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midterm-03-f

# midterm-03-f - University of Waterloo Waterloo Ontario...

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Unformatted text preview: University of Waterloo Waterloo, Ontario Mathematics 237 Mid-Term Test — Fall Term 2003 Duration: 1-;- hours Date: 21 October, 2003 NO AIDS PERMITTED NO CALCULATORS ALLOWED Family Name- Initials! ID. Number: SignaturezL Instructors: EZK 01 W. C. Lim (2.30pm class) 1:! 02/03 A. Kempf (11:30am class) Instructions: FOR EXAMINERS’ USE ONLY Question Maximum Mark 1. Complete the information section above, indicating your instructor’s name by a checkmark in the appro— priate box. 2. Attempt all questions, in the space provided. If you require more space, use the reverse of the pre— vious page. The marks for each question are in- dicated. Marks will be deducted for negligently presented work. Your grade will be inﬂuenced by how clearly you express your ideas, and by how well you organize your solu— tions. Justiﬁcation should be provided by referring to deﬁnitions and theorems where appropriate. 100 3. This examination has eight pages. The last page is for rough work. Math 237 - Mid—Term Test Page 2 of 8 Fall Term 2003 33y "H I)" x4+ly',<x,y>¢(o,o>. m (x 'v . 1. Let ﬂay) = [8] a) Show that lim f (:13, y) exists. Deﬁne f (0,0) so as to make f continuous at (0,0). La we a. WW) We Emu. XH+ L3) 2 )5) )(éﬁﬂﬂ) 7/ X'j‘ Xu+h| XH+I\3‘ X (Kidd) 7 Exﬂ\ r———‘— / X4 '+'-«J| X ’E‘)‘ MM 7 13M gin/I [x]? M ‘ I + ‘ “1‘53; u. Igwr “4’ a3 S g 5 'y', )im (x! :: [Em 4.2314. mm W WW '1' IX]: 0, ,ﬂ. en‘s}; 6M0! 6.74/01: a @ U W x“?! [12] b) With the f (0, 0) deﬁned above, determine Whether f is differentiable at (0,0). State the deﬁnition of differentiabﬂity as part of your answer. t} a «Chet-m fzm‘amttdiumtttt “4 a, m and ~§§ mu\$){¥{5£ 3) Jim 191.365. :gl 0le R510): £60“ Lafx) 35'ch! m Assumh-s3 4X: (3de RA sow NMHWLM' Q“) ~— Alﬁ), E\$ «C kﬁs Cm’hMOuS parka? dﬁyﬁv'l‘k‘wt’} 51+ 95, ~H~Qn -p (S d;‘p(4’ﬁ4*hbll 4+ (Q, Lw’nLj “4 S: Ea-szkﬁ (\$011))" +(">m) (x44 {drab-113)) : ,_.___L M Xena] m ﬁr (mﬂﬂqo) E K)’ ( "I -‘ ‘4 5‘3 A ' X x #3!) 4. (—Mﬁ (Hufth _ X a k x"+— (to) gum“? “(W gt“ CM m 40 «Cr (h wimp», 4% Am (rm-Juli: we (when; “elm” >93. ,; _ F (r 1,3334%» 2:.) 104,53; 3;) Iggy gag): gI/Y Tlms' “4’ q” (59.3), ?5=c: . x, 3‘1 nut {arwﬁaﬁwcuy t3 4L arm Mm! 4km“; ﬁ ,3 dome/amt «4 (0,03, Page 2 of 8 Math 237 — Mid-Term Test Page 3 of 8 Fall Term 2003 2. Suppose f : R2 —> R has continuous partial derivatives at a. [4] a) Write a formula for calculating the directional derivative of f at a in the direction of the unit vector 11. Du vi] 5:593 : (7(0le 0 1;! [8] b) Prove that the largest rate of change of f at a equals IIV f (a)||, and occurs in the direction Vf(a). m [0'31‘3 rat]: o‘[ flu“?! 6"] ‘c 4'] ,0 ls “[Le [6177“4 MM (‘4‘ D“ ml“ 9."; arc.” q Fosﬁaﬁl til (U' J u) :3! L' [Vii f 5 “1 "" “9;; a rfﬁ‘grﬂ‘ - a. I l/UC- [WAS] RN) Maxim...“ bur (9.)) (9:7) 5 (aqua) WLIH -[Uthuk’ 2 [ , 3f M ’5; (‘C’]‘k»{, + M-7(g\'u1 a)»; [8] c) Calculate the rate of change of f (a), y) = ye“! at a = (0,2) in the direction from a to the point (4, —1). 3- 3w = 316:”, : :: \OE '3 a). (513 L]- N )‘U o. . VHQ‘}: (4, i) ‘5 H v: (Li,—I\~ 91,; (L, -I‘:»-(o,2‘5= (LIV?) l «y c ~51...- ~ beh., 2 __L_ r t] U‘ “\l“ . {low V 5 V ‘ ( 5; ‘3 “I. bu [:(g‘B: v 'F (54‘) " ‘ (Lu) ’[lwxztlm {m pm! («.33 : M. 0. ’ L9. _ 3 e3 ‘3 . ‘8 ‘ a '24] \\V9mv Page 3 of 8 Math 237 - Mid—Term Test Page 4 of 8 Fall Term 2003 ‘ 3. Let g(a:,y) = \/1+x2 —y2. [8] a) Using the linear approximation, ﬁnd the tangent plane to the graph z = g(a:,y) at (a,b) = (2,1). g Labs) 2 0) (Q + 9X (gﬂx‘cﬂ i— 35(5‘101'9 #1. W: r (We 3 , ,__X_._ ( m:* 1 54) 2 .___3____ = 5’ JET—- “Jr ﬁalx‘i =‘ :lﬂwhwil ‘43?» : 4.2L... JJ'TT’I :: _'\*)( :3 \. J ‘1 ﬁg“ L065): 9+(xra\+;&(u,—|3 = ><+ 1% i / ’lﬁ’l'arfmlplam'l-o 669.)» 4+ {(3 ‘15 2: )(+ 31/31) ~i, [4] b) State a theorem relating the level surfaces of a function f : R3 -—> R and the gradient 0 vector Vf. Ci WW?) i 0 FM ‘l: (x i.) 2,) :k and 'c'hﬁ WK“; .Et, 4L; mrgw ‘CplC (an 134 gﬁdn L): ""361 («9- (2s ‘ J z (3% [8] c) Write a formula for calculating the tangent plane to a level surface of f : R3 ——> R at a. Use it to verify your result in part a). m l“an 00% lb 6‘ (Ml Wlm all {lrm‘am him ‘WLXk A mm 6; 91V” a), 4;, 9w = " \ Page 4 of 8 Math 237 — Mid-Term Test Page 5 of 8 Fall Term 2003 ‘ 4. Assume that f : R2 ——> R has continuous second partial derivatives. Suppose its gradient and Hessian matrix at (0,0) are given by Vf(o,0)=(2,—1), Hf(0,0)= . Consider g(t) = f(sin(t),sin(2t)). [10] a) Find g’(0). Lari“ L4:§ii\ V: 51MB?) 9 ‘. 3&3“ (“NB {1/ \V l 1 5-? wt i; ii ‘ ‘v - ' ‘ 3&3: .5: if + 5; fix CW“ I} mm 0 n 53oz, afar ' (“)9 aim LU» ﬁg 3 (as 1E ’ 31.x: 2C05t ’30“ (3- g (3 w o vcos ; (0).:- DCDS 0 3 D [10] b) Find g”(0). .2 1 . 5' (a: Elf-4w 8.5.15; — grew bu OCE+ a (7 EM» “4040\“4 @4114” H ’23} m 5 L“ ‘2‘)“ O’u g i ’1 'ﬁinz Li'OrL out 1 W. 9 Page 5 of 8 Math 237 — Mid-Term Test Page 6 of 8 Fall Term 2003 [6] 5. a) State Taylor’s theorem With second degree remainder for a function f : R2 —> R. Mun} gr 4“”?7-9 R, C1 (2+ apw'vﬁ‘ (Qt “FA {h Sent Mi'ﬁi‘bm‘ieao‘ 6(g3} e 8&2“), C "kt CQSNIA” ‘C’Lm ﬂick 4ka+f N £04 «9 2 43(9) + V-FCQOO (xv—19\$) + PHA (x3 WLM 2,915): 5'7er (£3 (x4334 gee) (gamma) + «Ewejarnj, b/l, [14] b) Let ﬂaky) = Show that WW!) — L<0,o)(\$7y)l S E0152 +y2) . L9 (253: 473134 «C; (5)0140 + C‘nyng; 60, bohkp} 51"} {7(J“‘tl5 :1“ 'F, M ﬂyGA): 3t: (l-FX143A)—i ’px [+0-+CJ)"1L (3-03 a go Sbmuﬁl I!" (93: {Y (\$331 "' {l " g)! {is}: "v Lﬁmﬁiﬁ : \$4on % @(xieﬁ + 00;— (2‘) -I\ g W "M AMM l “943’ 4} 4— 71 (Ar ‘\ \ \\1 Page 6 of 8 Math 237 — Mid-Term Test Page 7 of 8 Fall Term 2003 [4] 6. [BONUS] Prove that if f satisﬁes |f(a:,y)l S 51:2 + y2 for all (23,31) 6 R2, then f is differentiable at (0,0). ‘ . O Page 7 of 8 ...
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