Velocity. velocity. change. changing. limiting. object? velocity. nothing. limiting. Problems. lists. A Geometric Connection. Q : But what do we
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1 h 2 Tngent + Velocity Problems 01 Q : But wht do we men velocity o nothing chnge is by o n object? velocity the ( instntneous is rte but t prticulr instnt chnging!? nstntneous velocity will be deined to be the limiting vlue o the verge velocities ie ( D h Ct C 1 1 in More lter!! t t 1 * so lists re unvoidble! A Geometric Connection 02 The o the sty tngent line ( hence the line itsel will be deined s the limiting vlue o the slopes o the secnt lines * limits re unvoidble!
2 limit sided 2 2 Limit o unction Limit de 3 [ hve then identiy conusing prts 03 one de 4156 Notice tht The lign n L i nd only i hiy n L nd ti m+ d L 04 n de 1 04 V A del 2 5
3 lig lig li lin 2 3 Limit Lws : wht is time F? well s 22 but how 5 do we justiy 3 nd X3 8 this? so } 5 Limit Lws Suppose tht lj z ed L nd lxijn 84 M 1 nd 2 3 lxign [ ccµ} x L or climcd c ny constnt 4 [ n g ( L M ] THE [tg ]±m i Mto ;((D L < 5 tig[s µft ;±tigsl±m 6 7 c c x ( others ii ti; innt K provided L > 0 when n is even Et use the limit lws to ind line 3 io?steeoeyxn line 7 3 / : s :* 2 use lws to building brek into 2 blocks j yx
4 but ( see Nice dont wnt to tht relly gin g Direct Substitution is rtionl unction nd is in the domin o then lxign (x Fined just plug in * so DS teksus tht lxign 1 o * But DS does Not pply to ig why? E Find line K * DS does Not pply! notnnser it Try to mnipulte unction ix ti Ytti n e limit ignores 1 By 1 Cxti X 1 whenever / / 05 Strtegy ( ( b ( c 1 lth E Find lim n o h * DS yields so try to mnipulte lim u F ( Hh Ligo into h (1+4 Cler denomintor or combine rctions numertor in lim i u o ith h ( Hh
5 04 Yes o n i G w cll (e C Squeeze Theorem hcx de! my Trimurti Theorem cx Eg Cx E h c or ll x ner ( except possibly t AND Liy d ine sex L then six L E Recll tiny sin ( DN E MMM But wht bout x sin C? Find bigot x sin (
6 sin * wht to do? DS yields o Cot?? simpliy?? Try Squeeze Then Note tht! sin (E E sink ( x > o O since no+ X l io+ O the Squeeze Them tells us tht Xs in ( o F Y L i l l l immured / 7 yxsin h / L y x
7 06 Tht 2 5 continuity wht do think it : you to be continuous ( t mens or unction point? Gw 06 quick look Continuity ( ntuitive de A unction is Continuous i you cn drw its grph without liting your pen is the grph hs N * isncxttc o holes c removble discontinuities * Lig ix not * hif rel DN breks ininite discontinuities ( v A s jumps jump discontinuities other weird stu X ( 06 2 together G W together heven ( continuity Lws g re continuous t then so re t g g g nd G provided g c to g is continuous t nd is continuous t g C then og is continuous t too * The inl point requires the ollowing limit lw
8 Rtionl Root trig Tx nd Tx Tx Tx cb t Tx cc Theorem lim gcx7b nd i is continuous t b then x ig cgcx ( 7gc Theorem ( continuous Building Blocks The ollowing re continuous on their domins : unctions C nd polynomils Z unctions > inverse trig unctions 4 exponentil nd logrithmic unctions Determine where n ( is continuous * n ( is built rom unctions up on their tht re continuous domins so ncl is continuous or its domin * so wht is the domin o u (? Need > o nd Tx 30 XZO nd x L so O E x ntermedite Vlue Theorem VT Suppose is continuous on C b ] nd tht N is ny number b/w C { cb Then there is t lest one vlue c b/w nd b s N
9 10 : wnt cc N VL Cb l C could be b mny Ex Does x4 x hve ny rel solutions? * cn you ctor? grph? try VT N * Let Cx x X s t 0 VT C i is continuous everywhere C b/c it is poly pplies! i coy co (2 13 > o This by V T there is c b/w O nd 2 s t C c O
10 Limits t ninity i Wht should n Cx L men? Q : Given tie below im Cx grphs 7 how would you nswer x s limit o An limit Gw 07 2 E Find the ollowing s ntis (ln 7 x DN E W Sooitcos s s ight (b lion (c * cosio
11 Algebric techniques E Find the ollowing ndeterminte! s cllin y t lim x s O Hs \ TE highest power in denom 2 3 txt Zxt t (b him him x so sne ide : denom E xh 2 highest power x s t lim A > 70 7 x M 7 s 07 3 Q : wht did might you you notice in these exmples? be ble to nticipte some o the nswers? plese provide evidence o your Suggestion
12 we 2 7 Derivtives ie Rtes t Chnge Tngent lines The line to circle t point P is to deine esy Eet tent we to now try deine the line to n curve rbitrry y d t point t ( c strt by looking P lines C s gon je l : to x x 08
13 o o line or 08 Det tn line 2 Ey Drw the tn to 4541 t the point where 3 nd ind n ez the tn line Grph Tn line m o slope? im C3th_ h 70 h th t liu u so h (32+1 lim 9 t 6h th Answer L u : 70 Z 9 y ( X im ( 6th 6 u 70 y 6C 3 t 10 o point : ( 3 C 3 0 & 3 nstntneous Velocity Think bck to the beginning o the course 08 bet velocity 4 The derivtive o unction This is the uniying ide t inding slopes o tn lines inding velocities
14 line in ix 09 Det o derivtive TB Summry 1 C is the slope o the tn to y t x 2 C x is position t timex then C is velocity position t time den 3 velocity der 3 ccelertion generl Det * rte t y why? Cx chrge t C Liis then the ( instntneous with respect tox t is y H x limit o rtes o unitsor units or c : * Note units or X verge C chnge 09
15 2 his dierentibility t 2 8 The Derivtive Function Recll : G W 09 Det der unction Let ix T 1 Find ormul or Cx 2 use your ormul to ind ( 2 nd Cl 3 Grph Cx nd 1 t n nigo T Cxthl h Cx 3 ( Eth nox(xh C x th h 4 th X X XZ ( Lino o 2 C Q : Does it mke sense tht i Cx 20? ( 1 og 2 09 Des 3
16 it it ol O Non dierentibility Ee Sketch nd ( ix 1 1 L b CH Tx ix{ X 20 i < o o : 0 BE Grphiclly 9 is Not dierentible t i hs p corner t x 2 hs verticl tngent line t x 3 it hs discontinuity t x Thm is dierentible t then is continuous t Picture pls Assume C exists we wnt to show inn C x C one equivlently cont tht hiy ( d O observe * si den se / c? O why?
17 Nottion Let y Cx 1 mens tke the derivtive with respect to x ( un ex ( y is usully written 0 This ex d o Cl d X 2 Second derivtive mens ( 1 mens ( 4 Thus C DX 2
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