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Thederiva*ve
Analy*c,geometric,computa*onal
UBCMath102
Deriva*ve:Analy*cUsethedefini*onofthederiva*vetocomputethederiva*veofthefunc*ony=x3
Donefast?Trytoextendyourmethodtoy=xnWherenisanyposi*veinteger.Sols:seelastslides
MTtest-type
ques*on
Moreadvanced
MathLearningCenter
• TheMLCisaspaceforundergraduatestudentstostudymathtogether,withfriendlysupportfromtutors(gradstudentsinmath).
• LocatedatLSK310andLSK302• Hours:open5daysaweek.• Note:MLCisnotaplacetocheckHWanswersorreceivesolu*ons.MLCaimistoaidstudentsinbecomingbeVerlearnersandtodevelopcri*calthinkingskills.
• hVp://www.math.ubc.ca/~MLC/.
Diagnos*ctestresults
FinalGradeinM102≈DTscore+10%IfyourDT<40%àhighriskoffailingPossibleac*on:takeM180insteadofM102 OR :preparetoworkextra-hard
Helpwithhomework
UBCMath102
Assignment2:Problem16
• Evaluatethelimit,ifitexists.Ifalimitdoesnotexist,enterDNE.
• Hint:considerfactoringtopandboVom
UBCMath102
Assignment2:Problem18
• Evaluatethelimitby“guessing”
• Wehavenotlearnedenoughaboutthisfunc*ontoactuallycomputethelimit,butwecanuseavalueclosetox=0toguessthelimit
UBCMath102
“Guessing”
To“guess”avalueforthislimit:IwouldinputwhichofthefollowingintoWebWork?(A) (B) (C) (D) (E)Noneoftheabove
UBCMath102
Thederiva*ve
Geometric
UBCMath102
Tangentline
• Itisthelineweseewhenwezoomintothegraphofafunc*onatsomepoint
UBCMath102
o
Simplecases:
Whatisthederiva*veofthesefunc*ons?y=Cy=ax
Simplecases:
Thederiva*veisy=C y’=0y=ax y’=a
Rememberingslopes
Posi*ve zero nega*ve“infinite”(undefined)
Zoomingintothegraphof
• Atthepointx=1.5
UBCMath102Seecoursenotes
Zoomingintothegraphof
Atx=0
UBCMath102Seecoursenotes
Afunc*onwithacusp
Atthecusp,thederiva*vedoesnotexist.
UBCMath102Seecoursenotes
Whendoesthederiva*veexist?
Ifafunc*onisdiscon*nuousatx=athenitsderiva*veisnotdefinedatthatpoint.Ifafunc*onhasacuspatx=a,thenitsderiva*vedoesnotexist(isnotdefined)atthatpoint.Ifafunc*on“blowsup”(goestoinky)atx=a,thenitsderiva*vedoesnotexistatthatpoint.
1.Whatisthisfunc*on?
• A|x+1|• B|x-1|• C-|x+1|• D-|x-1|• E-(x-1)
UBCMath102
2.Thisfunc*onis
• Adiscon*nuous• Bnotdifferen*ableanywhere• Chasnolimitasx->0• Ddifferen*ableexceptatx=1• EIamconfused
UBCMath102
Sketchthederiva*veofthisfunc*on
UBCMath102
3.Thederiva*veofthisfunc*onlookslike:
• A
• B
UBCMath102
Dailytemprature(ForecastforWednSept20)Sketchtherateofchangeoftemperature.
Solu*oninclass
• Thelocalmaxandminareplaceswherethederiva*veiszero
Solarpower
Sustainabilityandcleanenergy
Rookopsolarcollectors
Solarpower(KW)
Whathappenedhere?
Notethedate
•
Sketchtherateofchangeofthisfunc*on
•
Solu*on(improvedfromtheoneinclass)
•
Predator-preypopula*oncycles
Wehaveagreatrecordofcyclesofpredatorsandprey(LynxandHare)inCanadianarc*cwhy??
Overalltrend
• PredatorandPrey(inthousands)
Justtherprey
SketchtherateofchangedP/dxofthepreypopula*on
Justtherprey
SketchtherateofchangedP/dxofthepreypopula*on
Justtherprey
SketchtherateofchangedP/dxofthepreypopula*on
Tangentlines
Nowweconsidertheslopesofthetangentlines
Tangentlines
Putthemallalongasingleline
Tangentlines
Figureoutwhereslopeiszero,posi*veornega*ve
Tangentlines
Figureoutwhereslopeiszero,posi*veornega*ve-0+0-0+0-0+
Tangentlines
Nowsketchtheslopesonanewgraph,carefultoaligntheplaceswhereitiszero-0+0-0+0-0+
Tangentlines
Addtherestofthesketch-0+0-0+0-0+
Tangentlines
Nowweconsidertheslopesofthetangentlines-0+0-0+0-0+
Rateofchangeotthepreypopula*on
Wecanusedesmostographthedervia*ve
Deriva*veondesmos
• Preypopula*on
• Rateofchangeofpreypopula*on
hVps://www.desmos.com/calculator/uenjl3dwk3
Desmos:tangentline• hVps://www.desmos.com/calculator/fs3lx8gptb
• Experimentwithslider• Experimentwithzoom!
UBCMath102
Answers
1.D2.D3.A
Solu*ons
Prac*cecalcula*onwedidinclass
UBCMath102
Challenge:
Deriva*veofy=xnfromthedefini*onofthederiva*ve:Bythebinomialtheorem,c1=n,sof’(x)=nxn-1
SeeAppendixE.1
Problemstotestyourskill
Aselec*onofproblemsfromtestsandexamstohelpyoustudyconceptsinthislecture
Relatedtestproblem
• Showninthegraphisthevelocityofapar*cle.Usethistosketchtheaccelera*onofthepar*cle.(Hint:theaccelera*onisthederiva*veofthevelocity)
UBCMath102
SeeP67prob2.20 Exam-
typeques*on
Graphthederiva*veoff(x)
SeeP85prob3.7a,b
Giventhederiva*vef’(x),graphtheoriginalfunc*onf(x)
SeeP85prob3.7c,d
Relatedtestproblem(MT1,2008)Microtubules(MT)arebiologicalpolymersimportantincellstructure,celldivision,andtransportofmaterialinsidecells.Thelengthofmicrotubules(“MTlength”)growsandshrinksasshowninthefollowingfigurefromJanuleviciusetal(2006)BiophysJ90:788-798.UsethisfiguretodrawasketchofMTgrowthrate(i.e.rateofchangeofmicrotubulelengthperunit*me)overthesame*meinterval.
Whichhasagreatermagnitude:therateofshrinkage(perunit*me)ortherateofgrowth(perunit*me)?
Exam-type
ques*on
Solu*onstopreviousproblemsFirsttryouttheproblemsattheendofthelastlectureslides.Onlythenshouldyou“peek”attheanswers.
Secantand
tangentlines
Asecantlineis
(A) Alinewhoseslopeisinstantaneousvelocity(B) Alineconnec*ngtwopointsonagraph(C) Thesameasanaveragevelocity(D) Thesameallalongthecurve(E) Notsure.
UBCMath102
Whichofthebelowisanaveragevelocity?
(A) Avveloc=(v1+v2+v3+v4)/4(B) Avveloc=(y1+y2+y3+y4)/4(C) Avveloc=(y1+y2+y3+y4)/(t1+t2+t3+t4)(D) Avveloc=(y1+y2)/(t1+t2)(E) Avveloc=(y2-y1)/(t2–t1)
UBCMath102