announcements - mcmaster universityclemene/1ls3/lectures/1ls3_week7.pdf · announcements topics: -...

AnnouncementsTopics:

-  sections6.1(extremevalues),6.4(L’Hopital’sRule),7.1(differentialequations)

*Readthesesectionsandstudysolvedexamplesinyourtextbook!

Homework:

-  reviewlecturenotesthoroughly-  workonpracticeproblemsfromthetextbookandassignmentsfromthecoursepackasassignedonthecoursewebpage(underthe“SCHEDULE+HOMEWORK”link)

MaximumandMinimumValues

isaglobal(absolute)maximumofifforallinthedomainofisalocal(relative)maximumofifforallinsomeintervalaround

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f (c) ≥ f (x)

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f (c)

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f

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f (c) ≥ f (x)

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x

€

c.

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f (c)

€

f

€

x

€

f .

MaximumandMinimumValues

isaglobal(absolute)minimumofifforallinthedomainofisalocal(relative)minimumofifforallinsomeintervalaround

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f (c) ≤ f (x)

€

f (c)

€

f

€

f (c) ≤ f (x)

€

x

€

c.

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f (c)

€

f

€

x

€

f .

Extrema

Identifythelabeledpointsaslocalmaxima/minima,globalmaxima/minima,ornoneofthese.

F

X

Y

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ExtremeValues

Notice:Extremevaluesoccurateitheracriticalnumberofforatanendpointofthedomain.

(However,notallcriticalnumbersandendpointscorrespondtoanextremevalue.)

Alsonote:Bydefinition,relativeextremevaluesdonotoccuratendpoints.

FindingLocalMaximaandMinima(FirstDerivativeTest)

Assumethatfiscontinuousatc,wherecisacriticalnumberoff.Iff’changesfrom+to-atx=c,thenfchangesfromincreasingtodecreasingatx=candf(c)isalocalmaximumvalue.Iff’changesfrom-to+atx=c,thenfchangesfromdecreasingtoincreasingatx=candf(c)isalocalminimumvalue.Iff’doesnotchangesignatx=c,thenfdoesn’thaveanextremevalueatx=c.

FindingLocalMaximaandMinima(FirstDerivativeTest)

Example:Findthelocalextremaof.

f (x) = ln xx

FindingLocalMaximaandMinima(SecondDerivativeTest)

Assumethatf’’iscontinuousnearcandf’(c)=0.Iff’’(c)>0thenthegraphoffisconcaveupatx=candf(c)

isalocalminimumvalue.Iff’’(c)<0thenthegraphoffisconcavedownatx=cand

f(c)isalocalmaximumvalue.Iff’’(c)=0orf”(c)D.N.E.thenthesecondderivativetest

doesn’tapplyandyouhavetousetheothermethod.

Application

Assignment53,#1(modified):Considerthefunction,where(a)Findthecriticalnumberoff.

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f (t) = Ate−βt

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A,β > 0.

Application

Assignment53,#1(modified):(b)Usethesecondderivativetesttodetermineifthecriticalnumberinpart(a)correspondstoalocalmaximum,localminimum,orneither.

Application

Assignment53,#1(modified):(c)Determinethevaluesofsuchthatfdescribesthegraphgivenbelow.

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A and β

ExtremeValueTheorem

Ifiscontinuousforall,thentherearepointssuchthatistheglobalminimumandistheglobalmaximumofon

Inwords:Ifafunctioniscontinuousonaclosed,finiteinterval,thenithasaglobalmaximumandaglobalminimumonthatinterval.

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f (x)

€

c1, c2 ∈ [a, b]

€

f (c1)

€

x ∈ [a, b]

€

[a, b].

€

f (c2)

€

f (x)

FindingAbsoluteExtremeValuesonaClosedInterval[a,b]

1.Findallcriticalnumbersintheinterval.2.Makeatableofvalues.

Thelargestvalueoff(x)istheabsolutemaximumandthesmallestvalueistheabsoluteminimum.

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A

B

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FindingAbsoluteExtremeValuesonaClosedInterval[a,b]

Example:Findtheabsoluteextremaofon

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g(x) = x13 (x − 2)2

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[−1, 1].

L’Hopital’sRule Anotherapplicationofderivativesistohelpevaluatelimitsoftheformwhereeither orIdea:Insteadofcomparingthefunctionsf(x)andg(x),comparetheirderivatives(rates)f’(x)andg’(x).

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limx→a

f (x)g(x)

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limx→a

f (x) = 0 and limx→a

g(x) = 0

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limx→a

f (x) = ±∞ and limx→a

g(x) = ±∞ .

L’Hopital’sRule

SupposethatfandgaredifferentiablefunctionssuchthatisanindeterminateformoftypeorIfneara(couldbe0ata)then

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limx→a

f (x)g(x)

= limx→a

# f (x)# g (x)

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limx→a

f (x)g(x)

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00

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∞∞ .

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" g (x) ≠ 0

L’Hopital’sRule

EvaluatethefollowinglimitsusingL’Hopital’sRule,ifitapplies.(a) (b)(c)

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limx→0

sin xx

limx→0

tan x − xx3€

limx→∞

ln xx3

L’Hopital’sRule

EvaluatethefollowinglimitsusingL’Hopital’sRule,ifitapplies.(a) (b) €

limx→∞

x1x

€

limx→∞

x 2e−3x

DifferentialEquations

Adifferentialequationisanequationthatinvolvesanunknownfunctionandoneormoreofitsderivatives.

Examples:

ʹy = 2+ y ʹy = x + yʹy = x2 + ex

DifferentialEquations

Asolutionofadifferentialequationisafunctionthat,alongwithitsderivatives,satisfiestheDE.

Example:Showthatisasolutionofthedifferentialequationandinitialcondition

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y'+3x 2y = 6x 2

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y = 2 + e−x3

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y(0) = 3.

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Pure-TimeDEs

Apure-timedifferentialequationisobtainedbymeasuringtherateofchangeoftheunknownquantityandexpressedasafunctionoftime.

Examples:Notethattheformulafortherateofchangedependspurelyonthetimet.

dsdt= t2 −3t +5 f '(x) = arctan x

Pure-TimeDEsExample:VolumeofaCellSupposeweobservethatofwaterentersacelleachsecond.DifferentialEquation:GeneralSolution:iscalledthe‘statevariable’

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2.0µm3

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V

Pure-TimeDEs

Example:VolumeofaCellSupposewearetoldthattheinitialvolumeofthecellisParticularSolution:

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150µm3.

AutonomousDEsAnautonomousdifferentialequationisderivedfromaruledescribinghowaquantitychangesandisexpressedasafunctionoftheunknownquantity.

Example:Supposethatthegrowthrateofapopulation,P,isproportionaltoitssize.

dPdt

= k ⋅P(t) or simply P ' = k ⋅P

AutonomousDEs

Example:PopulationModelIthasbeenobservedthattherelativerateofchangeofapopulationofwildfoxesinanecosystemis0.75babyfoxesperfoxpermonth.Initially,thepopulationis74thousand.DifferentialEquation:InitialCondition:ParticularSolution:

SolutionsforGeneralDEs

Ø  AlgebraicSolutionsØ  anexplicitformulaoralgorithmforthesolution(often,impossibletofind)

Ø  GeometricSolutionsØ  asketchofthesolutionobtainedfromanalyzingtheDE

Ø  NumericSolutionsØ  anapproximationofthesolutionusingtechnologyandandsomeestimationmethod,suchasEuler’smethod

GraphicalSolutionsofPure-TimeDEs

Example:Sketchthegraphofthesolutiontogiventheinitialcondition

s(1) =1.s '(t) = ln t

Euler’sMethod

Whatinformationdoesaninitialvalueproblemtellusaboutthesolution?

Example:DE:IC:

€

dydx

= x + y

€

y(0) =1

slopeofthesolutioncurvey(x)

anexactvalueofthesolution

Euler’sMethod

Euler’sIdea:First,usingtheinitialconditionasabasepoint,approximatethesolutioncurvey(x)byitstangentline.

FirstEulerapproximation

Euler’sMethod

Next,travelashortdistancealongthisline,determinetheslopeatthenewlocation(usingtheDE),andthenproceedinthat‘corrected’direction.

Euler’sapproximationwithstepsize

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Δx = 0.5

Euler’sMethod

Repeat,correctingyourdirectionmidcourseusingtheDEatregularintervalstoobtainanapproximatesolutionoftheIVP.Byincreasingthenumberofmidcoursecorrections,wecanimproveourestimationofthesolution.

Eulerapproximationwithstepsize

€

Δx = 0.25

Euler’sMethod

Summary:AnapproximatesolutiontotheIVPisgeneratedbychoosingastepsizeandcomputingvaluesaccordingtothealgorithm

€

tn+1 = tn + Δtyn+1 = yn +G(tn ,yn )Δt

€

dydt

=G(t,y), y(t0) = y0

€

Δt

Euler’sMethodAlgorithm:AlgorithmInWords:nexttime=currenttime+stepsizenextapproximation=currentapproximation+rateofchangeatcurrentvaluesxstepsize

€

tn+1 = tn + Δtyn+1 = yn +G(tn ,yn )Δt

Example

ConsidertheIVPApproximatethevalueofthesolutionatx=1byapplyingEuler’smethodandusingastepsizeof0.25.

y ' = x + y y(0) =1

Example

Calculations:

tn yn

x0=0 y0=1

TableofApproximateValuesfortheSolutiony(x)oftheIVP

QualitativeAnalysisofaDE

Example:ApopulationofcaribouismodeledbyInwhichofthefollowingsituationswillthepopulationincreaseintheimmediatefuture?(I)P(0)=100 (II)P(0)=200 (III)P(0)=3000

dPdt

= 2P(t) 1− P(t)2500

⎛

⎝⎜

⎞

⎠⎟ 1−

120P(t)

⎛

⎝⎜

⎞

⎠⎟, P(t) > 0.

AnnouncementsTopics:

-  sections7.1(differentialequations),7.2(antiderivatives),and7.3(thedefiniteintegral+area)

*Readthesesectionsandstudysolvedexamplesinyourtextbook!

Homework:-  reviewlecturenotesthoroughly-  workonpracticeproblemsfromthetextbookandassignmentsfromthecoursepackasassignedonthecoursewebpage(underthe“SCHEDULE+HOMEWORK”link)

Visitblood.catobookanappointment

DONORSNEEDED

TuesdayOctober30TuesdayNovember13ThursdayNovember15

@CIBCHall(StudentCenter3rdfloor)10am-4pm

SolutionsforGeneralDEs

Ø  AlgebraicSolutionsØ  anexplicitformulaoralgorithmforthesolution(often,impossibletofind)

Ø  GeometricSolutionsØ  asketchofthesolutionobtainedfromanalyzingtheDE

Ø  NumericSolutionsØ  anapproximationofthesolutionusingtechnologyandandsomeestimationmethod,suchasEuler’smethod

GraphicalSolutionsofPure-TimeDEs

Example:Sketchthegraphofthesolutiontogiventheinitialcondition

s(1) =1.s '(t) = ln t

Euler’sMethod

Whatinformationdoesaninitialvalueproblemtellusaboutthesolution?

Example:DE:IC:

€

dydx

= x + y

€

y(0) =1

slopeofthesolutioncurvey(x)

anexactvalueofthesolution

Euler’sMethod

Euler’sIdea:First,usingtheinitialconditionasabasepoint,approximatethesolutioncurvey(x)byitstangentline.

FirstEulerapproximation

Euler’sMethod

Next,travelashortdistancealongthisline,determinetheslopeatthenewlocation(usingtheDE),andthenproceedinthat‘corrected’direction.

Euler’sapproximationwithstepsize

€

Δx = 0.5

Euler’sMethod

Repeat,correctingyourdirectionmidcourseusingtheDEatregularintervalstoobtainanapproximatesolutionoftheIVP.Byincreasingthenumberofmidcoursecorrections,wecanimproveourestimationofthesolution.

Eulerapproximationwithstepsize

€

Δx = 0.25

Euler’sMethod

Summary:AnapproximatesolutiontotheIVPisgeneratedbychoosingastepsizeandcomputingvaluesaccordingtothealgorithm

€

tn+1 = tn + Δtyn+1 = yn +G(tn ,yn )Δt

€

dydt

=G(t,y), y(t0) = y0

€

Δt

Euler’sMethodAlgorithm:AlgorithmInWords:nexttime=currenttime+stepsizenextapproximation=currentapproximation+rateofchangeatcurrentvaluesxstepsize

€

tn+1 = tn + Δtyn+1 = yn +G(tn ,yn )Δt

Example

ConsidertheIVPApproximatethevalueofthesolutionatx=1byapplyingEuler’smethodandusingastepsizeof0.25.

y ' = x + y y(0) =1

Example

Calculations:

tn yn

x0=0 y0=1

TableofApproximateValuesfortheSolutiony(x)oftheIVP

QualitativeAnalysisofaDE

Example:ApopulationofcaribouismodeledbyInwhichofthefollowingsituationswillthepopulationincreaseintheimmediatefuture?(I)P(0)=100 (II)P(0)=200 (III)P(0)=3000

dPdt

= 2P(t) 1− P(t)2500

⎛

⎝⎜

⎞

⎠⎟ 1−

120P(t)

⎛

⎝⎜

⎞

⎠⎟, P(t) > 0.