Midterminforma-on

•  Whatdoesitcover?•  WhatshouldIexpect?See:MidtermInforma-onpageonthewiki

What’sdueandwhen?•  OSH4:

•  Pleasenotethatthedeadlineis8:59pm,Fri,Oct20Donotcountonbeingabletosubmitlater,ortosenditbyemailaMerthedeadline.

LeastsquaresdatafiRng;ChainRule

Leastsquares(Linearregression)datafiRngcont’d..

UBCMath102

Fromlast-me.

•  Wearegivensomedataandwanttodescribeitstrend.

•  Howdowefitthebestlinethroughthedata?

•  Thisisaprac-alapplica-onofcalculus,becauseitinvolvesminimiza-on.

•  “LeastsquaresfiRng”–auseful(andmathema-callysimple)procedure

UBCMath102

Learninggoals

•  UnderstandwhatdatafiRngmeansinthesimplest(linearleastsquares)seRng.

•  Understandtheconnec-ontoop-miza-on•  Beabletofitaliney=ax+bdatapoints•  Instruc-onsforhowtouseaspreadsheettofitalinetodata

Last-me:Example

•  Rainfalloverthreedays

•  Findafor“bestline”

UBCMath102

Day 1 2 3

Rain(cm) 2 3.3 4

Whatis“best”line?

•  Choselineforwhichtheresidualsareassmallaspossible!(minimize!)

UBCMath102

y=ax

(x1,y1)(xn,yn)

(xi,yi)

Whatisaresidual?

Residual=datavalue-theore-calvalue=yi–axi

UBCMath102

y=ax

(x1,y1)(xn,yn)

(xi,yi)

“Sumofsquareresiduals”

•  SSR=(y1-ax1)2+(y2-ax2)2+...+(yn-axn)2•  Short-handnota-on: SSR(a)=MinimizingSSR(a)isequivalenttofindingtheslopeofthelineforwhichthedevia-onsofdatafromthelinearesmallestoverall.

UBCMath102

Liney=ax;bestfitvalueofa:

•  WeshowedthatthiswasobtainedbyminimizingSSR

•  Theop-malvalueoftheslopeofthelinetofitthedata!

Defini-ons

•  Amodelisafunc-onusedtorepresentorfitdata.Forexample,somecommonones:f(x)=ax,f(x)=ax+b,f(x)=Ce-kx.

•  Residualsareameasureofhowfareachmodelvalueisfromthedatavalue:ri=yi-f(xi).

•  TheSumofSquaredResiduals(SSR)isameasureofhowwellthemodelfitsallthedata:SSR=∑(yi-f(xi))2.

•  SmallerSSRisbeher.UBCMath102

“Bestfit”

•  Thebestfitmodelisthemodelwithparametervalue(s)(a,aandb,etc)thatgivesthesmallestSSR.

UBCMath102

(1)Bestfitlinewithintercepty=ax+b

•  Theresidualsare(A)axi+b(B)(axi+b)2(C)yi-(axi+b)(D)yi2-(axi+b)2(E)yi-axi

Residualsforliney=ax+b

•  ri=yi-(axi+b)

(x1,y1)

(xn,yn)

(xi,yi)

y=ax+b

(2)Forbestfitlinewithintercepty=ax+b

•  Theresidualsareyi-(axi+b)andwewillminimize

(A)Σyi-(axi+b)(B)Σ|yi-(axi+b)|(C)Σ(yi-(axi+b))2(D)Σyi2-(axi+b)2(E)Noneoftheabove

(3)Forbestfitlinewithintercepty=ax+b

•  WewillminimizeSSR=Σ(yi-(axi+b))2Withrespectto

(A)a(B)b(C)xi(D)xiandyi(E)bothaandb

(4)Howwouldwedothat?

Findvalueofaandbsuchthat

(A) d(SSR)/da=0(B) d(SSR)/db=0(C) Both(A)and(B)(D) noneoftheabove(E) Noclue

UBCMath102

Challenge!

•  Forfun:Calculatevaluesofofaandbsuchthat

d(SSR)/da=0ANDd(SSR)/db=0WhereSSR=Σ(yi-(axi+b))2

UBCMath102

Bestfitlinewithintercepty=ax+b

•  RESULT:(SeeSupplementonwiki,noneedtomemorizethese!)

•  Where:

UBCMath102

Bestfitlinewithintercepty=ax+b

•  Manyspreadsheets(excel,Google-sheets)willcomputesuchlinesforyouautoma-cally

•  Someslidesfollowwithinstruc-ons.Orsee:M102Wiki

UBCMath102

Assignment7:Problem14

Usingaspreadsheettofitatrend-linetodata:yisthetotalvolumeofallchloroplastsinsideagivencellwithvolumexWanted:givendata,fittoit

Manyspreadsheetswillfitdata

Excel:HighlighttherowscontainingxanyvaluesInsert;chart;scaherplot

UBCMath102

Excelhasautoma-c“fitline”func-on

UBCMath102

y=0.5887x+0.0282R²=0.99493

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5 6 7

AxisTitle

AxisTitle

y

y

Linear(y)

Or,useGooglesheets

UBCMath102

Hereweshowhowtocalculatebestfitsfromscratch

•  CopythedatafromtheWebworkques-on

PasteintoGooglesheets(donotretype!)

Picksomevalueofaandcomputetheresiduals

UBCMath102

Residuals

UBCMath102

Squareresiduals

UBCMath102

SumofSquareResiduals(SSR)

UBCMath102

Computeslopeforbestfitliney=ax

UBCMath102

•  xiyi•  xi2•  Σxiyi•  Σxi2

• 

•  (Thisisbestslope)

UBCMath102

Computeslopeforbestfitliney=ax

TheSSRshouldbesmallforthebestfitline

•  LargeSSRforarbitraryvalueofa:

•  MuchsmallerSSRwhenweplugintheavaluewefound.

UBCMath102

Plotthedata

•  MakeascaherplotofthedataandgetGooglesheetstofitabestlinetoit.

y=ax+b•  Note:wecanalsodothisourselvesbycalcula-ngthequan--esaandbfromthedata

UBCMath102

Makeascaherplot

•  Highlightthecellswithdata,includinglabels

•  Insertchart:

UBCMath102

Formatchart

•  Start

•  Switchrowsandcolumns•  UsecolumnAasheaders

UBCMath102

•  Charts

•  Scaher•  (Selecttopchoice)

•  Insert

Dataplotwillappear

UBCMath102

Addtrendline•  {Controlclick}onchart

•  Advancededit

•  Customize•  Scrolldownmenu•  SelectTrendline;linear

UBCMath102

Trendlinewillappear

UBCMath102

Clickingonlinedisplaystrendline

UBCMath102

Lineagreeswithourowncalcula-ons

•  Calculatedvalues

Calcula-ons

•  Cells:

UBCMath102

ChainRule-Intro

UBCMath102

Func-oncomposi-on

•  Afunc-ony=g(f(x))

UBCMath102

Whatisthecorrectdecomposi-on?

•  Ifthefunc-onisThenthedecomposi-onis

Herearevaluesoff(x)atafewpoints

Thenf(f(3))isequalto:A.  3B.  -3C.  0D.  -2E.  2

(1)Func-oncomposi-on

UBCMath102

(2)Func-oncomposi-on

UBCMath102

ChainRuleofdifferen-a-on

•  Ify=g(u)andu=f(x)arebothdifferen-ablefunc-onsandy=g(f(x))isthecompositefunc-on,thenthechainruleofdifferen-a-onstatesthat

UBCMath102

CoffeeBudget

•  Rateofincreaseofmoneyspentoncoffee =

Rateofincreaseofpricepercup-mesRateofincreaseofcupsIdrink

(3)ChainRuleexample

UBCMath102

Solu-on

•  Thefunc-onis.Let

UBCMath102

(4)Chainruleexample

Computethederiva-veof)

UBCMath102

(4)ChainruleexampleThederiva-veofis

(A)(B)(C)(D)(E)Noneoftheabove

UBCMath102

Chainruleexample(2)

Thederiva-veofis:Quo-entrule:

Chainruleappliedtoop-miza-on

UBCMath102

Anttrails

Iloveants!(butnotinmykitchen)

UBCMath102

Anttrails

Ants:trailforminginanarena•  hhps://www.youtube.com/watch?v=-lz97WTM7aU

Ants:pheromonemarkings•  hhps://www.youtube.com/watch?v=tAe3PQdSqzg

Anttrails

•  Antscanfindtheshortestroutethatconnectstheirnesttoafoodsource.

•  eachantsecretesachemicalpheromonethatotherantswillfollow.

UBCMath102

Findtheminimumtraillengthconnec-ngnesttotwofoodsources

UBCMath102

Somepossibletrails

•  Wecanthinkofseveralwaystogetfromthenesttothefood

(1)AV-shapedanttrail

•  WhatisthetotallengthofthisVshapedtrail?

UBCMath102

(2)AT-shapedanttrail:

•  WhatisthetotallengthofthisTshapedtrail?

UBCMath102

(3)Whichpathisshorter?

(A) TheVpath.(B) TheTpath.(C)Theyarethesamelength.(D)ItdependsondandD.

UBCMath102

ItdependsondandD!

Example:supposeD=1ThelengthofTpathVpathàLargestdependsond!

Smallvslarged

•  larged smalld

(4)Yshapedtrail

Whatisthetotallengthofthetrailshown?

UBCMath102

(5)TheAnttrailproblem

WehavenowfoundthelengthLYoftheY-shapedpath.Whatarewegoingtodonext?

UBCMath102

(6)Whatistherangeofvaluesofx?

Ly(x)=

UBCMath102

Deriva-ve

Usethechainruletofindthederiva-veofthisfunc-on:

UBCMath102

(7)Deriva-veThederiva-veofthefunc-onfortheY-shapedpathlengthis

(8)Cri-calPoints

Thecri-calpointsare:

UBCMath102

Cri-calpoints

•  Set

UBCMath102

Annotherwaytogetthesolu-on

Minimalsurfaces(a.k.a.“soapbubbles”)

UBCMath102

Asolu-onbysoap!

Twoflatplexiglassplateswith“pegs”represen-ngthenestandfoodsources:Idea:credittoShawnDesaulniers

Asolu-onbysoap!

Soapbubblesformminimalsurfaces(i.e.shapesthathavethesmallestpossiblesurfaceareaforgivenconstraints:Herethreepegsand2flatplatesformtheconstraintsthatthesoapsurfacehastomeet.

Asolu-onbysoap!

Thepegsandflatplatessetaconstantheightinthezdirec-on:Hencethesoapbubbleselectsaconfigura-onwithshortestlengthinthexyplane

z

xy

Now-metofinishthecalculusproblem(forprac-ce)

UBCMath102

(9)Arewedoneyet?

(A) No,wes-llneedtofindthelengthL(x)atthecri-calpoint.

(B) Nowes-llneedtofindhowfarapartthefoodsourcesare,i.e.solveford

(C) No,weneedtocheckthetypeofcri-calpoint

(D) Yes,wearedone

UBCMath102

Checkingviathesecondderiva-ve

Findthesecondderiva-veofthefunc-onRecallthatthefirstderiva-veis

UBCMath102

Solu-on:

Seeanearlierchain-rulederiva-veproblem,andExample8.5intheM102Notes.(Sowhatdoesthistellus?)

UBCMath102

(10)Whatdoesthistellus?

(A) Thesecondderiva-vecanbeeitherposi-veornega-ve,sothecri-calpointcouldbeofeithertype

(B) Thesecondderiva-veisalwaysposi-vesothecri-calpointisalocalminimum.

(C) Thesecondderiva-veisalwaysposi-vesothecri-calpointisalocalmaximum.

UBCMath102

Sketchingtheoriginalfunc-on

Sketchthefunc-on(Remember,thisisanalterna-vetofirstorsecondderiva-vetestsfordiagnosingacri-calpoint.)

UBCMath102

Sketchingtheoriginalfunc-on

Sketchthefunc-on•  Smallx:L≈(D-x)+2d=C-x(straightline,slope-1)

UBCMath102 x

Sketchingtheoriginalfunc-on

Sketchthefunc-on•  Smallx:L≈(D-x)+2d=C-x•  Largex:L≈(D-x)+2x=D+x(straightline,slope+1)

UBCMath102 x

Sketchingtheoriginalfunc-on

Sketchthefunc-on•  Smallx:L≈(D-x)+2d=C-x•  Largex:L≈(D-x)+2x=D+xConnectthesesmoothly!verifiesaLOCALMIN

UBCMath102 x

Desmos

•  Graphofthefunc-onwithD=1

UBCMath102

Forfurtherstudy:Angles

WhatanglesdoestheYtrailform?Wefoundthat

UBCMath102

Angles

WhatangledoestheYtrailform?Seeifyoucananswerthisfornext-me.

AnswersDatafiRng•  1C•  2C•  3E•  4C

Anttrails•  1D•  2A•  3D•  4B•  5B•  6C•  7E•  8B•  9C•  10B

UBCMath102

Problemstotestyourskill

MC3

Eachofthegraphsrepresentsf(t),theamountoffoodgainedduring-metspentinafoodpatch.Whichofthesegraphsrepresentsthetypeofpatchwherefoodcanbecollectedrapidlyatfirst,butthereisonlysomemaximalamountthatcanbeobtainednomaherhowlongthesearchcon-nues?

UBCMath102

MC5

Theabsolutemaximumofthefunc-ony=f(x)=x+1/xontheinterval0.1≤x≤2occursat

UBCMath102

MC6

Lety=f(x)beasmoothfunc-on(deriva-vesofallordersexist)atx0.Whichofthefollowingstatementsiscorrect?

UBCMath102

Relatedproblems

Prac-cetheChainRule

UBCMath102

Solu-on

UBCMath102

Solu-onstopreviousproblemsFirsttryouttheproblemsattheendofthelastlectureslides.Onlythenshouldyou“peek”attheanswers.

• 

Op-malforaging

UBCMath102

Solu-on

Solu-on

HenceQ’(t)=R’(t)(sincepisaconstant).sothatmaximizingQturnsouttobethesameasmaximizingRasdoneinthelecture.Sot=

Moreop-malforaging

• 

Solu-on

• 

Cannonball

UBCMath102 Solu-on

Solu-on:

Reviewproblem(1)

Solu-on:

UBCMath102

ReviewProblem(2)

Solu-on:

UBCMath102

ReviewProblem(3)

UBCMath102 Solu-on

Solu-onto(3a)

UBCMath102

Solu-onto(3b)

UBCMath102