
![Page 21: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/21.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 6x2 − 6x− 12 = 6(x + 1)(x− 2)
Wecanformasignchartfromthis:
. .x− 2..2
.− .− .+
.x + 1..−1
.+.+.−
.f′(x)
.f(x)..2
..−1
.+ .− .+
.↗ .↘ .↗.max .min
![Page 22: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/22.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++.⌢ .⌣
.IP
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. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++.⌢ .⌣
.IP
![Page 24: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/24.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−−
.++.⌢ .⌣
.IP
![Page 25: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/25.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++
.⌢ .⌣
.IP
![Page 26: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/26.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++.⌢
.⌣
.IP
![Page 27: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/27.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++.⌢ .⌣
.IP
![Page 28: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/28.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++.⌢ .⌣
.IP
![Page 29: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/29.jpg)
. . . . . .
Step3: Onesigncharttorulethemall
Remember, f(x) = 2x3 − 3x2 − 12x.
.
.f′(x)
.monotonicity.
.−1..2
.+
.↗.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity.
.1/2
.−−.⌢
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." . . . "
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. . . . . .
Step3: Onesigncharttorulethemall
Remember, f(x) = 2x3 − 3x2 − 12x.
..f′(x)
.monotonicity.
.−1..2
.+
.↗.−.↘
.−.↘
.+
.↗
.f′′(x)
.concavity.
.1/2
.−−.⌢
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." . . . "
![Page 31: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/31.jpg)
. . . . . .
Step3: Onesigncharttorulethemall
Remember, f(x) = 2x3 − 3x2 − 12x.
..f′(x)
.monotonicity.
.−1..2
.+
.↗.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity.
.1/2
.−−.⌢
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." . . . "
![Page 32: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/32.jpg)
. . . . . .
Step3: Onesigncharttorulethemall
Remember, f(x) = 2x3 − 3x2 − 12x.
..f′(x)
.monotonicity.
.−1..2
.+
.↗.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity.
.1/2
.−−.⌢
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." . . . "
![Page 33: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/33.jpg)
. . . . . .
Combinationsofmonotonicityandconcavity
.
.I.II
.III .IV
.decreasing,concavedown
.increasing,concavedown
.decreasing,concave up
.increasing,concave up
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. . . . . .
Step3: Onesigncharttorulethemall
Remember, f(x) = 2x3 − 3x2 − 12x.
..f′(x)
.monotonicity.
.−1..2
.+
.↗.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity.
.1/2
.−−.⌢
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
."
. . . "
![Page 35: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/35.jpg)
. . . . . .
Step3: Onesigncharttorulethemall
Remember, f(x) = 2x3 − 3x2 − 12x.
..f′(x)
.monotonicity.
.−1..2
.+
.↗.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity.
.1/2
.−−.⌢
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." .
. . "
![Page 36: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/36.jpg)
. . . . . .
Step3: Onesigncharttorulethemall
Remember, f(x) = 2x3 − 3x2 − 12x.
..f′(x)
.monotonicity.
.−1..2
.+
.↗.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity.
.1/2
.−−.⌢
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." . .
. "
![Page 37: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/37.jpg)
. . . . . .
Step3: Onesigncharttorulethemall
Remember, f(x) = 2x3 − 3x2 − 12x.
..f′(x)
.monotonicity.
.−1..2
.+
.↗.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity.
.1/2
.−−.⌢
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." . . . "
![Page 38: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/38.jpg)
. . . . . .
Step4: Graph
.
.f(x) = 2x3 − 3x2 − 12x
.x
.f(x)
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." . . . "
..(3−
√105
4 , 0) .
.(−1, 7)
..(0, 0)
..(1/2,−61/2)
..(2,−20)
.
.(3+
√105
4 , 0)
![Page 39: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/39.jpg)
. . . . . .
Step4: Graph
.
.f(x) = 2x3 − 3x2 − 12x
.x
.f(x)
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." . . . "
..(3−
√105
4 , 0) .
.(−1, 7)
..(0, 0)
..(1/2,−61/2)
..(2,−20)
.
.(3+
√105
4 , 0)
![Page 40: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/40.jpg)
. . . . . .
Step4: Graph
.
.f(x) = 2x3 − 3x2 − 12x
.x
.f(x)
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." . . . "
..(3−
√105
4 , 0) .
.(−1, 7)
..(0, 0)
..(1/2,−61/2)
..(2,−20)
.
.(3+
√105
4 , 0)
![Page 41: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/41.jpg)
. . . . . .
Step4: Graph
.
.f(x) = 2x3 − 3x2 − 12x
.x
.f(x)
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." . . . "
..(3−
√105
4 , 0) .
.(−1, 7)
..(0, 0)
..(1/2,−61/2)
..(2,−20)
.
.(3+
√105
4 , 0)
![Page 42: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/42.jpg)
. . . . . .
Step4: Graph
.
.f(x) = 2x3 − 3x2 − 12x
.x
.f(x)
.f(x)
.shapeof f.
.−1.7
.max
..2
.−20
.min
..1/2
.−61/2
.IP
." . . . "
..(3−
√105
4 , 0) .
.(−1, 7)
..(0, 0)
..(1/2,−61/2)
..(2,−20)
.
.(3+
√105
4 , 0)
![Page 43: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/43.jpg)
. . . . . .
Graphingaquartic
ExampleGraph f(x) = x4 − 4x3 + 10
(Step0)Weknow f(0) = 10 and limx→±∞
f(x) = +∞. Nottoomany
otherpointsonthegraphareevident.
![Page 44: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/44.jpg)
. . . . . .
Graphingaquartic
ExampleGraph f(x) = x4 − 4x3 + 10
(Step0)Weknow f(0) = 10 and limx→±∞
f(x) = +∞. Nottoomany
otherpointsonthegraphareevident.
![Page 45: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/45.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
.
.4x2..0.0.+ .+ .+
.(x− 3)..3.0
.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0
.− .− .+
.↘ .↘ .↗.min
![Page 46: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/46.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
.
.4x2..0.0.+ .+ .+
.(x− 3)..3.0
.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0
.− .− .+
.↘ .↘ .↗.min
![Page 47: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/47.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0
.+ .+ .+
.(x− 3)..3.0
.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0
.− .− .+
.↘ .↘ .↗.min
![Page 48: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/48.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+
.+ .+
.(x− 3)..3.0
.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0
.− .− .+
.↘ .↘ .↗.min
![Page 49: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/49.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+
.+
.(x− 3)..3.0
.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0
.− .− .+
.↘ .↘ .↗.min
![Page 50: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/50.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0
.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0
.− .− .+
.↘ .↘ .↗.min
![Page 51: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/51.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0
.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0
.− .− .+
.↘ .↘ .↗.min
![Page 52: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/52.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0.−
.− .+
.f′(x)
.f(x)..3.0.
.0
.0
.− .− .+
.↘ .↘ .↗.min
![Page 53: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/53.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0.− .−
.+
.f′(x)
.f(x)..3.0.
.0
.0
.− .− .+
.↘ .↘ .↗.min
![Page 54: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/54.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0
.− .− .+
.↘ .↘ .↗.min
![Page 55: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/55.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0
.− .− .+
.↘ .↘ .↗.min
![Page 56: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/56.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0.−
.− .+
.↘ .↘ .↗.min
![Page 57: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/57.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0.− .−
.+
.↘ .↘ .↗.min
![Page 58: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/58.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0.− .− .+
.↘ .↘ .↗.min
![Page 59: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/59.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0.− .− .+
.↘
.↘ .↗.min
![Page 60: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/60.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0.− .− .+
.↘ .↘
.↗.min
![Page 61: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/61.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0.− .− .+
.↘ .↘ .↗
.min
![Page 62: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/62.jpg)
. . . . . .
Step1: Monotonicity
f′(x) = 4x3 − 12x2 = 4x2(x− 3)
Wemakeitssignchart.
. .4x2..0.0.+ .+ .+
.(x− 3)..3.0.− .− .+
.f′(x)
.f(x)..3.0.
.0
.0.− .− .+
.↘ .↘ .↗.min
![Page 63: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/63.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
.
.12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 64: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/64.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
.
.12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 65: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/65.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0
.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 66: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/66.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.−
.+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 67: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/67.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+
.+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 68: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/68.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0
.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 69: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/69.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0
.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 70: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/70.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.−
.− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 71: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/71.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.− .−
.+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 72: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/72.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 73: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/73.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0
.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 74: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/74.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++
.−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 75: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/75.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−−
.++.⌣ .⌢ .⌣
.IP .IP
![Page 76: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/76.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++
.⌣ .⌢ .⌣
.IP .IP
![Page 77: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/77.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣
.⌢ .⌣
.IP .IP
![Page 78: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/78.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢
.⌣
.IP .IP
![Page 79: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/79.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 80: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/80.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP
.IP
![Page 81: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/81.jpg)
. . . . . .
Step2: Concavity
f′′(x) = 12x2 − 24x = 12x(x− 2)
Hereisitssignchart:
. .12x..0.0.− .+ .+
.x− 2..2.0.− .− .+
.f′′(x)
.f(x)..0.0 .
.2
.0.++ .−− .++.⌣ .⌢ .⌣
.IP .IP
![Page 82: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/82.jpg)
. . . . . .
Step3: GrandUnifiedSignChart
Remember, f(x) = x4 − 4x3 + 10.
.
.f′(x)
.monotonicity..3.0.
.0
.0.−.↘
.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity..0.0 .
.2
.0.++.⌣
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shape..0.10
.IP
..2.−6
.IP
..3
.−17
.min
. . . . "
![Page 83: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/83.jpg)
. . . . . .
Step3: GrandUnifiedSignChart
Remember, f(x) = x4 − 4x3 + 10.
.
.f′(x)
.monotonicity..3.0.
.0
.0.−.↘
.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity..0.0 .
.2
.0.++.⌣
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shape..0.10
.IP
..2.−6
.IP
..3
.−17
.min
.
. . . "
![Page 84: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/84.jpg)
. . . . . .
Step3: GrandUnifiedSignChart
Remember, f(x) = x4 − 4x3 + 10.
.
.f′(x)
.monotonicity..3.0.
.0
.0.−.↘
.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity..0.0 .
.2
.0.++.⌣
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shape..0.10
.IP
..2.−6
.IP
..3
.−17
.min
. .
. . "
![Page 85: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/85.jpg)
. . . . . .
Step3: GrandUnifiedSignChart
Remember, f(x) = x4 − 4x3 + 10.
.
.f′(x)
.monotonicity..3.0.
.0
.0.−.↘
.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity..0.0 .
.2
.0.++.⌣
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shape..0.10
.IP
..2.−6
.IP
..3
.−17
.min
. . .
. "
![Page 86: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/86.jpg)
. . . . . .
Step3: GrandUnifiedSignChart
Remember, f(x) = x4 − 4x3 + 10.
.
.f′(x)
.monotonicity..3.0.
.0
.0.−.↘
.−.↘
.−.↘
.+
.↗.f′′(x)
.concavity..0.0 .
.2
.0.++.⌣
.−−.⌢
.++.⌣
.++.⌣
.f(x)
.shape..0.10
.IP
..2.−6
.IP
..3
.−17
.min
. . . . "
![Page 87: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/87.jpg)
. . . . . .
Step4: Graph
.
.f(x) = x4 − 4x3 + 10
.x
.y
.f(x)
.shape..0.10
.IP
..2.−6
.IP
..3
.−17
.min
. . . . "
..(0, 10)
..(2,−6) .
.(3,−17)
![Page 88: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/88.jpg)
. . . . . .
Step4: Graph
.
.f(x) = x4 − 4x3 + 10
.x
.y
.f(x)
.shape..0.10
.IP
..2.−6
.IP
..3
.−17
.min
. . . . "
..(0, 10)
..(2,−6) .
.(3,−17)
![Page 89: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/89.jpg)
. . . . . .
Step4: Graph
.
.f(x) = x4 − 4x3 + 10
.x
.y
.f(x)
.shape..0.10
.IP
..2.−6
.IP
..3
.−17
.min
. . . . "
..(0, 10)
..(2,−6) .
.(3,−17)
![Page 90: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/90.jpg)
. . . . . .
Step4: Graph
.
.f(x) = x4 − 4x3 + 10
.x
.y
.f(x)
.shape..0.10
.IP
..2.−6
.IP
..3
.−17
.min
. . . . "
..(0, 10)
..(2,−6) .
.(3,−17)
![Page 91: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/91.jpg)
. . . . . .
Step4: Graph
.
.f(x) = x4 − 4x3 + 10
.x
.y
.f(x)
.shape..0.10
.IP
..2.−6
.IP
..3
.−17
.min
. . . . "
..(0, 10)
..(2,−6) .
.(3,−17)
![Page 92: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/92.jpg)
. . . . . .
Outline
TheProcedure
SimpleexamplesA cubicfunctionA quarticfunction
MoreExamplesPointsofnondifferentiabilityHorizontalasymptotesVerticalasymptotesTrigonometricandpolynomialtogetherLogarithmic
![Page 93: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/93.jpg)
. . . . . .
ExampleGraph f(x) = x +
√|x|
Thisfunctionlooksstrangebecauseoftheabsolutevalue. Butwheneverwebecomenervous, wecanjusttakecases.
![Page 94: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/94.jpg)
. . . . . .
ExampleGraph f(x) = x +
√|x|
Thisfunctionlooksstrangebecauseoftheabsolutevalue. Butwheneverwebecomenervous, wecanjusttakecases.
![Page 95: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/95.jpg)
. . . . . .
Step0: FindingZeroes
f(x) = x +√
|x|I First, lookat f byitself. Wecantellthat f(0) = 0 andthat
f(x) > 0 if x ispositive.
I Aretherenegativenumberswhicharezeroesfor f?
x +√−x = 0
√−x = −x
−x = x2
x2 + x = 0
Theonlysolutionsare x = 0 and x = −1
![Page 96: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/96.jpg)
. . . . . .
Step0: FindingZeroes
f(x) = x +√
|x|I First, lookat f byitself. Wecantellthat f(0) = 0 andthat
f(x) > 0 if x ispositive.I Aretherenegativenumberswhicharezeroesfor f?
x +√−x = 0
√−x = −x
−x = x2
x2 + x = 0
Theonlysolutionsare x = 0 and x = −1
![Page 97: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/97.jpg)
. . . . . .
Step0: FindingZeroes
f(x) = x +√
|x|I First, lookat f byitself. Wecantellthat f(0) = 0 andthat
f(x) > 0 if x ispositive.I Aretherenegativenumberswhicharezeroesfor f?
x +√−x = 0
√−x = −x
−x = x2
x2 + x = 0
Theonlysolutionsare x = 0 and x = −1
![Page 98: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/98.jpg)
. . . . . .
Step0: Asymptoticbehavior
f(x) = x +√
|x|I lim
x→∞f(x) = ∞, becausebothtermstendto ∞.
I limx→−∞
f(x) isindeterminateoftheform −∞ + ∞. It’sthe
sameas limy→+∞
(−y +√y)
limy→+∞
(−y +√y) = lim
y→∞(√y− y) ·
√y + y
√y + y
= limy→∞
y− y2√y + y
= −∞
![Page 99: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/99.jpg)
. . . . . .
Step0: Asymptoticbehavior
f(x) = x +√
|x|I lim
x→∞f(x) = ∞, becausebothtermstendto ∞.
I limx→−∞
f(x) isindeterminateoftheform −∞ + ∞. It’sthe
sameas limy→+∞
(−y +√y)
limy→+∞
(−y +√y) = lim
y→∞(√y− y) ·
√y + y
√y + y
= limy→∞
y− y2√y + y
= −∞
![Page 100: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/100.jpg)
. . . . . .
Step0: Asymptoticbehavior
f(x) = x +√
|x|I lim
x→∞f(x) = ∞, becausebothtermstendto ∞.
I limx→−∞
f(x) isindeterminateoftheform −∞ + ∞. It’sthe
sameas limy→+∞
(−y +√y)
limy→+∞
(−y +√y) = lim
y→∞(√y− y) ·
√y + y
√y + y
= limy→∞
y− y2√y + y
= −∞
![Page 101: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/101.jpg)
. . . . . .
Step1: Thederivative
Remember, f(x) = x +√
|x|.Tofind f′, firstassume x > 0. Then
f′(x) =ddx
(x +
√x)
= 1 +1
2√x
NoticeI f′(x) > 0 when x > 0I lim
x→0+f′(x) = ∞
I limx→∞
f′(x) = 1
![Page 102: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/102.jpg)
. . . . . .
Step1: Thederivative
Remember, f(x) = x +√
|x|.Tofind f′, firstassume x > 0. Then
f′(x) =ddx
(x +
√x)
= 1 +1
2√x
NoticeI f′(x) > 0 when x > 0
I limx→0+
f′(x) = ∞
I limx→∞
f′(x) = 1
![Page 103: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/103.jpg)
. . . . . .
Step1: Thederivative
Remember, f(x) = x +√
|x|.Tofind f′, firstassume x > 0. Then
f′(x) =ddx
(x +
√x)
= 1 +1
2√x
NoticeI f′(x) > 0 when x > 0I lim
x→0+f′(x) = ∞
I limx→∞
f′(x) = 1
![Page 104: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/104.jpg)
. . . . . .
Step1: Thederivative
Remember, f(x) = x +√
|x|.Tofind f′, firstassume x > 0. Then
f′(x) =ddx
(x +
√x)
= 1 +1
2√x
NoticeI f′(x) > 0 when x > 0I lim
x→0+f′(x) = ∞
I limx→∞
f′(x) = 1
![Page 105: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/105.jpg)
. . . . . .
Step1: Thederivative
Remember, f(x) = x +√
|x|.If x isnegative, wehave
f′(x) =ddx
(x +
√−x
)= 1− 1
2√−x
Again, thislooksweirdbecause√−x appearstobeanegative
number. Butsince x < 0, −x > 0.
NoticeI lim
x→0−f′(x) = −∞
I limx→−∞
f′(x) = 1
![Page 106: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/106.jpg)
. . . . . .
Step1: Thederivative
Remember, f(x) = x +√
|x|.If x isnegative, wehave
f′(x) =ddx
(x +
√−x
)= 1− 1
2√−x
Again, thislooksweirdbecause√−x appearstobeanegative
number. Butsince x < 0, −x > 0. NoticeI lim
x→0−f′(x) = −∞
I limx→−∞
f′(x) = 1
![Page 107: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/107.jpg)
. . . . . .
Step1: Thederivative
Remember, f(x) = x +√
|x|.If x isnegative, wehave
f′(x) =ddx
(x +
√−x
)= 1− 1
2√−x
Again, thislooksweirdbecause√−x appearstobeanegative
number. Butsince x < 0, −x > 0. NoticeI lim
x→0−f′(x) = −∞
I limx→−∞
f′(x) = 1
![Page 108: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/108.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞.+ .− .+
.↗ .↘ .↗. max. min
![Page 109: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/109.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.
I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞.+ .− .+
.↗ .↘ .↗. max. min
![Page 110: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/110.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞
.+ .− .+
.↗ .↘ .↗. max. min
![Page 111: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/111.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞.+
.− .+
.↗ .↘ .↗. max. min
![Page 112: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/112.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞.+ .−
.+
.↗ .↘ .↗. max. min
![Page 113: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/113.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞.+ .− .+
.↗ .↘ .↗. max. min
![Page 114: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/114.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞.+ .− .+
.↗
.↘ .↗. max. min
![Page 115: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/115.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞.+ .− .+
.↗ .↘
.↗. max. min
![Page 116: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/116.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞.+ .− .+
.↗ .↘ .↗
. max. min
![Page 117: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/117.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞.+ .− .+
.↗ .↘ .↗. max
.min
![Page 118: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/118.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞.+ .− .+
.↗ .↘ .↗. max. min
![Page 119: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/119.jpg)
. . . . . .
Step1: Monotonicity
I Wherearethecriticalpoints? Weseethat f′(x) = 0 when
1− 12√−x
= 0 =⇒√−x =
12
=⇒ −x =14
=⇒ x = −14
I Weknow f isnotdifferentiableat 0 aswell.I Wecan’tmakeamulti-factorsignchartbecauseoftheabsolutevalue, butwecantestpointsinbetweencriticalpoints.
..f′(x)
.f(x).
.−14
.0 ..0
.∓∞.+ .− .+
.↗ .↘ .↗. max. min
![Page 120: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/120.jpg)
. . . . . .
Step2: ConcavityI If x > 0, then
f′′(x) =ddx
(1 +
12x−1/2
)= −1
4x−3/2
Thisisnegativewhenever x > 0.
I If x < 0, then
f′′(x) =ddx
(1− 1
2(−x)−1/2
)= −1
4(−x)−3/2
whichisalsoalwaysnegativefornegative x.
I Inotherwords, f′′(x) = −14|x|−3/2.
Hereisthesignchart:
..f′′(x)
.f(x)..0
.−∞.−−.⌢
.
..−−.⌢
![Page 121: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/121.jpg)
. . . . . .
Step2: ConcavityI If x > 0, then
f′′(x) =ddx
(1 +
12x−1/2
)= −1
4x−3/2
Thisisnegativewhenever x > 0.I If x < 0, then
f′′(x) =ddx
(1− 1
2(−x)−1/2
)= −1
4(−x)−3/2
whichisalsoalwaysnegativefornegative x.
I Inotherwords, f′′(x) = −14|x|−3/2.
Hereisthesignchart:
..f′′(x)
.f(x)..0
.−∞.−−.⌢
.
..−−.⌢
![Page 122: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/122.jpg)
. . . . . .
Step2: ConcavityI If x > 0, then
f′′(x) =ddx
(1 +
12x−1/2
)= −1
4x−3/2
Thisisnegativewhenever x > 0.I If x < 0, then
f′′(x) =ddx
(1− 1
2(−x)−1/2
)= −1
4(−x)−3/2
whichisalsoalwaysnegativefornegative x.
I Inotherwords, f′′(x) = −14|x|−3/2.
Hereisthesignchart:
..f′′(x)
.f(x)..0
.−∞.−−.⌢
.
..−−.⌢
![Page 123: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/123.jpg)
. . . . . .
Step2: ConcavityI If x > 0, then
f′′(x) =ddx
(1 +
12x−1/2
)= −1
4x−3/2
Thisisnegativewhenever x > 0.I If x < 0, then
f′′(x) =ddx
(1− 1
2(−x)−1/2
)= −1
4(−x)−3/2
whichisalsoalwaysnegativefornegative x.
I Inotherwords, f′′(x) = −14|x|−3/2.
Hereisthesignchart:
..f′′(x)
.f(x)..0
.−∞.−−.⌢
.
..−−.⌢
![Page 124: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/124.jpg)
. . . . . .
Step3: Synthesis
Nowwecanputthesethingstogether.
f(x) = x +√
|x|
..f′(x)
.monotonicity.
.−14
.0 ..0
.∓∞.+1
.↗.+
.↗.−.↘
.+
.↗.+1
.↗.f′′(x)
.concavity..0
.−∞.−−.⌢
.−−.⌢
.−−.⌢
.−∞.⌢
.−∞.⌢.f(x)
.shape.
.−1.0
. zero
..−1
4
.14
. max
..0.0
.min
.−∞ .+∞
." ." . ."
![Page 125: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/125.jpg)
. . . . . .
Step3: Synthesis
Nowwecanputthesethingstogether.
f(x) = x +√
|x|
..f′(x)
.monotonicity.
.−14
.0 ..0
.∓∞.+1
.↗.+
.↗.−.↘
.+
.↗.+1
.↗.f′′(x)
.concavity..0
.−∞.−−.⌢
.−−.⌢
.−−.⌢
.−∞.⌢
.−∞.⌢.f(x)
.shape.
.−1.0
. zero
..−1
4
.14
. max
..0.0
.min
.−∞ .+∞."
." . ."
![Page 126: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/126.jpg)
. . . . . .
Step3: Synthesis
Nowwecanputthesethingstogether.
f(x) = x +√
|x|
..f′(x)
.monotonicity.
.−14
.0 ..0
.∓∞.+1
.↗.+
.↗.−.↘
.+
.↗.+1
.↗.f′′(x)
.concavity..0
.−∞.−−.⌢
.−−.⌢
.−−.⌢
.−∞.⌢
.−∞.⌢.f(x)
.shape.
.−1.0
. zero
..−1
4
.14
. max
..0.0
.min
.−∞ .+∞." ."
. ."
![Page 127: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/127.jpg)
. . . . . .
Step3: Synthesis
Nowwecanputthesethingstogether.
f(x) = x +√
|x|
..f′(x)
.monotonicity.
.−14
.0 ..0
.∓∞.+1
.↗.+
.↗.−.↘
.+
.↗.+1
.↗.f′′(x)
.concavity..0
.−∞.−−.⌢
.−−.⌢
.−−.⌢
.−∞.⌢
.−∞.⌢.f(x)
.shape.
.−1.0
. zero
..−1
4
.14
. max
..0.0
.min
.−∞ .+∞." ." .
."
![Page 128: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/128.jpg)
. . . . . .
Step3: Synthesis
Nowwecanputthesethingstogether.
f(x) = x +√
|x|
..f′(x)
.monotonicity.
.−14
.0 ..0
.∓∞.+1
.↗.+
.↗.−.↘
.+
.↗.+1
.↗.f′′(x)
.concavity..0
.−∞.−−.⌢
.−−.⌢
.−−.⌢
.−∞.⌢
.−∞.⌢.f(x)
.shape.
.−1.0
. zero
..−1
4
.14
. max
..0.0
.min
.−∞ .+∞." ." . ."
![Page 129: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/129.jpg)
. . . . . .
Step3: Synthesis
Nowwecanputthesethingstogether.
f(x) = x +√
|x|
..f′(x)
.monotonicity.
.−14
.0 ..0
.∓∞.+1
.↗.+
.↗.−.↘
.+
.↗.+1
.↗.f′′(x)
.concavity..0
.−∞.−−.⌢
.−−.⌢
.−−.⌢
.−∞.⌢
.−∞.⌢.f(x)
.shape.
.−1.0
. zero
..−1
4
.14
. max
..0.0
.min
.−∞ .+∞." ." . ."
![Page 130: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/130.jpg)
. . . . . .
Graph
f(x) = x +√
|x|
.
.f(x)
.shape.
.−1.0
. zero
.−∞ .+∞..−1
4
.14
. max
.−∞ .+∞..0.0
.min
.−∞ .+∞." ." . ."
.x
.f(x)
..(−1, 0) .
.(−14 ,
14)
..(0, 0)
![Page 131: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/131.jpg)
. . . . . .
Graph
f(x) = x +√
|x|
.
.f(x)
.shape.
.−1.0
. zero
.−∞ .+∞..−1
4
.14
. max
.−∞ .+∞..0.0
.min
.−∞ .+∞." ." . ."
.x
.f(x)
..(−1, 0) .
.(−14 ,
14)
..(0, 0)
![Page 132: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/132.jpg)
. . . . . .
Graph
f(x) = x +√
|x|
.
.f(x)
.shape.
.−1.0
. zero
.−∞ .+∞..−1
4
.14
. max
.−∞ .+∞..0.0
.min
.−∞ .+∞." ." . ."
.x
.f(x)
..(−1, 0) .
.(−14 ,
14)
..(0, 0)
![Page 133: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/133.jpg)
. . . . . .
Graph
f(x) = x +√
|x|
.
.f(x)
.shape.
.−1.0
. zero
.−∞ .+∞..−1
4
.14
. max
.−∞ .+∞..0.0
.min
.−∞ .+∞." ." . ."
.x
.f(x)
..(−1, 0) .
.(−14 ,
14)
..(0, 0)
![Page 134: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/134.jpg)
. . . . . .
Graph
f(x) = x +√
|x|
.
.f(x)
.shape.
.−1.0
. zero
.−∞ .+∞..−1
4
.14
. max
.−∞ .+∞..0.0
.min
.−∞ .+∞." ." . ."
.x
.f(x)
..(−1, 0) .
.(−14 ,
14)
..(0, 0)
![Page 135: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/135.jpg)
. . . . . .
ExampleGraph f(x) = xe−x2
Beforetakingderivatives, wenoticethat f isodd, that f(0) = 0,and lim
x→∞f(x) = 0
![Page 136: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/136.jpg)
. . . . . .
ExampleGraph f(x) = xe−x2
Beforetakingderivatives, wenoticethat f isodd, that f(0) = 0,and lim
x→∞f(x) = 0
![Page 137: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/137.jpg)
. . . . . .
Step1: Monotonicity
If f(x) = xe−x2 , then
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
.−.↘
.+
.↗.−.↘
![Page 138: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/138.jpg)
. . . . . .
Step2: Concavity
If f′(x) = (1− 2x2)e−x2 , weknow
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
.−−.⌢
.++.⌣
.−−.⌢
.++.⌣
![Page 139: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/139.jpg)
. . . . . .
Step3: Synthesis
f(x) = xe−x2
..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
. . . " ." . .
![Page 140: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/140.jpg)
. . . . . .
Step4: Graph
.
.x
.f(x)
.f(x) = xe−x2
.
.(−
√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
. . . " ." . .
![Page 141: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/141.jpg)
. . . . . .
Example
Graph f(x) =1x
+1x2
![Page 142: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/142.jpg)
. . . . . .
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 143: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/143.jpg)
. . . . . .
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 144: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/144.jpg)
. . . . . .
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 145: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/145.jpg)
. . . . . .
Forhorizontalasymptotes, noticethat
limx→∞
x + 1x2
= 0,
so y = 0 isahorizontalasymptoteofthegraph. Thesameistrueat −∞.
![Page 146: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/146.jpg)
. . . . . .
Step1: 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
![Page 147: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/147.jpg)
. . . . . .
Step1: 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
![Page 148: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/148.jpg)
. . . . . .
Step1: 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
![Page 149: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/149.jpg)
. . . . . .
Step1: 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
![Page 150: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/150.jpg)
. . . . . .
Step1: 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
![Page 151: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/151.jpg)
. . . . . .
Step1: 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
![Page 152: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/152.jpg)
. . . . . .
Step1: 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
![Page 153: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/153.jpg)
. . . . . .
Step1: 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
![Page 154: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/154.jpg)
. . . . . .
Step2: 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
![Page 155: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/155.jpg)
. . . . . .
Step2: 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
![Page 156: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/156.jpg)
. . . . . .
Step2: 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
![Page 157: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/157.jpg)
. . . . . .
Step2: 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
![Page 158: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/158.jpg)
. . . . . .
Step2: 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
![Page 159: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/159.jpg)
. . . . . .
Step2: 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
![Page 160: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/160.jpg)
. . . . . .
Step2: 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
![Page 161: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/161.jpg)
. . . . . .
Step2: 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
![Page 162: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/162.jpg)
. . . . . .
Step2: 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
![Page 163: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/163.jpg)
. . . . . .
Step2: 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
![Page 164: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/164.jpg)
. . . . . .
Step3: 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
![Page 165: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/165.jpg)
. . . . . .
Step3: 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
![Page 166: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/166.jpg)
. . . . . .
Step3: 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
![Page 167: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/167.jpg)
. . . . . .
Step3: 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
![Page 168: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/168.jpg)
. . . . . .
Step3: 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
![Page 169: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/169.jpg)
. . . . . .
Step3: 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
![Page 170: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/170.jpg)
. . . . . .
Step3: 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
![Page 171: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/171.jpg)
. . . . . .
Step3: 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
![Page 172: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/172.jpg)
. . . . . .
Step3: 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
![Page 173: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/173.jpg)
. . . . . .
Step3: 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
![Page 174: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/174.jpg)
. . . . . .
Step3: 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
![Page 175: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/175.jpg)
. . . . . .
Step3: 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
![Page 176: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/176.jpg)
. . . . . .
Step4: Graph
. .x
.y
..(−3,−2/9)
..(−2,−1/4)
![Page 177: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/177.jpg)
. . . . . .
ProblemGraph f(x) = cos x− x
. .x
.y
![Page 178: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/178.jpg)
. . . . . .
ProblemGraph f(x) = cos x− x
. .x
.y
![Page 179: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/179.jpg)
. . . . . .
ProblemGraph f(x) = x ln x2
. .x
.y
![Page 180: Lesson 22: Graphing](https://reader034.fdocuments.in/reader034/viewer/2022042816/559379a81a28ab6f2d8b45a6/html5/thumbnails/180.jpg)
. . . . . .
ProblemGraph f(x) = x ln x2
. .x
.y