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