ChainRule

RelatedRatesImplicitdifferen4a4on

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Teams points

11

3

19

10

TeamsizeTeams points

2132

11

3

19

10

TeamsizeTeams points/person

2132

5.5

3

5

6.3

Growingsphere:

•  Radiusincreaseswith4me•  =>volumealsoincreaseswith4me•  =>surfaceareaalsoincreaseswith4me

•  Informa4onaboutoneofthesecanbeusedtocharacterizealltheotherones!

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ChainRule

Timetà

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r(t) V(r)

(1)GrowingsphereIftheradiusofasphereincreasesataconstantratethenthevolumeofthatsphere

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Growingsphere

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Growingsphere

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Growingsphere

Iftheradiusofasphereincreasesataconstantratethenthevolumeofthatspheresa4sfiessothevolumeofthesphereincreasesataratepropor4onaltoitssurfacearea.

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(2)Growingsphere:

Supposethevolumeofthesphereisincreasingataconstantrate.Then

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•  Given:

•  Butfrombefore:

•  So:

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(inverselypropor4onaltosurfacearea!)

GrowingsphereII

•  Smallsphere:rchangesfast•  Largesphere:rchangesslowly

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Growingsphere

Supposethevolumeofthesphereisincreasingataconstantrate.Then

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(3)Changingdistance:

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(4)Changingdistance:

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(5)Changingdistance:

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Changingdistance

•  Thevariablesarex(t),y(t),L(t)

•  Given:

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Changingdistance

Chainrule:

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Changingdistance

•  Nowuse

•  Thedistancebetweenthetwopeopleisincreasingattherate

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m/s

Take-homemessage:

•  Drawandlabeldiagram•  Writedowntherela4onshipsneeded•  Determinewhichquan44esarechanging,whichareconstant.

•  Differen4ate(carefully),andremembertousethechainrulewhereneeded.(Thisgivesyouageneralequa4onthatholdsforall4me)

•  Pluginnumbersattheend.(Thisdeterminesarateofchangeatoneinstant)

ChainRule

Implicitdifferen4a4on

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Implicitdifferen4a4on

•  Notallcurvescanbedescribedby1func4on•  Example:acircle–need2func4ons

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Implicitdifferen4a4on

•  However,xandyarelinkedbytherela4onship

•  “Locally”(closetosomepoint)thisrela4onshipimpliesthatydepends

(uniquely)onx•  Wecallthisan“implicit”func4on

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.(x,y)

.(x,y)

Implicitfunc4on

•  Locally,ifxchanges,thenyalsochanges:.

•  Locallyydependsonx.•  Wesome4meswritey(x)todenotethat.

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.(x,y) .(x,y)

Slope

•  Wecanaskwhatistheslopeofthecurveatthepoint(x,y).

Slopeofsecantline Slopeoftangentline

.(x,y).(x,y)

Wecanfindusingthechainrule

•  Rememberthatweconsideryasavariablethatdependsonx

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Implicitdifferen4a4on(chainrule)

•  Chainruleappliedatthisstep!!

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Example9.5:

Findtheslopeofthetangentlinetothepointx=1/2inthefirstquadrantonacircleofradius1andcenterattheorigin.

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(6)Example9.5:

Findtheslopeofthetangentlinetothepointx=1/2inthefirstquadrantonacircleofradius1andcenterattheorigin.

Theslopeis:(A)(B)(C)(D)(E)

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Example9.5:solu4on

Findtheslopeofthetangentlinetothepointx=1/2inthefirstquadrantonacircleofradius1andcenterattheorigin.

•  Atx=1/2gety=

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Inversefunc4ons

(Anexampleofwhyimplicitdifferen4a4onissohelpful)inversefunc4ons

Useimplicitdifferen4a4ontoverifythepowerruleforfrac4onalpower

Considerthefunc4onFinditsderiva4vebyimplicitdifferen4a4onaferrewri4ngitas

Solu4on:Werewritetherela4onshipintermsoftheinversefunc4onWeconsideryasavariablethatdependsonxandwedifferen4atebothsidesNowusethechainrule:Finally,isolatedy/dx:

Soln,contd:

•  Simplifywherepossible,usingtherela4onshipofyandx:

Comment

•  Implicitdifferen4a4onisveryusefulforfindingderiva4veofinversefunc4ons

•  Wewillseemanyexampleswherethisisso.

•  Thepresentexample:usedknownderiva4veofthefunc4ony=xn(powerrule)tofindthederiva4veofy=x1/n

Example:

Findtheslopeofthetangentlineatthepoint(1,1)onthecurve

•  Hint:aproductruleisneeded!

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(7)Example:

Findtheslopeofthetangentlineatthepoint(1,1)onthecurve

Theslopeis(A)1(B)-2/3(C)-3(D)-1(E)3

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Example:

Findtheslopeofthetangentlineatthepoint(1,1)onthecurve

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Productrule

Exponen4alfunc4onsandexponen4algrowth

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Thefunc4on2n

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(1)Whichofthefollowingapproxima4onsisreasonable?

(A) 210≈100(B) 210≈500(C) 210≈1000(D) 210≈5000(E) 210≈10000

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(2)Andromedastrain

hjps://www.youtube.com/watch?v=gEwzDydciWc

Thisstatementis (A)TRUE (B)FALSE

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Exponen4algrowthapplica4on

POLYMERASECHAINREACTION-  AwayofamplifyingDNAforresearch,gene4cs,crime-sceneforensics

-  hjps://www.youtube.com/watch?v=HMC7c2T8fVk

-  SeealsothePCRSong:-  hjps://www.youtube.com/watch?v=x5yPkxCLad

Asmoothfunc4on2x

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Otherbases:y=ax

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Whatisthederiva4veofy=ax?

Usethedefini4onofthederiva4vetocalculatedy/dx.

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(3)Whatisthederiva4veofy=ax?

(A)

(B)

(C)whereCissomeconstant

(D)

(E)whereCissomeconstant

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(3)Whatisthederiva4veofy=ax?

Usethedefini4onofthederiva4vetocalculatedy/dx.

Thisthingisaconstant

(3)Whatisthederiva4veofy=ax?

Usethedefini4onofthederiva4vetocalculatedy/dx.

Thisthingisaconstant

IDEA:pickaconvenientbasesuchthatthisconstantisC=1

Introducingaspecialbase

What’ssospecial?

Thederiva4veofexisex

Comparison

POWERFunc4ony=f(x)=xnf’(x)=nxn-1f’’(x)=()xn-2…f(n)(x)=n(n-1)…1f(n+1)(x)=0

EXPONENTIALFunc4ony=f(x)=exf’(x)=exf’’(x)=ex…

Howdoesexcomparewithotherbases?

Itsgraphlooksalotlikethatofothercommonbases!

Sowhyshouldwecareaboutit?

•  Weusethisexponen4alfunc4onmostlybecauseitsderiva4veismoreconvenientthanthederiva4vesoftheotherexponen4alfunc4ons.

Ln(x)istheinversefunc4onforex

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Thenaturallogarithmisaninversefunc4onfor

• 

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•  Restatetherela4onshipintheform

•  Nowuseimplicitdifferen4a4ontofinddy/dx

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•  Restatetherela4onshipintheform

•  Nowuseimplicitdifferen4a4ontofinddy/dx

•  Result:

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Answers

•  ChainRule&ImplicitDiff

•  1A•  2A•  3E•  4B•  5C•  6B•  7D

•  Exponen4als•  1C•  2B•  3E

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QUESTIONSTOTESTYOURSKILLS

Implicitdifferen4a4onexample:Astroid

Thecurvehastheshapeofanastroid.Findtheslopeofthetangentlinetoapointontheastroid.

(SeeExample9.8)hjps://www.youtube.com/watch?v=CeQDxV1vGqM&feature=youtu.be

(10)Thegraphofexlookslikewhichofthese?

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Findtheequa4onofthistangentline

Tangentlinetothecurvey=exatx=0.

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Exponen4al:Examques4on

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Note:Part(a)useslogarithminfinalstep..Seenextlecture

Prac4ce:conceptualmidtermques4on

Hereisthegraphofy=CektforsomeconstantsC,k,andatangentline.UsedatafromthegraphtodetermineCandk.

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Prac4cemidtermques4on

Sketchthegraphofthefunc4on

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Prac4ceExamques4on

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Prac4ceExamQues4on:

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Prac4ceExamQues4on

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Prac4ceExamques4onforImplicitdifferen4a4on

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Solu4onstopreviousproblems

Conicalpileofsugar

Problemfromlast4meSugarispouredataconstantflowrate,1cm3/stoformaconicalpile.Atwhatrateisthebaseareaincreasing?

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Solu4on:firstexpressVintermsofA

NowuseChainRule

Timetà

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A(t) V(r)

Solu4on,cont’d

Relatedtestproblem

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Solu4on:

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