what is calculus? motivating the need for a “limit” · 2020. 6. 1. · (2.1) 1.5 and 1.7: the...
TRANSCRIPT
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: