Tangentlines,cont’d

Linearapproxima5onandNewton’sMethod

Last5me:

A“challenging”tangentlineproblem,becausewehadtofigureoutthepointoftangency.

ground

??(A)Igetit!(B) IthinkIseehowwedidit(C) I’mnotsure(D) I’mconfused(E) I’mtotallylost

Last5me:

A“challenging”tangentlineproblem,becausewehadtofigureoutthepointoftangency.

ground

??(A)Igetit!(B) IthinkIseehowwedidit(C) I’mnotsure(D) I’mconfused(E) I’mtotallylost

SeeSec5on5.6,Try

problem5.10

Last5me

“Uselinearapproxima5ontocalculate105”Solu5on:-  figureoutthefunc5on-  Figureoutapointnearbywhereweknowthevalueofthefunc5on

-  Uselinearapproxima5on

Last5me

“Uselinearapproxima5ontocalculate105”Solu5on:-  figureoutthefunc5onf(x)=x-  Figureoutapointnearbywhereweknowthevalueofthefunc5on.Weknow100=10

-  Uselinearapproxima5on

Geometry:

“S5ckatangentline”onthefunc5onatx0=100,readthevalueonTLnearby(atx=105).

x0

LinearApprox:

-Advantage:easytocalculate:==10+(5/20)=10.25

-  Onestepmethod

-  Disadvantage:notveryaccuratefurtheraway

Anotherway(Newton’smethod)

Converttheproblemtotheformf(x)=0Useintui5onordesmostogetaroughes5mateforthezeroofthefunc5on.UseNewton’smethodtogetbe_erandbe_erapproxima5on.Disadv:Needsomeintui5onAdvantage:Outstandinglevelofaccuracy!

1.Converttorightform

Finddecimalapproxtox=105.TouseNewton’smethodIwouldconvertthistosolvingtheproblemf(x)=0whereA.  f(x)=x2-105B.  f(x)=x-105C.  f(x)=x-10.5D.  f(x)=105

Converttorightform

Finddecimalapproxtox=105.-  Sameas:findxsuchthatx2=105x2-105=0-  Nowtheproblemisintheformf(x)=0

forf(x)=x2-105

2.My“ini5alguess”forthezerooff(x)=x2-105wouldbe

A.x0=100B.  x0=10C.  x0=105D.  x0=105E.  I’mlost.

Ini5alguess

Findvalueofxsomewherearoundzerooff(x)=x2-105Thatwillbethestar5ngvalueforNewton’smethod.x0=10

Newton’sMethod

Awaytofinddecimalapproxima5onforthezeroofafunc5on

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Plan

•  WhatisNewton’sMethod•  Thegeometrybehindit•  Aprac5calexamplewhereweneedtouseit•  HowtouseaspreadsheettoimplementNewton’sMethod.

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WhatisNewton’sMethod?

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Newton’smethod

Solvef(x)=0

(3)WhichoftheseisproblemsisnotsuitableforNewton’smethod?

•  (A)Solvef(x)=0forx.•  (B)Findthezerosofafunc5onf(x).•  (C)Findwherethegraphoff(x)crossesthexaxis.

•  (D)Approximatethevalueofafunc5onclosetoaknownpointx0.

•  (E)Findrootsofanequa5ong(x)=C.

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Newton’smethodcanbeusedto

•  (A)Solvef(x)=0forx.•  (B)Findthezerosofafunc5onf(x).•  (C)Findwherethegraphoff(x)crossesthexaxis.

(E)ItcanalsofindrootsofF(x)=g(x)-C=0

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Hammerandnail

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Newton’smethod

Solvef(x)=0

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Thegeometrybehindit

Tangentline

•  Equa5onoftangentlineatx0:

•  Findthepointx1atwhichthetangentlineintersectsthexaxis.

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(4)Thetangentlineintersectsthexaxisat:

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DerivingNewton’sFormula

•  EqnofTL:•  TLintersectsxaxiswhen

Improvingtheroughes5mate..

•  Repeattheprocess:

•  Atthek’thstep:

Putitonaspreadsheet!

Letthespreadsheetdothoserepe55vecalcula5onsforyou!Goal:finddecimalapproxtosquarerootof105accurateto10decimalplaces!

Spreadsheetsetup

•  Thefunc5onisf(x)=x2-105•  Weneeditsderiva5vef’(x)=2x•  Ini5alguessx0=10

•  Formulatorepeat:

•  x1=x0–f(x0)/f’(x0)

Spreadsheet

•  Thenextfewslidesshowhowtosolvethisproblemonaspreadsheet.

Setup

• 

Transferx1toColumnA

• 

Highlightthecellsanddragdown

Itshouldlooklike:

Dragdowntheen5rerowtorepeat

• 

PartII:Lookingatallthisgeometrically

Recap

•  Letuslookagainatlinearapproxima5onandatNewton’smethod

•  Wewillusedesmosandthefunc5on

Toillustrateconcepts.

PART1

•  Ademooflinearapproxima5on

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Desmos

Graphthefunc5on:Usedesmostographitsderiva5ve

Itshouldlooklike:

• 

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Desmoscont’d

Addthevaluex0=5Aslidershouldappearforx0soyoucanshiftheloca5onofthattangentline.Addthepoint(x0,f(x0))toyourgraph.

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Itshouldlooklike:

• 

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Desmoscont’d

Addtheequa5onofagenerictangentlineatx=x0Youshouldbeableto“animate”x0andseethatlinesweepacrossyourcurve.

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Itshouldlooklike:

• 

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UsingDesmos

Considerthetangentlineatxo=5.Useyourgraphtofindthevaluesbelow:f(5),f(6)andthelinearapproxtof(6)basedatxo=5

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(5)UsingDesmos

Igotthevalues.f(5),f(6)andthelinearapproxtof(6)(A)4.37,4,5(B)4.37,5,2(C)4.37,2,2.5(D)huh??

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PART2

•  AdemoofNewton’sMethodonDesmos

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Ondesmos

UseNewton’sMethodtofindthelargestzeroofthefunc5onStartwithroughini5alguessx0=5.

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Tangentlineatx0=5

•  TLandf(x)

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Newton’sformula

•  Addtheformula

•  Plotthepoint

Looklikethis:(hereisx1)

• 

Actualzeroofthefunc5on–wewanttofindanaccuratedecimalexpansionforit

Desmosdoesthework

WegotDesmostocomputex1usingtheNewton’sMethodformula:•  (Wegotthisbyfindingwhereagenerictangentlinecrossesthexaxis,seepreviouslectures)

Repeattheprocess

Togetclosertothezerooff:•  Constructatangentlineatthepointx1•  Findthepointwherethenewtangentlineintersectsxaxis(Findx2).

h_ps://www.desmos.com/calculator/1bcshd1qot

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Weusethesameideatofindx2:

• 

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ToNo5ce:

•  Wearegejngclosertotheactualzeroofthefunc5on.

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Effectofini5alguess

Gotothesliderforx0andshifittothevalue3.Whathappenstothetangentlines?Whathappenstotheapproxima5onforthezeroofthefunc5on?

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Badini5alguess!

•  Anini5alguessx0=2.3leadsastray!Doesnotconvergetothezerooff!Seebelow:

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Importantcomments:

1.  TheDesmosdemonstra5onisonlytoshowtheGEOMETRYthatexplainshowNewton’sMethodworks.Youdonotneedthis(elaborate)construc5oninaprac5calproblem.

2.  HowdoIfindanini5alguess?ßUseDesmos,orsketchthefunc5on.Ensurethereisnomax/minbetweenx0andthetruezero..Wewillseehowtosketchfunc5onsinfuturelectures.

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Answers

•  1.A•  2.B•  3.D•  4.A•  5.C

Solu5onstopreviousproblemsFirsttryouttheproblemsattheendofthelastlectureslides.Onlythenshouldyou“peek”attheanswers.

Differen5ate,usingpowerorquo5entrule

solu5on

Calcula5onwedidinclass

Findtheequa5onofthetangentlinetothegraphofatx=0.Wheredoesthetangentlineintersectthexaxis?

solu5on

Solu5on:(a)Eqnoftangentline

Relatedtestques5on(MT1,2014)(Tangentlines)

Considerthefunc5on(a) Atwhichpoints(a,f(a))doesthegraphofthis

func5onhavetangentlinesparalleltotheliney=−x.

(b)  Whatistheequa5onofthetangentlinesateachofthesepoints.

solu5on

Solu5on:

(a)

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Solu5on

(b)

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Solu5ontoRelatedtest-typeproblemfromlast5me:

Findalinearapproxima5onthatprovidesaroughes5mateforthevalueof(1.1)8.Explainwhytheapproxima5onisan(over/under)-es5mate.

•  (Note:acalculatorvalueis2.1435)

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solu5on

Testques5on(Quo5entrule,ra5onalfunc5ons)MT12014

Atanall-you-can-eatbuffet,thetotalcaloriesyougaincanberepresentedbythefunc5onwheret≥0isthe5meinminutesyouspendattherestaurantandAandbareposi5veconstants.•  (a)Ifyoustayedforalong5me,whatasymptotewouldyourtotalcaloricgainapproach?

•  (b)Aferhowmuch5medoyougainexactlyhalfofthatasympto5ccaloricamount?

•  (c)At5met,whatistheinstantaneousrateatwhichyourcaloricgainchanges? solu5on

Solu5on:

(a)(b)(c)

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Testyourskillontheseproblems

LinearApproxima5on

Findalinearapproxima5onthatprovidesaroughes5mateforthevalueof(1.1)8.Explainwhytheapproxima5onisan(over/under)-es5mate.

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FindthepreydensitysuchthatP(x)=G(x)

Note:thisproblemissimilartotheaphid–ladybugproblemfromweek1butthefunc5onP(x)isnotthesame.

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Newton’sMethod