sol5 - Math 2163 Jeff Merrnin’s sections, Quiz 5, October...

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Unformatted text preview: Math 2163 Jeff Merrnin’s sections, Quiz 5, October 7 Solo‘l'ions : 1. (2 points each) Indicate whether the following statements are true or false. (“True” means “Always true”, “false” rneans “sometimes false” or “possibly falSe”.) Justify your answers. :3 : x(s,t), y = y(s,t), and all functions appearing here are continuous and differentiable with continuous derivatives of all orders. (a) There is a function g(:1:, y) such that g$(:1:, y) : 1:2 +y2 and gy(:1:, y) = 2 2 m — y . SOIU‘i-l‘OA ((DR. FUNc‘HOA :11]: (a) There is a function g(z, 3;) such that g;(z, y) = :::2+y2 and 9,,(22, y) = 32—y2. / 7 / l y, *6 +29% fag =47X7— 7.1ng -~ /} V y ' 7M % a /‘ A. ’I//#"& tlyfifi'f’ "'1 ( HEM/l / I “a / & VIVA M15, “If/“M! W44“ ”* W31”? 47¢ 47% 7%.“ 5 51-5 ,2; . axisi'co‘. “I M7 5010*;0fl: sufvosl- Scalac‘ $vn¢+fl>fi ] Then 3% Add qr; Lon'i'Mva’ so a" 237’. = 35'0‘4 f) = 7.7 - gr 0‘7") 3 2X - s ‘5 («0 §ino¢ 'Hacse. arc. unulj‘“ [no 0° 3 (mi—{*1 “6 Gigi‘cmcn‘l' f5 $4M. (b) If w : h(z) is a differentiable function of 2, and z = f(m,y), then fl _ @995 8:: — 62 Soiu‘i'lbn Kyle Kern! PAM, because 9w - 32:91.2 9X, 099?; dz 9W; cm W easel SO\O+;OO 4'0 +LQ )\n'}eflr]€c‘ yroLlc/nl ZacL E'Hcr‘! rut W "5 0 ~FJ/1(nl-t'n 0mg Jag/2 :ry' ¥V J‘ A; 939:2.ng W n 'e, W WIN a< 3? E\ E Z :5 0‘ $UnC+faxy a; 440 Van‘gb/(sl 5° +0 441$ X J .‘ V J I z, . j (“WC 6? u/ Mn rm/fim :5 X) gov flee ow T11; (Mil/044W) pl? 'Fuflc/xU/vs w% )< firm; and 1’70, w: . . 'j-o 1",:- 0W :fézL! ‘floo; “on/j do fix/Y V’IC (c) If 2 is défimfi iriplicitly as a function of :1: and y by F( 13:11., 2) = 0, then 7 OF (9.x _ EI 62 Solu+ian L7 om qnoay mws s+uclen+ : gifil C 706/412 M 9175/7! Finch?) That/1,4 3,131,1c5 Tx‘ 12. 32 (d) Thé dy, dz) isinorinlgl some“ by w, FALgE. TM vahriclxjdmdg 7 (e) If 2 = f(a:,y), then 21, = 2.1% + Zyyt- . 1 ‘3 Solu‘lfim (5+evc Perry 3 7 may, :l’lflaioi1'ifieteinln vté‘humfi rt :59 Gimulfliflv {’0 A1 a 0“ A1 015 W'MZW" XERT {W ( 2. (6 points) Suppose that m2 » +cos(t+et—‘/§£:r—5)—(t+et)( L Find %. (Hint?: Use the chain rule.) goko'l’hm Chan+q\ RoHersonI z = (t+et)e K gig , ‘ I (1+5 ‘ l 2 ,I 5:.(1 + e‘)c ' +(tos(f+c‘— 6‘2”) ~(I—r-é’ 3—,— . \ - - ~ J Find (Hint?: Use the chain rule.) I, X, -M 4 ,(r p ..A \ . ~—-‘ ' is 4 ~ J ".2; \j % .fi‘ X ) X 5;) {» L'- x 04 :1; a}? f " ' ‘1 -A » , 1. "‘x ‘2 . \ i ,2 i ,. \ V . t J L r ‘-i~ h .2” "fl lu’\v x) 1"-) a 1 LC I" - {,V uni"), ) ) X ~31»; r A; 9.25:; y’é l 21 c- / h" ‘ Z L. 4'. :3 ' 4- '1 x I d I 5': "" \7 «J I p: Y a"); {J A M§~_ ‘ > f - A. 23 gt 5 6 my 3. (4 points) Find an equation for the tangent plane to the graph of z = e at (may) = (1:0)- So\o+;0q (USMJ +6\n¢-JI¢I;+ Vcc+or5> z : WW +";<“a,%3<><~a> 45.5mm Solul‘lon (amaler +lm3r4>t a; a luggage“) L7 sqm wag: Xbrl \J U Extra Credit (3 points) Explain the error in the following “proof” that all differentiable functions are constants: Suppose that z = f (2;,y) and y = g(:1:). Then, by the chain rule, 3—; = %%+g—;% = %+g—Z%. Thus 3—;% = 0, so either Q = 0 or $3 , ,, = 0. Thus either f or g is a constant function in By (1.1. If f (:L', y) = $+y, which is not constant in 1:, it must follow that g(.’1:) is constant. Since we could have chosen any differentiable g with this choice of f , it follows that all differentiable functions are constants. My Solol'ioni Tl“: Frau?!“ is ‘hk‘l' +l\{ Del-«+130 gor- Pq/‘Hc‘l AUI‘Vr-AH'V‘S 1‘: {- aw€c\. ,5, indcfowlch'f- a; ‘l’k Va‘egnggs in 'H‘g ‘Gormolq 491' 2. F9" “Verls SUPPDSC i= )H)! owl w:x-y. Wm EC¥,y)=¥*-y, bo’r %(x,w)=Zx—w: 3 In +kc 'c“’.")- C‘kkllaizl 1'30")" \‘n +14: 5(a)“) ~53"352. X ‘ 2> .. a :1 ba- d . Hue, ‘ch, v07 Q'i‘cyl' Inc f5 3% ._ 7-2;? gig-r (3—; .i; B 'l" 2L .n +5; ((Q')’ is wl'Hw raped 4-9 “M: 6l‘nalc cpbflJMg‘lC 2x}; (I c a} . o fl (At *‘5 g g OLAA 'l’l‘L 3%” 0“ 41¢ rral‘l- I} WI‘H‘ as?“ D c (.90 ma, )9), ‘ ‘ D): “V? «(‘5 ACSQWK“, Qua Caflocllfa] 'H'Lm h? ’0‘ 23'. A 30 “s 37 wrmfl‘ To be Nor: Ok’c'nvl/wc 910le “We. wi‘iH‘“ X=é )3?!“ l a A .31— 21:4 A . . rl’bm %:%G¥E+§f5— 7)? + HE?" «A 5"“ MM "C~mJl-J7a1w'~‘ 9094M. V I? g, o 0 fl )— $— T 9 < r- 3 9 9 3 ...
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sol5 - Math 2163 Jeff Merrnin’s sections, Quiz 5, October...

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