announcements - mcmaster universityclemene/1ls3/lectures/1ls3_week7.pdf · announcements topics: -...
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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)
€
f
€
f (c) ≥ f (x)
€
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)
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f (c)
€
f
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f (c) ≤ f (x)
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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
€
A,β > 0.
Application
Assignment53,#1(modified):(b)Usethesecondderivativetesttodetermineifthecriticalnumberinpart(a)correspondstoalocalmaximum,localminimum,orneither.
Application
Assignment53,#1(modified):(c)Determinethevaluesofsuchthatfdescribesthegraphgivenbelow.
€
A and β
ExtremeValueTheorem
Ifiscontinuousforall,thentherearepointssuchthatistheglobalminimumandistheglobalmaximumofon
Inwords:Ifafunctioniscontinuousonaclosed,finiteinterval,thenithasaglobalmaximumandaglobalminimumonthatinterval.
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f (x)
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c1, c2 ∈ [a, b]
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f (c1)
€
x ∈ [a, b]
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[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
€
g(x) = x13 (x − 2)2
€
[−1, 1].
L’Hopital’sRule Anotherapplicationofderivativesistohelpevaluatelimitsoftheformwhereeither orIdea:Insteadofcomparingthefunctionsf(x)andg(x),comparetheirderivatives(rates)f’(x)andg’(x).
€
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)
€
limx→a
f (x)g(x)
€
00
€
∞∞ .
€
" 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
€
y'+3x 2y = 6x 2
€
y = 2 + e−x3
€
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:
€
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
€
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.