lesson 22: graphing

. . . . . .

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

http://www.imageofsurgery.com/

. . . . . .

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



. . . . . .

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)


. .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)


. .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)


. .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)


. .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)


. .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)


. .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)


. .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)


. .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)


. .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

. . . . . .

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

." . . . "

. . . . . .


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

." . . . "

. . . . . .


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

. . . . . .


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

."

. . . "

. . . . . .


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

." .

. . "

. . . . . .


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

." . .

. "

. . . . . .


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)

. . . . . .

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

. . . . . .

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

. . . . "

. . . . . .


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

.

. . . "

. . . . . .


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

. .

. . "

. . . . . .


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

. . .

. "

. . . . . .


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)

. . . . . .

Outline

TheProcedure



. . . . . .

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: 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

. . . . . .




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

. . . . . .




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

. . . . . .



|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

. . . . . .



|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


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


1− 12√−x

= 0 =⇒√−x =

12

=⇒ −x =14

=⇒ x = −14


..f′(x)

.f(x).

.−14

.0 ..0

.∓∞

.+ .− .+

.↗ .↘ .↗. max. min

. . . . . .

Step1: Monotonicity


1− 12√−x

= 0 =⇒√−x =

12

=⇒ −x =14

=⇒ x = −14


..f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+

.− .+

.↗ .↘ .↗. max. min

. . . . . .

Step1: Monotonicity


1− 12√−x

= 0 =⇒√−x =

12

=⇒ −x =14

=⇒ x = −14


..f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .−

.+

.↗ .↘ .↗. max. min

. . . . . .

Step1: Monotonicity


1− 12√−x

= 0 =⇒√−x =

12

=⇒ −x =14

=⇒ x = −14


..f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .− .+

.↗ .↘ .↗. max. min

. . . . . .

Step1: Monotonicity


1− 12√−x

= 0 =⇒√−x =

12

=⇒ −x =14

=⇒ x = −14


..f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .− .+

.↗

.↘ .↗. max. min

. . . . . .

Step1: Monotonicity


1− 12√−x

= 0 =⇒√−x =

12

=⇒ −x =14

=⇒ x = −14


..f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .− .+

.↗ .↘

.↗. max. min

. . . . . .

Step1: Monotonicity


1− 12√−x

= 0 =⇒√−x =

12

=⇒ −x =14

=⇒ x = −14


..f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .− .+

.↗ .↘ .↗

. max. min

. . . . . .

Step1: Monotonicity


1− 12√−x

= 0 =⇒√−x =

12

=⇒ −x =14

=⇒ x = −14


..f′(x)

.f(x).

.−14

.0 ..0

.∓∞.+ .− .+

.↗ .↘ .↗. max

.min

. . . . . .

Step1: Monotonicity


1− 12√−x

= 0 =⇒√−x =

12

=⇒ −x =14

=⇒ x = −14


..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

.−∞.−−.⌢

.

..−−.⌢

. . . . . .

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


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


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


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


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)

. . . . . .

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

.


limx→0

x + 1x2

= ∞,

so x = 0 isaverticalasymptoteofthegraph.

Wecanmakeasignchartasfollows:

. .x + 1..0.−1

.− .+

.x2..0.0

.+ .+

.f(x)..∞.0

..0.−1

.− .+ .+

. . . . . .

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

f(x) =1x

+1x2

=x + 1x2

.


limx→0

x + 1x2

= ∞,

so x = 0 isaverticalasymptoteofthegraph. Wecanmakeasignchartasfollows:

. .x + 1..0.−1

.− .+

.x2..0.0

.+ .+

.f(x)..∞.0

..0.−1

.− .+ .+

. . . . . .

Forhorizontalasymptotes, noticethat

limx→∞

x + 1x2

= 0,

so y = 0 isahorizontalasymptoteofthegraph. Thesameistrueat −∞.

. . . . . .

Step1: 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


x2− 2

x3= −x + 2

x3.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−

.↘ .↗ .↘.min .VA

. . . . . .

Step1: Monotonicity


x2− 2

x3= −x + 2

x3.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘

.↗ .↘.min .VA

. . . . . .

Step1: Monotonicity


x2− 2

x3= −x + 2

x3.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗

.↘.min .VA

. . . . . .

Step1: Monotonicity


x2− 2

x3= −x + 2

x3.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min .VA

. . . . . .

Step1: Monotonicity


x2− 2

x3= −x + 2

x3.


. .−(x + 2)..0.−2

.+ .−

.x3..0.0

.− .+

.f′(x)

.f(x)..∞.0

..0.−2

.− .+ .−.↘ .↗ .↘

.min

.VA

. . . . . .

Step1: Monotonicity


x2− 2

x3= −x + 2

x3.


. .−(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.


. .(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.


. .(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.


. .(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.


. .(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.


. .(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.


. .(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.


. .(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.


. .(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′


..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′


..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′


..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′


..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′


..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′


..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′


..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′


..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′


..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′


..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′


..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) = x ln x2

. .x

.y

lesson 22: graphing

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