几何英文

Themaintopictodayiswhatisderivative.

Andwe’regoingtolookatthisfromveraldifferentpointsofview,andthefirstoneisthe

geometricinterpretation.

That’swhatwe’llspendmostoftodayon.

Andthen,we’llalsotalkaboutaphysicalinterpretationofwhataderivativeis.

Andthenthere’sgoingtobesomethingelwhichIguessismaybethereasonwhyCalculus

issofundamental,andwhywealwaysstartwithitinmostscienceandengineeringschools,which

istheimportanceofderivatives,ofthis,toallmeasurements.

ansinscience,inengineering,ineconomics,

inpoliticalscience,etc.

Now,that’swhatwe’llbegettingstartedwith,andthen,there’sanotherthingthatwe’re

gonnadointhisunit,whichiswe’regoingtoexplainhowtodifferentiateanything.

So,ngyoucanthinkof,anythingyou

canwritedown,wecandifferentiateit.

Allright,sothatwhatwe’regonnado,andtodayasIsaid,we’regonnaspendmostofour

timeonthisgeometricinterpretation.

Solet’sbeginwiththat.

Soherewegowiththegeometricinterpretationofderivatives.

And,whatwe’regoingtodoisjustaskthegeometricproblemoffindingthetangentlineto

somegraphofsomefunctionatsomepoint.

Sothat’stheproblemthatwe’readdressinghere.

Allright,sohere’sourproblem,andnowletmeshowyouthesolution.

Let’’’sabovethepointx0.x0

’sthegeometricproblem.I

taskthatwehavenowistofigureouthowtodoit.

So,whatdidwelearnaboutwhatatangentlineis?Atangentlinehasanequation,andany

linethroughapointhastheequationy-y0iqualtom,theslope,isthe

equationforthatline,andnowtherearetwopiecesofinformationthatwe’regoingtoworkout

whatthelineis.

’pecifyP,givenx,weneedto

knowthelevelofy,whichisofcourjustf(x0).Sothat’sthefirstthingweneedtoknow.

t’uluswe

,’sthetrickypart,

andthat’sthepartthatwehavetodiscussnow.

Sojusttomakethatexplicithere,I’mgoingtomakeadefinition,whichisthatf’(x0),which

isknowasthederivative,off,atx0,istheslopeofthetangentlinetoy=f(x)atthepoint,let’s

justcallitP.

That’swhatitis,butstillIhaven’tmadeanyprogressinfiguringoutanybetterhowIdrew

etosaysomethingthat’smoreconcrete,becauIwanttobeabletocookup

wayofthinking

aboutthat,letmejusttrythis,soIcertainlyamtakingforgrantedinsortofnon-calculuspartthat

therpossibilitymightbe,

thislinehere,howdoIknowthatthisorangelineisnotatangentline,butthisotherlineisa

tangentline?Infact,thislinecrossatsomeotherpoint,’snotreally

thefactthatthelinecrossattwopoints,becauthelinecouldbewiggly,thecurvecouldbe

wiggly,’snotwhatdistinguishesthe

tangentline.

SoI’mgonnahavetosomehowgraspthis,andI’llfirstdoitinlanguage.

Andit’sthefollowingidea:it’sthatifyoutakethisorangeline,whichiscalledacantline

（割线）,andyouthinkofthepointQasgettingclorandclortoP,thentheslopeofthatline

edrawitclorenough,thenthat’s

gonnabethecorrectline.

Now,sothetangentlineiqualtothelimitofsocalledcantlinePQ,

herewe’rethinkingofPasbeingfixed（不变）andQasvariable（变化）.

Thenwe’regonnabeabletoputsymbolsandformulas（符号和公式）tothiscomputation.

Andwe’’sdothat.

Firstofall,I’’mthinkingofthisline

,gradually,we’

thestepswillintroduceustothebasicnotationswhichareudthroughoutCalculus.

Sothefirstnotationthat’sudisyouimaginehere’sthex-axisunderneath,andhere’sthex0,

’retravelinghereahorizontaldistancewhichis

denotedbyΔ’sdeltax,’sone

otherthingisthis

’sthisdistancehere,whichwedenoteΔf,whichisthechangeinf.

Andthen,theslopeisjusttheratio,Δf/Δtheslopeofthecant.

AndthewaywewritethatisthelimitasΔ’

isslopeofthetangentline.

rtodothat,I’mgoingtowriteΔfmore

ngeinf,sorememberthatthepointPisthepoint(x0,f(x0)).That’swhatwe

rdertocomputethedistancesandinparticularthe

verticaldistancehere,I’ishorizontaldistance

isΔx,thenthislocationis(x0+Δx).Andsothepointabovethatpointhasaformula,whichis

(x0+Δx,f(x0+Δx)).AndnowIcanwriteadifferentformulaforthederivative,whichisthe

following:sothisf’(x0),whichisthesameasm,isgoingtobethelimitasΔxgoestooofthe

changeinf,wellthechangeinfisthevalueoffattheupperpointhere,whichis(x0+Δx)and

minusitsvalueatthelowerpointP,whichiff(x0),dividedbyΔht,sothisisthe

formula.I’mgoingtoputthisinalittlebox,becauthisisbyfarthemostimportant

formulatoday,sisthewaythat

we’’ample,we’ll

’lltakethefunctionf(x),whichis1/’ssufficiently

complicatedtohaveaninterestinganswer,andsufficientlystraightforwardthatwecan

isthatwe’regonnadohere?Allwe’regoing

toiswe’’sallwe’regoingtodo,and

visuallywhatwe’reaccomplishingissomehowtotakethehyperbola,andtakeapointonthe

hyperbola,’swhatwe’reaccomplishingwhenwedo

’reaccomplishingthisgeometricallybutwe’t,

weconsiderthisdifferenceΔf/Δ

I’mgonnamakeitagainabovethispointx0,’llmakethe

alueoffatthetop,whenwemovetotherightbyf(x),soIjust

readofffromthis,mula,thefirstthingIgethereis1/(x0+Δx).

That’1/x0,that’nIhavetodivide

thatbyΔ,sohere’ingis

’sprettycomplicated,becauthere’salwaysadifferencein

isgui,thedenominatorisadifference,becauit’sthedifference

betweenthevalueontherightsideandthevalueontheleftsidehere.

Ok,sonowwe’’is

equalto,let’equalto1/Δxtimes……AllI’mgoing

ommondenominatoris(x0+Δx)*

sointhenumeratorforthefirstexpressionI,havex0,andforthecondexpressionIhave

x0+ΔisthesamethingasIhadinthenumeratorbefore,factoringoutthis

there

stoneisthatx0andx0cancel,n

thecondstepisthatthetwoexpressionscancel,thenumeratorandthedenominator.

’s

iquals-1/(x0+Δx)ntheverylaststepistotakethelimitasΔxtendsto0,and

nowwecandoit.

Beforewecouldn’?Becauthenumeratorandthedenominatorgaveus0/0.

ButnowthatI’vemadethiscancellation,thathappensisIt

thisΔxto0,andIget-1/’ht,soinorderwordswhatI’ve

shown-letmeputituphere-isthatf’(x0)=-1/,let’slookatthegraphjustalittle

bittocheckthisforplausibility,allright?What’shappeninghere,isfirstofallit’snegative.

It’slessthan0,’sthesimplest

condthingthatIwouldjustliketopointoutisthatas

xgoestoinfinity,thataswegofarthertotheright,0goesto

infinity(notzero),’salsoconsistenthere,whenx0isverylarge,this

isasmallerandsmallernumberinmagnitude,althoughit’’salways

slopingdown.

Allright,soI’eIshouldstopforaquestionor

.学生提问.ormula

herwords,whyisitthatthixpression,

whenΔxtendsto0,iqualto-1/x02?Letmeillustrateitbystickinginanumberforx0to

ht,soforinstance,letmestickinhereforx0thenumber3.

Thenit’s-1/(3+Δx)’sthesituationthatwe’thequestioniswhat

happensthisnumbergetssmallerandsmallerandsmaller,andgetstobepractically0?Well,

literallywhatwecandoisjustplugin0there,andyouget(3+0)

tendsto-1/9(over3*3).Andthat’swhatI’msayingingeneral

withthixtranumberhere.

Otherquestions?Sothequestioniswhathappenedbetweenthisstepandthisstep,right?

t,stisthisΔxwhichis

sittinginthedenominator,hat’sintheparenthes

issuppodtobethesameaswhat’n,at

thesametimeasdoingthat,Iputthatexpression,whichisthedifferenceoftwofractions,I

edenominatorhere,youetheproductof

nIjustfiguredoutwhatthenumeratorhad

tobewithoutreally……

Otherquestions?OK.

SoIclaimthatonthewhole,calculusgetsabadrap,thatit’sactuallyeasierthanmostthings.

vetomake

thingsharder,becauthat’sisactuallywhatmostpeopledoincalculus,andit’s

ecretisthatwhenpeopleaskproblemsin

calculus,rearemany,

sothelittlepieceoftheproblemwhichiscalculusisactuallyfairlyroutineandhastobeisolated

therestofit,reliesoneverythingelyoulearnedinmathematicsup

tothisstage,’sthecomplication.

Sonowwe’ingaboutwordproblem.

Weonlyhaveonesortofwordproblemthatwecanpo,becauallwe’vetalkedaboutisthis

hoaretheonlykindsofwordproblemswecanpo.

Sowhatwe’theareasoftriangles,enclod

bytheaxesandthetangenttoy=1/,sothat’medrawapicture

’considerthefirstquadrant.

Here’ht,it’re’smaybeoneofourtangentlines,which

nwe’,sothere’sourproblem.

Sowhydoesithavetodowithcalculus?Ithastodowithcalculusbecauthere’satangentinit,

sowe’ou’lle,thecalculusis

theeasypart.

Solet’fall,I’

importantthingtorememberofcour,isthatthecurveisy=1/’sperfectlyreasonabletodo.

Andalso,we’regonnacalculatetheareasofthetriangles,andyoucouldaskyourlf,intermsof

what?Well,we’ceweneedanumber,we

’regonnahavetodosomeofthisanalysisjustaswe’ve

’mgonnapickapointand,consistentwiththelabeling,we’vedonebefore,I’m

gonnatocallit(x0,y0).Sothat’salmosthalfthebattle,havingnotations,xandyforthevariables,

andx0andy0,,onceyouethatyouhavethelabeling,Ihopeit’s

tofall,thisisthepointx0,andoverhereisthepointy0.

That’ssomethingthatwe’rdertofigureouttheareaofthistriangle,

it’sprettyclearthatweshouldfindtheba,

weshouldfindtheheight,soweneedtofindthatvaluethere.

Let’rewegoingtodothis?Well,solet’

whatisthatweneedtodo?Iclaimthatthere’sonlyonecalculusstep,andI’mgonnaputstarhere

’vefiguredoutwhatthe

tangentlineis,’

what’stheformulaforthetangentline?Putthatoverhere,it’sgoingtobey-y0iqualto,and

here’sthemagicnumber,’’s-1/x0x0(x-x0).So

we’

havetofigureoutalltherestofthequantitiessowecanfigureoutthearea.

owedothat?Well,tofindthispoint,’regonnafind

’sthefirstthingwe’that,whatweneedtodo

equationforthexinterceptis

y=uginy=0,that’sthishorizontalline,’sdothatintostar.

Weget0minus,thaty0isf(x0),andf(x)is1/x,so

thisthingis1/t’qualto-1/e’sx,andhere’ht,soinorderto

findthisxvalue,

–x/sisplus1/x0becauthex0andx0x0cancelsomewhatAndsoifIputthison

theotherside,Igetx/x0x0iqualto2/thenmultiplythrough-sothat’swhatthis

implies-andifImultiplythroughbyx0x0Igetx=,soIclaimthatthispointwe’vejust

calculatedit’,I’malmostdone.

’mgonnauaverybigshortcut

ht,IclaimIcanstareatthis

andIcanlookatthat,’ht.

That’reasonIknowthisisthefollowingsohere’s

thesymmetryofthesituation,’sakindofmirrorsymmetry

lvestheexchangeof(x,y)with(y,x);sotradingtherolesofxandy.

SothesymmetrythatI’musingisthatanyformulaIgetthatinvolvesx’sandy’s,ifItradeallthe

x’sandreplacethembyy’sandtradeallthey’sandreplacethembyx’s,thenI’llhaveacorrect

erywhereIeayImakeitanx,andeverywhereIeanxI

makeitay,sthat?That’sjustanaccidentofthiquation.

That’sbecau,sothesymmetryexplained…isthattheequationisy=1/t’sthesame

thingasxy=1,ifImultiplythroughbyx,whichisthesamethingasx=1/’swherethex

fyoudon’ttrustthixplanation,yougetalsogetthey-intercept

bypluggingx=gedy=

candothesamethinganalogouslytheotherway.

AllrightsoI’malmostdonewiththegeometryproblem,andlet’,let

’dliketosayisjustmakeonemoretiny

ardestpartofcalculusis

thatwecallitonevariablecalculus,butwe’reperfectlyhappytodealwithfourvariablesatatime

orfive,problem,Ihadanx,ay,’salreadyfour

themanipulations

wedowiththemarealgebraic,andwhenwe’redoingthederivativeswejustconsiderwhat’s

llytherearemillionsofvariablefloatingaround

’swhatmakesthingscomplicated,andthat’ssomethingthatyouhavetoget

re’ssomethingelwhichismoresubtle,andthatIthinkmanypeoplewhoteach

thesubjectoruthesubjectaren’taware,becauthey’vealreadyenteredintothelanguageand

they’resocomfortablewithitthattheydon’’ssomething

’twannacreatesixnamesforvariablesoreightnamesfor

lippedthembyyou.

Sowhyisthat?WellnoticethatthefirsttimethatIgotformulafory0here,itwasthispoint.

Andsotheformulafory0,whichIpluggedinrighthere,wasfromtheequationofthecurve,

y0=1/ondtimeIdidit,Ididnotuy=1/hiquationhere,sothisisnot

y=1/’’saneasymistaketomakeiftheformulasareallablurto

youandyou’thatx-intercept

calculationthereinvolvedwherethishorizontallinemetthisdiagonalline,andy=0reprented

othisconstantly

becauit’sway,’smuchconvenientforustoallow

ourlvestheflexibi

similarly,lateron,ifIhaddonethisbythismorestraightforwardmethod,forthey-intercept,I

wouldhavetxequalto0,Thatwouldhavebeenthisverticalline,whichisx=dn’t

changetheletterxwhenIdidthat,isoneofthemain

ankeepyourlfstraight,you’realotbetteroff,andasIsaythis

isoneofthecomplexities.

Allright,sonowlet’,actually

I’reaofthetriangleis,wellit’

ba2x0theheightis2y0,’s1/2(2x0)(2y0),whichis(2x0)(y0),whichis,lo

andbehold,musingthinginthiscaisthatitactuallydidn’tmatterwhatx0andy0are.

’sjustaccidentofthefunction1/enstobethe

ht,sowehavesomemorebusinesstoday,someriousbusiness.

,firstofall,arejust

firstoneisthefollowing:we

alreadywrotey=f(x).AndsowhenwewriteΔy,thatmeansthesamethingasΔ’sa

viouslywewrotef’forthederivative,sothisisNewton’snotation

ofthemisdf/dx,andanotheroneis

dy/dx,etimesweletthefunctionslipdownbelow

sothatbecomesd/dx(f)andd/dx(y).Sotheareallnotationsthatareudforthederivative,

notationsareinterchangeably,sometimes

eomits-noticethat

thisthingomits-theunderlyingbapoint,’n’tgive

rearelotsofsituationslikethatwherepeopleleaveoutsome

oftheimportantinformation,’sanothercouple

edoutthiscalculation

ofthederivativeofthefunction1/’sdothat.

Soexample2isgoingtobethefunctionf(x)=x∧n,n=1,2,3;

whatwe’retryingtofigureoutisthederivativewithrespecttoxofx∧ninournewnotation,

n,we’regoingtoformthixpression,Δf/Δ’regoing

wepluginforΔfis((x+Δx)∧n-x∧n)/Δ

ore,letmejuststickthisinthenI’,Iwrotex0hereandx0

I’mgoingtogetridofit,becauinthisparticularcalculation,it’sanuisance.

Idon’thaveanxfloatingaround,st

don’’sawasteofblackboardenergy.

There’satotalamountofenergy,andI’vealreadyfilledupsomanyblackboardsthat,

there’,,no0’kofx

ca,Δhtnow,inorderto

simplifythis,inordertounderstandalgebraicallywhat’sgoingon,Ineedtounderstand

t’needalittletingbitof

it,,thebinomialtheoremwhichisinyourtextandexplained

inanappendix,saysthatifyoutakethesumoftwoguysandyoutakethemtothenthpower,

thatofcouris(x+Δx)hefirsttermx∧n,that’swhen

n,youcouldhavethisfactorofΔxandalltherestx’s.

Soatleastonetermoftheform(x∧(n-1))Δmanytimesdoesthathappen?

Well,ithappenswhenthere’safactorfromhere,fromthenextfactor,andsoon,andsoon.

There’thegreatthingisthat,withthis

alone,alltherestofthetermsarejunkthatwewon’more

specific,there’kiswhat’scalledbigO((Δx)

∧2).Whatthatmeansisthatthearetermsoforder,sowith(Δx)∧2,(Δx)∧

ht,that’citing,,this

istheonlyalgebrathatweneedtodo,andnowwejustneedtocombineittogether

tgetourresult.

So,nowI’

Δf/Δx,whichrememberwas1/Δxtimesthis,whichisthistimes,nowthis

is(x∧n+nx∧(n-1)Δx+...thisjunkterm)-x∧’swhatwehavesofarbad

,I’mgoingtodothemaincancellation,which

,that’s1/Δx(n∧(n-1)Δx+thistermhere).AndnowIcan

divideinbyΔnx∧(n-1)+nowit’sO(Δx).There’satleastonefactorofΔxnot

twofactorsofΔx,Icanjusttakethelimit.

’swhyIcallitjunkoriginally,becauit

ath,tendsto,asΔxgoesto0,

nx∧(n-1).AndsowhatI’veshownyouisthatd/dxofxton,iqualtonx

∧(n-1).Sonowthisisgonnabesuperimportanttoyourightonyourproblemt

ineverypossibleway,andIwanttotellyouonething,onewayinwhichit’s

thingextendsto

,ifItaked/dx

ofsomethinglike(x∧3+5x∧10)that’sgonnabeequalto3x∧2,thatapplyingthis

ruletox∧nhere,I’llgets5*10so50x∧9.

Sothisisthetypeofthingthatwegetoutofit,andwe’regonnamakemore

haywiththatnexttime.

dmylfoff.

ThequestionwasthebinomialtheoremonlyworkswhenΔ,the

binomia’svery

’tcarewhatallthecrazy

’sjunkforourpurposnow,becauwedon’thappenedtoneedanymore

,becauΔxgoesto0.

OK,eyounexttime.