. . . . . .
Section4.4CurveSketching
V63.0121.034, CalculusI
November16, 2009
Announcements
I Wednesday, November25isaregularclassdayI nextandlastquizwillbetheweekafterThanksgivingI FinalExam: Friday, December18, 2:00–3:50pm
. . . . . .
Outline
TheProcedure
SimpleexamplesA cubicfunctionA quarticfunction
MoreExamplesPointsofnondifferentiabilityHorizontalasymptotesVerticalasymptotesTrigonometricandpolynomialtogetherLogarithmic
. . . . . .
Objective
Givenafunction, graphitcompletely, indicating
I zeroesI asymptotesifapplicableI criticalpointsI local/globalmax/minI inflectionpoints
.
.Imagecredit: ImageOfSurgery
. . . . . .
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.
. . . . . .
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)
. . . . . .
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
TheProcedure
SimpleexamplesA cubicfunctionA quarticfunction
MoreExamplesPointsofnondifferentiabilityHorizontalasymptotesVerticalasymptotesTrigonometricandpolynomialtogetherLogarithmic
. . . . . .
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.
. . . . . .
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.
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++.⌢ .⌣
.IP
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++.⌢ .⌣
.IP
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−−
.++.⌢ .⌣
.IP
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++
.⌢ .⌣
.IP
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++.⌢
.⌣
.IP
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++.⌢ .⌣
.IP
. . . . . .
Step2: Concavity
f′′(x) = 12x− 6 = 6(2x− 1)
Anothersignchart: .
.f′′(x)
.f(x).
.1/2
.−− .++.⌢ .⌣
.IP
. . . . . .
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
." . . . "
. . . . . .
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
." . . . "
. . . . . .
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
." . . . "
. . . . . .
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
." . . . "
. . . . . .
Combinationsofmonotonicityandconcavity
.
.I.II
.III .IV
.decreasing,concavedown
.increasing,concavedown
.decreasing,concave up
.increasing,concave up
. . . . . .
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
."
. . . "
. . . . . .
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
." .
. . "
. . . . . .
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
." . .
. "
. . . . . .
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
." . . . "
. . . . . .
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)
. . . . . .
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)
. . . . . .
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)
. . . . . .
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)
. . . . . .
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)
. . . . . .
Graphingaquartic
ExampleGraph f(x) = x4 − 4x3 + 10
(Step0)Weknow f(0) = 10 and limx→±∞
f(x) = +∞. Nottoomany
otherpointsonthegraphareevident.
. . . . . .
Graphingaquartic
ExampleGraph f(x) = x4 − 4x3 + 10
(Step0)Weknow f(0) = 10 and limx→±∞
f(x) = +∞. Nottoomany
otherpointsonthegraphareevident.
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . "
. . . . . .
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
.
. . . "
. . . . . .
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
. .
. . "
. . . . . .
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
. . .
. "
. . . . . .
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
. . . . "
. . . . . .
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)
. . . . . .
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)
. . . . . .
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)
. . . . . .
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)
. . . . . .
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)
. . . . . .
Outline
TheProcedure
SimpleexamplesA cubicfunctionA quarticfunction
MoreExamplesPointsofnondifferentiabilityHorizontalasymptotesVerticalasymptotesTrigonometricandpolynomialtogetherLogarithmic
. . . . . .
ExampleGraph f(x) = x +
√|x|
Thisfunctionlooksstrangebecauseoftheabsolutevalue. Butwheneverwebecomenervous, wecanjusttakecases.
. . . . . .
ExampleGraph f(x) = x +
√|x|
Thisfunctionlooksstrangebecauseoftheabsolutevalue. Butwheneverwebecomenervous, wecanjusttakecases.
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
= −∞
. . . . . .
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
= −∞
. . . . . .
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
= −∞
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
.−∞.−−.⌢
.
..−−.⌢
. . . . . .
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
.−∞.−−.⌢
.
..−−.⌢
. . . . . .
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
.−∞.−−.⌢
.
..−−.⌢
. . . . . .
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
.−∞.−−.⌢
.
..−−.⌢
. . . . . .
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
.−∞ .+∞
." ." . ."
. . . . . .
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
.−∞ .+∞."
." . ."
. . . . . .
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
.−∞ .+∞." ."
. ."
. . . . . .
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
.−∞ .+∞." ." .
."
. . . . . .
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
.−∞ .+∞." ." . ."
. . . . . .
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
.−∞ .+∞." ." . ."
. . . . . .
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)
. . . . . .
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)
. . . . . .
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)
. . . . . .
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)
. . . . . .
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)
. . . . . .
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
. . . . . .
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
.−.↘
.+
.↗.−.↘
. . . . . .
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
.−−.⌢
.++.⌣
.−−.⌢
.++.⌣
. . . . . .
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
. . . " ." . .
. . . . . .
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
. . . " ." . .
. . . . . .
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 −∞.
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
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
. . . . . .
Step4: Graph
. .x
.y
..(−3,−2/9)
..(−2,−1/4)
. . . . . .
ProblemGraph f(x) = cos x− x
. .x
.y
. . . . . .
ProblemGraph f(x) = cos x− x
. .x
.y
. . . . . .
ProblemGraph f(x) = x ln x2
. .x
.y
. . . . . .
ProblemGraph f(x) = x ln x2
. .x
.y
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