1

WhatisCalculus?MotivatingtheNeedfora“Limit”SeeMath5Awebpage:WhatisCalculus?ConnectingPrecalculustoCalculus.1) AreaofRectanglevs.AreaUnderaCurve https://www.desmos.com/calculator/pesw8ofqvi2) LengthofLineSegmentvsLengthofCurve https://www.desmos.com/calculator/6zdb5tjl6d3) TheTangentProblem(2.1)4) InstantaneousVelocity.(2.1)1.5and1.7:TheLimitTheideaofnumbers“approachinganumber”isfundamentaltoallofcalculus.Inthisunitwewillconsider: 1)Whatdoweformallymeanbylimit? 2)Howdowecomputethevalueofalimit? 3)Usingthelimittocomputeinstantaneousrateofchange.IntuitiveDefinitionofLimit(pg51)

2

Intuitively,howmightwecomputealimit?

Example:(pg50)Howmightwecompute:

�

limx→2

(x2 − x + 2)

https://www.desmos.com/calculator/q1ytl9yrbv 1)NumericalApproach.

Ifxapproaches2fromtheright(

�

x→ 2+)Ifxapproaches2fromtheleft(

�

x→ 2−)

NOTEONDESMOS:MakingtablesinDesmoscanbehelpfulandcansaveyoucomputationtime.Youareallowedtouseitforyourhomework,justmakesureyoucanalsodothecalculationwithacalculatorforthetest.Justclickonthe+symbolintheupperleftcorner.2)GraphicalApproach 3)AlgebraicApproach-Couldwejustevaluatef(2)?

3

Caution:Theabovemethodsdonotalwayswork.

Ex2page52:Find

�

limt→0

t2 + 9 − 3t2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

https://www.desmos.com/calculator/ung82vock41)Numerical:

2)Graphical3)Algebraic.

4

Weneedtoformalizewhatweactuallymeanbylimitinordertofindwaystocomputethevaluemoreeffectively

Meaningof

�

f (x) − L < ε Seehorizontalbandongraphbelow.Meaningof

�

x − a <δ Sketchongraphbelow

Meaningof

�

0 < x − a

5

ThisδdoesNOTsatisfytherequirement …………..thissmallerδdoes.See5ApageDesmosandGeogebrademonstrations.ExampleProblems

6

Prove:

�

limx→1

(4 x + 2) = 6

7

Prove:

�

limx→3

x2 + x − 12x − 3

= 7SeeExample4page78.

8

OneSidedLimitsSee“Delta-EpsilonApplet”witha=3on5Apage.Whatwouldyousayinthatexampleabout

�

limx→3

f (x) =________________________________________________

Whathappenswhen

�

x→ 3+ _______________________________________________________Whathappenswhen

�

x→ 3− _______________________________________________________

Intuitively,whatmightwemeanby

�

limx→a +

f (x) = L

Sketchafunctionshowing

�

limx→a +

f (x) = L andcreateanumericaltablewhichdescribesthespecificexample

�

limx→2+

f (x) = 5 .Usethesetodeveloptheformaldefinition.

9

Whatpartofthepreviousdefinitionisaffectedofweonlyconsider

�

x→ a+

Howdoyouthinkthedefinitionwouldchangeifweweretolookat

�

limx→a −

f (x) = L ?

10

Exanple:

Seeexample3page77foraproofofaonesidedlimit.

11

InfiniteLimits

�

limx→a

f (x) = ±∞ limx→a +

f (x) = ±∞ limx→a −

f (x) = ±∞

Sketchafunctionsuchshowing

�

limx→a

f (x) = ∞ andcreateanumericaltablewhichdescribesthespecificexample

�

limx→2

f (x) = ∞ .Usethesetodeveloptheformaldefinition

Whatpartofdefinitionchanges?

12

Determinethedefinitionfor

�

limx→a

f (x) = −∞ .Itmayhelptosketchagraphormakeatableaswehavebeendoing,

Whatwouldbethedefinitionfor

�

limx→a +

f (x) = −∞ ?

Prove

�

limx→0

1x2

= ∞

13

ComputingInfiniteLimitsatx=a–VerticalAsymptotesCompute:

�

limx→2+

3x − 2

�

limx→2−

3x − 2

�

limx→2

3x − 2

Ingeneral,if

�

limx→a

f (x) yieldsanexpressionoftheform

�

nonzero#0 ,thegraphof

�

f (x) hasaverticalasymptote.And

�

limx→a

f (x) =∞ _________________________−∞ _________________________DNE ________________________

⎧ ⎨ ⎪

⎩ ⎪

SoweneedonlydeterminetheSIGN.

EX:Compute

�

limx→5+

4x2 +15 − x

14

3.4iLimitsatInfinity-TheoryWhatmightthislooklikegraphically?

�

limx→∞

f (x) = L

Recallgraphingrationalfunctions:

�

f (x) = 2xx − 2 Considergraphical,numerical.

https://www.desmos.com/calculator/dikrtjbb6p

Developtheformaldefinitionfor

�

limx→∞

f (x) = L

Definition:Given_________________________________theremustbeacorresponding_______________________suchthatif________________________________

then_____________________________________.

15

Prove:

�

limx→∞

1x2

= 0

Otherinfinitelimits:

�

limx→∞

f (x) = L

�

limx→∞

f (x) = ±∞

�

limx→−∞

f (x) = ±∞

Sketchafunctionsuchshowing

�

limx→−∞

f (x) = L anddeveloptheformaldefinition

16

Sketchafunctionsuchshowing

�

limx→∞

f (x) = ∞ anddeveloptheformaldefinition

17

LIMITDEFINITIONSSUMMARIZEDGivenany____(1)challengeonfvalues___,thereisacorresponding__(2)__restrictiononthexvalues_suchthatif__(3)___xisinthedescibedregion_________,then____(4)_____satisfiesthechallenge__

Tryit:Whatisthedefinitionfor

�

limx→−∞

f (x) = −∞ Agraphoratablemayhelp.ThegoalisNOTtomemorize,buttounderstand.Givenany____________________-__,thereisacorresponding_____________________suchthatif________________,then___________________________________________

18

1.6CalculatingLimitsSofarwehaveconsideredthreemethodsofcalculating/approximatinglimits.1)__________________________________________2)_______________________________________3)______________________________________Hereweseektofindadditionalmethods.LimitLaws:

19

ProveProperty1:Suppose

�

limx→a

f (x) = L .Thistellsusthatforevery__________thereisa________________suchthatif________________then_________Likewise,ifand

�

limx→a

g(x) = M thenforevery__________thereisa________________suchthatif________________then_________Weneedtoshow

�

limx→a

f (x) + g(x)( ) = L +M ,sowmustshowthatevery__________thereisa________________suchthatif________________then_________Properties1-5leadto

20

AdditionalProperties:

Whatdowedointhecaseswhere

�

limx→a

f (x) = f (a)?

�

limx→5

2xx − 5

�

limx→2

x2 − 4x − 2

�

limx→3

1x −

13

x − 3

�

limt→0

t2 + 9 − 3t2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

21

OneSidedLimits

�

limx→4

f (x) where

�

f (x) = x2 if x > 23x if x ≤ 2

⎧ ⎨ ⎪

⎩ ⎪

TheoremsonLimits

Example:

�

limx→0

x2 sin 1x

⎛ ⎝ ⎜

⎞ ⎠ ⎟

22

3.4iiCalculatingLimitsatInfinity

Examples:

�

limx→∞

2xx2 − 7x

�

limx→∞

5x + 32x +1

�

limx→∞

3x3

2x2 + 5x

23

�

limx→∞

2x2 +13x − 5

�

limx→−∞

2x2 +13x − 5

�

limx→∞

x2 + x

�

limx→∞

x2 − x

24

1.8ContinuityIntuitiveideaofacontinuousfunction:

Prove:f(x)=2x+3iscontinuousatx=1:

25

Continuityonaninterval:fissaidtobecontinuousontheinterval(c,d)ifforevery

�

a∈(c,d) ,f(x)iscontsata

Sincewefoundthatforanypolynomialorrationalfunctionthatifaisinthedomainoffthen

�

limx→a

f (x) = f (a) ,weknowthat_____________________________________________________________Example:Findtheintervalswherethefollowingfunctionsarecontinuous:

�

f (x) = 4 x3 + 2x +1

�

g(x) =5x − 93x + 5

Continuityatendpointsofaninterval

Considerthecontinuityof

�

f (x) = x

We include endpoints in the interval of continuity in the special case that f is only defined in that interval and one sidedcontinuityisimplied.

26

Itturnsoutthatallthebasicfunctionsweusearecontinuous______________________________________________________________.(seebookformoredetails)Example:Findtheintervalswherethefollowingfunctionsarecontinuous:

�

f (x) = cos x + 3

�

g(x) =3x

sin x − 1

�

f (x) = x2 − x − 12 ContinuityofPiecewiseDefinedFunctions:

�

f (x) = x2 − 1 if x ≥ 02x if x < 0

⎧ ⎨ ⎪

⎩ ⎪

27

Whatissospecialaboutcontinuousfunctions?

fcontinuouson[a,b] fnotcontinuous notclosedintervalProvethat

�

x3 + x − 1 = 0 hasatleastonesolution,

28

1.4and2.1:Usingthelimittocomputeinstantaneousrateofchange. (1)TheTangentProblemand (2)TheInstantaneousVelocityProblemsTheTangetProblem:(SimilartoEx1pg45)

Findtheslopeofthetangentlineto

�

f (x) =12x4 atthepoint

�

P 1, 12( ) .Firstofall,whatdowemeanbyatangentline?

Computingslopealgebraically,giventwopointsP(x1,y1)andQ(x2,y2)onaline,wecomputetheslopeorthelineas

�

m =y2 − y1x2 − x1

.Whywontthisworkhere?Sowhatcanwedo?

29

Let’sexamine3approaches.Method1)GraphicalApproach:Wecanestimatetheslopegraphicallybydrawinganeatgraphtoscale,drawingthe“tangentline”andcomputing“riseoverrun”usingtheLINEsketched.Whatarepossibledownsidestothisapproach?

ORMethod2)AverageApproach:Asecondapproachforestimatingtheslopeistousetwodifferentpoints,Q1andQ2onf(x)suchthatthetangentlineliesbetweenthetwolinesPQ1,PQ2(sooneofPQ1,PQ2issteeperthanourtangentline,oneisflatter),findtheslopesofPQ1andofPQ2.Approximatetheslopeofthetangentlinebyaveragingthetwoslopes.Ifwehavediscretedata,thisisusuallytheapproachweuse.(seelaterexample)

30

Method3)“QapproachP“or“limit”approach.Wecanestimatetheslopebyintroducingasecondpointandusingtheslopeformula

�

m =y2 − y1x2 − x1

.Randomly,letthatsecondpointQ(2,8),apointonf(x).ApproximatetheslopeofthetangentusingmPQ

mtan

�

≈mPQ=

�

8 − 12

2 − 1= 7.5

Doyouthinkthatisanoverestimateorunderestimate.Howcanwegetabetterestimate?

Let’schooseoursecondpointasapointonf(x)whichisclosertoP.LetQbe

�

32 , f

32( )( ) = 3

2 ,( ) Then

mtan

�

≈mPQ=

�

− 12

32 − 1

=

31

WatchtheanimationaswecontinuetoletthepointQonf(x)getclosertoP.https://www.desmos.com/calculator/oj5kla60tl (Tangent Secant Desmos on 5A page)(Dothisbyslidingthebuttonfromh=1slowlytowardh=0)NoticethehowthelinePQ(thesecantline)morecloselyapproximatesthetangentline.Noticethecalculationofslopeeachtimeshownonthegraph.Dothesecomputationsforslopeappeartobeapproachingsomevalue?TheslopecomputationscanalsobeseeninthetablewherethexvalueofQ,(x1)isshownintheleftcolumnandmPQisshownintherightcolumn.Seeingtheseslopecomputationsinatablemayhelpyouanswerwhetherthesecomputationsforslopeappeartobeapproachingsomevalue?

IfweformalizetheprocessofMethod3inthisexample,weseeonewaywecanuselimits.LetQbeageneralpointonf(x)so

testQbe

�

(x, f (x)) whichinthiscaseis

�

(x, 12 x4 ) Then

�

mPQ =f (x) − f (1)

x − 1=12 x

4 − 12

x − 1.InordertoletQmoveclosetoPwewould

needtoletxmovecloseto1.Wewillwritethisas

32

_____________________________________________Andwenowknowhowtocomputethislimitexactly.Ifweapplythisprocessoffindingthetangentlinetoageneralfunctionf(x)atsomefixedpointP(a,f(a))byintroducingasecondpointQonthecurve,Q(x,f(x))

�

m tan at x=a = limQ→P

mPQ = limx→a

f (x) − f (a)x − a

OR

Ifwere-labeltheabovegraphwithpointQbeing____________________________,theforumulacanequivalentlybewrittenas

�

m tan at x=a = limQ→P

mPQ = limh→

33

Note:Thisishowwedefinethetangentline.

34

Example:Findtheslopeofthetangentlineto

�

f (x) = x2 atx=2.Usingthefirstformofthedefinition: Twoapproaches: 1)Puta=2inatthebeginning,thencomputelimit. 2)Computelimitwith“a”andthenputina=2attheend.

35

Findingslopeofatangentlineifonlydiscretedataisavailable.Ifwearegivendiscretedata(atableasopposedtoaformulaforf(x)),thelimitapproachtofindingthetangentlineexactlyisnotpossiblesoweuseMethod1or2toestimatetheslopeofthetangentline.Example:Estimatetheslopeofthetangentlinetof(x)atx=3forthefunctiongiven.

Method1 Method2

36

TheVelocityProblem:

Ifcartravels20milesin30minutesandweknowitisgoingaconstantrate,atanygiventimethevelocityis

�

r =dt

=201/2

= 40 milesperhour.ButwhatiftherateisNOTconstantandwewanttoknowthevelocitypreciselyatthe10thminute?Whatmightwedo?Thisproblemiscalledfindingtheinstantaneousvelocity.Ex3pg48.Giventhatthedistancefallenaftertsecondsiss(t)=4.9t2(meters),findtheinstantaneousvelocityat5seconds.First,whatistheequationtellingus?Whatifwewanttocomputetheaveragevelocityoverthefirsttwoseconds?Computetheaveragevelocityoverthetimeintervalfrom4to5seconds.

37

Ingeneral,theAverageVelocityovertimeinterval

�

t1, t2[ ] =

�

dt

= s(t2 ) − s(t1)t2 − t1

Howdowedeterminetheinstantaneousvelocityatt=5seconds?

Smallertimeinteval… AveVelontimeinteval[4,5]=

�

s(5) − s(4)5 − 4

=122.5 − 78.4

1= 44.1m /sec

AveVelon[4.5,5]=

�

s(5) − s(4.5)5 − 4.5

=122.5 − 99.225

0.5= 46.55m /sec

AveVelon[4.9,5]=

�

s(5) − s(4.9)5 − 4.9

=122.5 − 117.649

0.1= 48.51m /sec

AveVelon[4.99,5]=

�

s(5) − s(4.99)5 − 4.99

=122.5 − 122.01049

0.01= 48.951m /sec

AveVelon[4.9999,5]=

�

s(5) − s(4.9999)5 − 4.9999

=122.5 − 122.459100

0.0001= 48.99951m /sec

Seevaluesonpage48forintervalsonthelargersideof5.Whatdoyounotice.Generalizing,theinstantaneousvelocityforthisfunctionatt=5is

�

= limt→ 5

s(t) − s(5)t − 5

= limt→ 5

4.9t2 −122.5t − 5

Showrelationtotangentproblem.

38

Theproblemoffindingthetangentlineandfindinginstantaneousvelocity,thoughseeminglyphysicallyunrelatedareexactlythesameprocess.Infact,anytimeweseektofindaninstantaneousrateofchange,werepeatthisprocess.Soweintroduceanewnotationforthisprocess.(Fornowwejustareviewingitasashorthandnotationforthisprocess,wewillexaminethismoreinthenextunit)

�

′ f (a) = limx→a

f (x) − f (a)x − a

= limh→0

f (a + h) − f (a)h

Thisrepresentstheinstantaneouschangeoff(x)inrelationtoachangeinxwhenx=a.Thus,theslopeofthetangenttof(x)atx=acanbedenotedby

�

′ f (a) andtheinstantaneousvelocityofthepositionfunctionofanobjectinrectilinearmotions(t)att=ais

�

′ s (a) = limx→a

s(t) − s(a)t − a

= limh→0

s(a + h) − s(a)h

Example:Let

�

s(t) =1tbetheposition(inmeters)ofanobjectinrectilinearmotion.Findtheinstantaneousvelocitywhent=1

second.

39

Example:Usingthegraphoff(x),estimatethefollowing:

�

′ f (2) ≈ _____________________

Findasuchthat

�

′ f (a) = 0 Example: