MATH 201 Notes Part 3, Spring 2019

Example: Hn = oh 3y ) Lompute det Alu) x eSin(py). Fingal aad B So that Df’ = © Noke: whtn Df’ com putes to aero, thenf) iS Ca (lef Qh@fnonle function, Examples off ch functions Fb! C*Ciny % 2 (IoC T20b) peoture 17 [oSVov rele] | as rrr Nott on not tion’ ; OP fy Bran fens 2 Fcneny yon fi = e% ty” a , (20- y Cos ty fe, --- “(ayy 00: fferentiability of functions "Pa yarlabtet fis dc ffereatiable a4 P (4or4 0) if é fone) = flterto)t Ac Cher 4o) A% tfy(toyo® yt E;aAat 24y where % 6, 30 as Ox79,4Y?%O and oe. xe ~To fest ff is differm Hable af (ergo) « J * Ide 4°E Cheep coutinuity & Clocy) Le if Lnuit doesn he ovrt or Limit exertr but fern) # #Otee4o) then Pis NOT contimeoul at (Mery o) Howey or tf f is caatimuou sf: Ms compe Ay (torje) & A Horo). TF oneor woth pestrel’s Ouits don't exest > not d(ftereatrosie, tf both Oxcyt: Y (ompule the difference ’ f (re ~(£CW 0049) % thy C400 oft 4 0/4) 20. Then acvide he dc ffEre nce. dylan) + (Oy)* = (ae). C9-Yo)* lea take Opmcts es (xy) — Cos Yo) If Limit (sy Bere the f or db fferenfiable at Creryo) .. the vector Cty, by) is the “der (Va tire” OF Ff. gp hat egal . rary (44) #(0-¢) fry: O (a4) = (%o) (1) Ts P woatinuous/ ; , Check y) fcmey) > ary? | ‘hares aah n-€ 2 Cy 300) Py es i) Frrd fx( 9°) and fy 00) [vse the deftal ttoa] UP ES atau #| ra A foe hee 9s0O Ln pg -O om 2) he? h po h hoo Slowclerty 9f = . 02) [ii Ts f gPferentasle at C¢8)? Compute: Lexy) Alors) -FCtorpeJO*- Ayl*ergo) AY which (a A*y™ _ (ofa) (COMP = BI" (Pleoryo)=0.9'Ve0) yu reye arty? Now givide by fareyr < a? y" zak calculate ite Dimit. (ary? eye : NE: a maeyrt ne “eg -0| te BF a ty) (are : a’ ty?) y Rane 24 7 ; \ out qe fram = Woke: fore) =f too) yen] topo) PH Application TP ox Ay eres mail andf is diftereniizble at (Xe ge) then we Mey v6 the gp proximation £1 y) LP (t0cYo) ¢feCree Jo) x +fy Ga Yo)OY » thre is called the Linear appraiia tion of F. 1psS (6) Ca-->elyn)o =F D w2= dost , yt =2sint a So (42 Lt2eest pt are trisation j= 200 t-| aA) E depends [aay dhe hilt we og thai Pull ciecle , A 4” 2 4 dé oct — tS =| >) =o oes, ie She X = dost aFind parametiisetion of arcof ¥ = 2* fiom CO,0) fe 0,4) Hsien: (Feu oft d2 Miscellaneous’ + dt eat’! ° pas et . at (Ne Tf poy i5 déftereatoable at (aocyo) then fis conk MAGS 49) ax J ey the deriva hve ° gras f - = (h, fy) . This vector is the derivetive 44 The chary rule ford variable: For 2 variables - Th P= Au) Spy We Pica y) add 1a z&) y¥ay¥) u> U(x) Draws dig tam! QW, fea) fect) ee df . df du O = dx oe oe dt ne +fy 4 Proof ! Yt 9) ~ fever) = Per) +fyldyHe Art G AY Repleee = r@) ,ye 9) Hen’ PAG), YH) ~ Prot) » Yolk) = fe O% HP y eAe FH AY HK - fab), 9)~ fr, pol) - fv st + fy a +6, tt, Now 2 SE ‘4 9 de tt bE 0 then: go. Uy r sae OS tone Example? fry.y)emrg> , d= 3esb ond ye 35k Use w for Hag, 4o)) , thea fing 2% , “Answer? dey dk de 0 oa de ‘sit (cha Rule) dw _ , Tt 7 2a (—Ssin t) + dy 3cosb) = 2730s ENBIME) E IEZSMEIC3C0St) = O Al Fern atively. F we (cosE)’ + (3Sint) = 4 sw ae 7 O a= 2(Y%V) , y 2) ) fet we Poacuv) , 9 ly %)) u Eyarplel: wf (29% _ ~vnr Trtiolucee urt > V =2ay Then W> Pi w) We ply o8CS 1G -ApyeHo) TS" {2 prey) Ty or oY Or \y Chan ral (2 (oc T2olb) decture [7 [a@ur> T pew evs] 1. det fra) ="[thao't) at 0 (v) Find Taylor Cries genetated b ri about 4=O (Wi) Tf f ‘ ? heerror2 (0.[) j5 Ipprex mated by Ps (0]), Whack & thee TOr ¢ Answers: oo (t) First _d a “opt ae (tant) =a = Z,( (ee) a) Ellin) 4 \ 7, 20 Go gay panel cr dé = YJ db = barry es SS | ar aco ° An 2a tt tp > ~() May Panel PO = C0" (2) = 2 _ ape yrp 2et! 020 ane be fey fine Yared . CUE ae zo = & erat? _ xr at a? azo (2tt (2442) (2. 3p FS arora W ha) i (Vole! the index of Pra) coimthic cose 1? doen't k Cxtst thus Py lar)= Pal) A? When We Approwuae y 2 ts) (o\)? he § rips NSA icHy Q [Hey peed) {, 2 , ferror] < Coy incetlag , Lerrer| < Last onu sed be7 8.4 d det fierj2 fu () U6 Taylo s/s theorem jo find the Series of f about a= (it) Estimate efror When appl oxs MAti 49 Joa) by p,(aJ for 36" $5 Answers: greet Step 2 STEPS fen) = at Pa): 2 write ee | il ceries ( Expand on = hy? Pie OO ")s Ja? ps fh ( Fy GEN< )(#) (a) 5 (s)(-4 rae oa dl" 2)a * Ply “Ul [a.m Me eee i (4) = Chem “fe)> oF whe) pela go ol 3, 2) (Tf We imo ca) by p,C4), shen erfoy US. (W ali y ' Topstinate errer for 36 XE 5 , bound fo) Ld Vy geeks OAV = (4/ 2 (-z sA Zz re Te ) ( XM oon ae 4 p *y (14) ~ (e») aty~ ate? F 3y) 47 1 Lie pene (sey 'Y Cy ty 5 det ae jry 7 95 420 J Jjre =f (urg2yty’? = [-i) =1 420 5 But along azo 2 L 8 20 ge L#o s\ Pimit deewt 430 ory” exist. (240CT2014 Lecture 20 Don't fase: Wi: rh, i Sy" Kwari” = vectty Fax 1 vartable : For dvarfables: ZN & aprox mathn My (14,0022 PP (A) X k a ( to. 4o) . J / , bth) - A@ ACeny L Path POY) prayn ¢ Herel Fear y + (4-bth) fy) ~ ut ~ Directional deriva fiver’. ~ det us ult 4, yj be @uattyector, MAAC; [2 l= [urea = / Cy # £(fOS Ore (Sia es k ca | = fos "Ot Sin & = | ey U =~ a tbs 2 Mae = fareh* thea uact vector = 5) which ; § Q eb rte * foreo™ J Example : Find aunrt vector fa the direction from thepornd Cl, 3) fo RB (-1, 9). . a > 32 ~ = 2.7 AB= 22-37 [AB] efigs @e Et eS & HveEn a fascthan of & pot Pi(me,y4o) on a unk VECFF ~ Mey Pel, net Mitte, gst04)~ MtI9). 9 fay s> S Which is called the Ie Lveal d2NVvattve of fm the dicechion U 4 Qt pom es Example : fry mt Poll, -!), Find DA(te) aos wets ff P(l+tu, -lt5ur)- ACL SQ =f tS) + Cs (1 Hin) ~ CEU) S20 oo S : - f IF2su, 45 %4/?-J—-Su, #5y, + S44 S30 Ay = f YU, +4) es ee S20 | & Theorem : Tf f iy differentiab| e at Po(ter Yo) then dees Me dues Neh fy past but Dt) for ony difectlon >, F ra Pot the D. Apo) = fy (to), U, A(Poh, = yaw? a? py = Ain)? U thoy the Ta a) > Vel, P= A@y ¢ fy ehy, Example * : fury, 2)= ay + Zinn 17> Find D> (m1) 6), whefe > Bite +P Arnswer: hy = yt 2Cos% A (fo) = It oc0os 7 =| y= 2 fy (Po) = 7 (Of ole let HA) + LF tas c SINK Po (to )= O aie V3 Questions regarding VP, pf L. Do these behave Dike: a Sat trey en ap action malt icetion ana division Fit) 2H 8 2W):? (P45), P+ (YF Hh): Belly) e Fels of i" a =a tI) O+ hy Sli (PEA +9 i rey 19 9f ; 3 esl tO HIS = #23 re ane po +36 rf = Vv # 4 =f, +4 , Exercise be do by yoursel: et) - For 9, (Peg); = Vifeg) «iP (Fe Fle) = vA@ + Pq % - BP + Dg 2. How lage oa, D flr) be 7 and inwhat divective/ Answer: Recall dot ite of 2 vectors - v Beyotrhy + bsF A= 4, P+ay Fak ; len RBx ab, t% bs thpdg Caraly He) Geometric aly’ So B= fr Bese Ee % ’ ofly® THp.)* ai, (Po) |W] eo 2 ~ U2~- » 4 : fe] -1 Hew | ry [os danger? 010) «1 ony] thatis when f= O , oC when ve pe ia the same disrectl(on of the GP Adlent os vf 7 Siorlarly the smallest DACP) =-| Pf (Pa)] CO= 01 when a isn the % ose (260CT2016) lecture 2 | pe divection of "PH det Us go back to the Brevet years)... finding eguation of fang eat lo curve at A, (4) Fend te a x rf) 2) Fed 44 at Altecyo) ag red? ent d @) y-ye mata) Dowe | Now ih Math 2 Ol--- _ Given a function of Lyvan ables FCmy) | a Level curve C of f (i.e fom 4) =C) and g poMt onthe Level Curve (0c Yo) how to find epeation of he tangent bine to Cat (Xorg) Xo y-frayco 9 Feny)=0 K J Ht) gtr) Lfwe Can find q vector we of A ortkegora) to the vector AG tren! FB, 2-0 un - Given aLevel curve Reta a tenes Pray=c fl Atego) on it, fiad a vecte nofwal to fhe Curve at/t, To See what thir poral es, Led the curve be paramet rived by. S a&) ac L<b ie -F (ad), yk) ) = for allt. Mow adperent ves ce (pee Gyo) (2 Pedy 9) Tove ter