lesson 21: curve sketching (section 10 version)
DESCRIPTION
The increasing/decreasing test and test for concavity allow us to spread out all the information we need about a function to reconstruct its graph.TRANSCRIPT
. . . . . .
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)