TangentLineequa,on

Findtheequa,onofthetangentlinetothegraphofthefunc,ony=5x2atthepointx=1.Wheredoesthetangentlineintersectthexaxis?Finishedfast?Findtheequa,onofthetangentlinetothegraphofthefunc,ony=f(x)atthepointx=x0.Wheredoethetangentlineintersectthexaxis?

SimpleExam-type

ques,on

Officehoursthisweek

•  Tu:3:30-5PM

•  Iamatan“OpenBook”seminaronFridaymorning,sowhichreplacementofficehoursarebestforyou?–  (A)Wed10-11:30–  (B)Wedn3:30-5pm–  (C)Thurs10-11:30

Midtermsignup

•  Signupforoneofthethreeop,ons.•  Checkbackforyourroomassignment–maynotbesameasyourfriend’s

•  Declareanyconflicts(lecture,anothermidterm,etc)

“StartOffStrong”

•  Spac.science.ubc.ca Sept26-27,5-7pm•  Learn,psfromprofsabouthowtostudyformidtermexams

Helpwithhomework(OSH2)

d1(x)thatgivesthedistancefromapointonthex-axis(x,0)tothepoint(2,3).Usedistanceformula

OSH2cont’d

Whatisthereare2pointsandyouwanttheshortestdistance?

OSH2cont’d

Whatisthereare2pointsandyouwanttheshortestdistance?

HelpwithHW

In1938A.V.Hillproposedamathmodelfortherateofshorteningofamuscle,v,(cm/sec)whenitisworkingagainstaloadp(gms).Hissocalledforce-velocitycurveisgivenbytherela,onship

a,bposi,veconstants.HINT:Rewriteintheformv=f(p).Note:v>0meansmusclecontracts.v<0meansmuscleexpands

Toworkon

•  Prac,cetheProductandQuo,enRules•  Prac,cesketchingderiva,vesandan,deriva,ves

•  StartstudyingfortheMidTerm•  Note:onlinecopyofbookjustuploaded:nowhasanswerstomostoftheproblemsinthebook.

Tangentlines

Andlinearapproxima,ons

UBCMath102

GenericTangentLineequa,on

Findtheequa,onofthetangentlinetothegraphofthefunc,ony=f(x)atthepointx=x0.

UBCMath102

Solu,on(inclass):

Tangentlinescanbeusedto(locally)approximateafunc,on

UBCMath102

• 

•  Func,onà•  Nearx=0TangentLineisclosetothefunc,on.

•  TL(lowerthanf(x)nearx=0)“underes,mates”thefunc,on”atthatpoint

1.Linearapproxima,on

Findanapproxima,onforf(1.5)WhatistheslopeoftheTL?(A) 2(B) 3(C) 3/2(D) -3(E)  1/2

2.Linearapproxima,on

Findanapproxima,onforf(1.5)Whatisthepointoftangency??(A) (0,-1)(B) (2,1)(C) (1,1)(D) (1,2)(E) Noneoftheabove

3.Linearapproxima,on

Theeqnofthetangentlineis(A) y=2x-1(B) y=2x+1(C) y=2x-0.5(D) y=x-1(E) y=-1-2x

4.Linearapproxima,on

TheLINEARAPPROXIMATIONtof(1.5)is(A) f(1.5)≈1(B) f(1.5)≈1.5(C) f(1.5)≈2(D) f(1.5)≈2.5(E) f(1.5)≈3

Linearapproxima,on

TheLINEARAPPROXIMATIONtof(1.5)is(A) f(1.5)≈1(B) f(1.5)≈1.5(C) f(1.5)≈2(D) f(1.5)≈2.5(E) f(1.5)≈3

1.5

2

5.Linearapproxima,on

TheLINEARAPPROXIMATIONtof(1.5)is(A) Anoveres,mate(B) Anunderes,mate

1.5

2

Rescueme!

UBCMath102

Rapunzel,Rapunzel,Let

downyourhair!

Rescueme!

UBCMath102

Noway!You’renotclimbingupmyhair!Gogetaladder!

Rescue!WhatistheshortestladderthatyoucanusetorescueRapunzelfromthewindowat6metersheightifthereisananthillintheway?

UBCMath102

ground

anthillSeeP111Featuredproblem5.1

Exam-type

ques,on

Rescue!Whatistheshortestladderthatyoucanusetorescuesomeonefromthewindowat6metersheightifthereisahillintheway?

UBCMath102

ground

TangentLine

groundladder

anthill

Solu,on:

Weneedtofindthelengthofthetangentlineshowninthediagram.Todoso,weshouldsolvethefollowing(restatedmathema,cal)problem:

UBCMath102

Solu,on,cont’d

UBCMath102

Solncont’d:UsethefactthattheTLgoesthroughthepoint(0,6)

Nowsolvethisequa,ontofindx0

UBCMath102

Nowgettheequa,onoftheTL:

•  Slope:

•  TLhasy-intercept6andslope-1soitseqn:

Is:

UBCMath102

Solncont’dNowthatwehavetheequa,onofthetangentline,wecandetermineitslength.(Seenextslide).Theproblemhasbeensolved..Andthisonewaslesstrivialthantheearlierproblems,sincewehadtosolveforthepointoftangency.

UBCMath102

Solu,on:

UBCMath102

6

66√2

y=6-x

Thelengthoftheshortestladderis:6sqrt(2)

6

66√2

Tangent line y = 6 - x

(2,4)

(6,0)

(0,6)

Linearapproxima,on

•  Usingatangentlinetoapproximatethevalueofafunc,on(nearsomepoint).

UBCMath102

(6)Tangentlineequa,on

Whichofthefollowingistheequa,onofthetangentlinetof(x)atthepointx0

UBCMath102

(7)Linearapproxima,onClosetothepointx0wecan

approximatethefunc,onf(x)by

(8)Linearapproxima,on

Theapproxima,onisgoodprovided•  (A)xissmall•  (B)x0issmall•  (C)(x-x0)islarge•  (D)(x0-x)issmall•  (E)Noneoftheabove

UBCMath102

(9)Alinearapproxima,on..

approximatesthetruevalueofthefunc,onatsomepointclosetox0bythevalueshownin

•  A,B,C,D,orE UBCMath102

A

B

C

DE

Geometry

UBCMath102

f(x0)

Δf

Geometry

UBCMath102

f(x0)

(x-x0)

f’(x0)(x-x0)

Δf

Δf

(10)Example:

UBCMath102

(11)Examplecont’d

UBCMath102

(12)Examplecont’d

UBCMath102

(13)Overorunder-es,mate?Theapproxima,onwemadewill(A) Over-es,matethetruevalue(B) Under-es,mate“““

UBCMath102

FromLastWeek

Moreaboutdisplacement,velocity,andaccelera,on–---sincewedidnothaveachancetoprac,cethisonFriday:tryoutthisques,on

UBCMath102

Howtooutrunacheetah

hvps://www.youtube.com/watch?v=e51TlGBTyVg

I’mpuwngmymoneyonthe“livleguy”

UBCMath102 hvps://en.wikipedia.org

Exam-type

ques,on

SeeP100Featuredproblem4.3

HowtooutrunacheetahAcheetahcanrunattopspeedvcbutithastoslowdown(decelerate)tokeepfromgewngtoohot.(Assumeaconstantdecelera,ona=-ac).Agazellerunsataslowerspeedvgbutitcanmaintainthatconstantspeedforalong,me.Ifthegazelleisini,allyadistancedfromthecheetah,andrunningaway,whenwoulditbecaught?Istherea(farenough)distancedsuchthatitnevergetscaught?

UBCMath102

Answers

1.A2.C3.A4.C5.B6.C

7.E8.D9.B10.D11.B12.A13.A

UBCMath102

Solu,on:toproblemdoneontheboard

UBCMath102Tangentlineintersectsxaxisaty=0,soatx=10/5=1/2

Solu,onstopreviousproblemsFirsttryouttheproblemsattheendofthelastlectureslides.Onlythenshouldyou“peek”attheanswers.

Solu,onto:deriva,veof|sin(x)|

Ondesmos:(notethatderiva,vedoesnotexistatallthecusps!)

Testyourskills

Problemstoworkon

Differen,ate,usingpowerorquo,entrule

Calcula,onandprac,ce

Findtheequa,onofthetangentlinetothegraphofatx=0.Wheredoesthetangentlineintersectthexaxis?

Relatedtestques,on(MT1,2014)(Tangentlines)

Considerthefunc,on(a) Atwhichpoints(a,f(a))doesthegraphofthis

func,onhavetangentlinesparalleltotheliney=−x.

(b)  Whatistheequa,onofthetangentlinesateachofthesepoints.

Testques,on(Quo,entrule,ra,onalfunc,ons)MT12014

Atanall-you-can-eatbuffet,thetotalcaloriesyougaincanberepresentedbythefunc,onwheret≥0isthe,meinminutesyouspendattherestaurantandAandbareposi,veconstants.•  (a)Ifyoustayedforalong,me,whatasymptotewouldyourtotalcaloricgainapproach?

•  (b)A{erhowmuch,medoyougainexactlyhalfofthatasympto,ccaloricamount?

•  (c)At,met,whatistheinstantaneousrateatwhichyourcaloricgainchanges?