. . . . . .

Section4.4CurveSketchingI

V63.0121, CalculusI

March30, 2009

Announcements

I Quiz4thisweek(Sections2.5–3.5)I Officehoursthisweek: M 1–2, T 1–2, W 2–3, R 9–10

..Imagecredit: FastEddie42

. . . . . .

OfficeHoursandotherhelpInadditiontorecitation

Day Time Who/What WhereinWWHM 1:00–2:00 LeingangOH 718/618

3:30–4:30 KatarinaOH 7075:00–7:00 CurtoPS 517

T 1:00–2:00 LeingangOH 718/6184:00–5:50 CurtoPS 317

W 1:00–2:00 KatarinaOH 7072:00–3:00 LeingangOH 718/618

R 9:00–10:00am LeingangOH 718/6185:00–7:00pm MariaOH 807

F 2:00–4:00 CurtoOH 1310

. . . . . .

CIMS/NYU professorwinsAbelPrize

I MikhailGromov, born1943inRussia

I contributionstogeometryandtopology

I discoveredthepseudoholomorphiccurve

I AbelPrizeisthehighestinmathematics

. . . . . .

OntheproblemsassignedfromSection2.8Announcementsweremadeinclassbutnotonline

I ResubmityourProblemSet6onWednesday, April1withProblemSet 8

I WewillpicktwoadditionalproblemstogradefromProblemSet 8

I IfthescoresonthemakeupproblemsfromPS 8exceedthescoresoftheSection2.8problemsfromPS 6, themakeupscoreswillbesubstituted.

I Thisalsotakescareoftheproblematicproblem2.8.28.

Thisofferisonlygoodthisweek.

. . . . . .

Outline

TheProcedure

TheexamplesA cubicfunctionA quarticfunction

. . . . . .

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.Itworksthesameasthelasttheorem. Picktwopoints x and y in(a,b) with x < y. Wemustshow f(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

TheexamplesA cubicfunctionA quarticfunction

. . . . . .

Graphingacubic

ExampleGraph f(x) = 2x3 − 3x2 − 12x.

First, let’sfindthezeros. Wecanatleastfactoroutonepowerofx:

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.

First, let’sfindthezeros. Wecanatleastfactoroutonepowerofx:

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.

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

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

. . . . . .

Concavity

f′′(x) = 12x− 6 = 6(2x− 1)

Anothersignchart: .

.f′′(x)

.f(x).

.1/2

.−− .++.⌢ .⌣

.IP

. . . . . .

Concavity

f′′(x) = 12x− 6 = 6(2x− 1)

Anothersignchart: .

.f′′(x)

.f(x).

.1/2

.−− .++.⌢ .⌣

.IP

. . . . . .

Concavity

f′′(x) = 12x− 6 = 6(2x− 1)

Anothersignchart: .

.f′′(x)

.f(x).

.1/2

.−−

.++.⌢ .⌣

.IP

. . . . . .

Concavity

f′′(x) = 12x− 6 = 6(2x− 1)

Anothersignchart: .

.f′′(x)

.f(x).

.1/2

.−− .++

.⌢ .⌣

.IP

. . . . . .

Concavity

f′′(x) = 12x− 6 = 6(2x− 1)

Anothersignchart: .

.f′′(x)

.f(x).

.1/2

.−− .++.⌢

.⌣

.IP

. . . . . .

Concavity

f′′(x) = 12x− 6 = 6(2x− 1)

Anothersignchart: .

.f′′(x)

.f(x).

.1/2

.−− .++.⌢ .⌣

.IP

. . . . . .

Concavity

f′′(x) = 12x− 6 = 6(2x− 1)

Anothersignchart: .

.f′′(x)

.f(x).

.1/2

.−− .++.⌢ .⌣

.IP

. . . . . .

Onesigncharttorulethemall

.

.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

." . . . "

. . . . . .

Onesigncharttorulethemall

..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

." . . . "

. . . . . .

Onesigncharttorulethemall

..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

." . . . "

. . . . . .

Onesigncharttorulethemall

..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

." . . . "

. . . . . .

Onesigncharttorulethemall

..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

."

. . . "

. . . . . .

Onesigncharttorulethemall

..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

." .

. . "

. . . . . .

Onesigncharttorulethemall

..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

." . .

. "

. . . . . .

Onesigncharttorulethemall

..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

." . . . "

. . . . . .

Graph

. .x

.f(x)

..(3−

√105

4 , 0) .

.(−1, 7)

..(0,0)

..(1/2,−61/2)

..(2,−20)

.

.(3+

√105

4 , 0)

. . . . . .

Graph

. .x

.f(x)

..(3−

√105

4 , 0) .

.(−1, 7)

..(0,0)

..(1/2,−61/2)

..(2,−20)

.

.(3+

√105

4 , 0)

. . . . . .

Graph

. .x

.f(x)

..(3−

√105

4 , 0) .

.(−1, 7)

..(0,0)

..(1/2,−61/2)

..(2,−20)

.

.(3+

√105

4 , 0)

. . . . . .

Graph

. .x

.f(x)

..(3−

√105

4 , 0) .

.(−1, 7)

..(0,0)

..(1/2,−61/2)

..(2,−20)

.

.(3+

√105

4 , 0)

. . . . . .

Graph

. .x

.f(x)

..(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

Weknow f(0) = 10 and limx→±∞

f(x) = +∞. Nottoomanyother

pointsonthegraphareevident.

. . . . . .

Graphingaquartic

ExampleGraph f(x) = x4 − 4x3 + 10

Weknow f(0) = 10 and limx→±∞

f(x) = +∞. Nottoomanyother

pointsonthegraphareevident.

. . . . . .

Monotonicity

f′(x) = 4x3 − 12x2 = 4x2(x− 3)

Wemakeitssignchart.

. .4x2..0.0 .+ .+.+

.(x− 3)..3.0.− .+.−

.f′(x)

.f(x)..3.0

.min

..0.0.−

.↘.−.↘

.+

.↗

. . . . . .

Concavity

f′′(x) = 12x2 − 24x = 12x(x− 2)

Hereisitssignchart:

. .12x..0.0.− .+ .+

.x− 2..2.0.− .− .+

.f′′(x)

.f(x)..0.0

.IP

..2.0

.IP

.++.⌣

.−−.⌢

.++.⌣

. . . . . .

GrandUnifiedSignChart

.

.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

. . . . "

. . . . . .

GrandUnifiedSignChart

.

.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

.

. . . "

. . . . . .

GrandUnifiedSignChart

.

.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

. .

. . "

. . . . . .

GrandUnifiedSignChart

.

.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

. . .

. "

. . . . . .

GrandUnifiedSignChart

.

.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

. . . . "

. . . . . .

Graph

. .x

.y

..(0, 10)

..(2,−6) .

.(3,−17)