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test2sol

# test2sol - University of Waterloo MATH 237’ — Calculus...

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Unformatted text preview: University of Waterloo MATH 237’ — Calculus 3 Short Test 1 — Fall 2006 Version 2 Sections 01, 02, 03 Date: January 24, 2007 Time: 5:30 p.m. - 6:20pm Duration: 50 minutes , NO AIDS PERMITTED N O CALCULATORS ALLOWED Family Name: M Initials: __ Id. Number: Signature: Instructors: C] 01 L. Krivodonova {:1 02 R. Moraru D 03 B. Ingalls Instructions: 1. Complete the information section above, indicat— ing your instructor’s name by a check-mark in the FOR EXAMINERS, USE omy appropriate box 2. Attempt all problems in the space provided. If you require more space, use the reverse s1de of the pre- vious page. 3. 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 solutions. Justiﬁcation should be provided by referring to deﬁnitions and theorems where appro- priate. 4. This test has 4 pages. 1. M O l‘“ {(1,0) 2 5:30 ’32? (a) Verify that the function has no limit at the point (0, 0). I; 1‘1 "W ﬁx 0 = "‘ "‘2 X40 I X40 17‘ Page 2 of 4 (b) Let 1: = 2—537 (\$,y)#(0,0) 9(a) {L (55,9):(050) i) Find a value of L so that g is continuous on the plane R2. 5‘.an a U raj-Md 1'" ‘IJ Oo‘vJ‘ii/luuuf 5% a.” lamb [A {‘ltf 3‘ (Lemma féWraWLlr-V- €3<¢I7¢+ pal? (0(6). . “M L _ I Aﬂonwakiwg (0,9). alum; 7:0 ) \-Q ‘FU‘A 1 — Oi 3 Sinai we dirk +011 +1; (Mi)? exiJ‘H, we prune] 41 vent? W H1 [land {3 111m: 1 .231. \‘ -_— lxi “’¥‘ml in+71 10— 35%), = £ Ix. a! 171) 5 llxl'ﬁjL! Sina hm 1*130 H4 Squeek WW M‘A‘rwf (mﬁeaw) ) W “H: limit LI 0. SQHU‘U L30 HIUHJ in ' I % \J’l‘NJ Cax+2nuﬁus A71 (016), i ii) Is your answer in part (i) the only such value of L? Explain. (OVEN/‘43 03" (0,0) regutr-CJ W +Lz VILLA of ) iJ aﬁL/wl +0 ‘Fﬁl lim‘f. 5mm ‘H4 [Ind 15' 1-04 L WWI“ LE 8?!ou 4-2: 3.2m. 010:!) I “lam-1L3 L Page 3 of 4 2. (a) Consider the function u(a:, t) = 008(5): + at). By taking partial derivatives, verify that the function u satisﬁes the partial differential equation Bu Bu _ = a— at 81: ' ' - '2 U6 '[ﬂvul 93%: = 51 C41(¥*4{) s . If t * Sin (7c +a{) 3t (X+A“\ : -a m (W10 “VJ "i": : —- fin (x+4+3 531(5‘4-“0 If : ~5in (7‘+a{.) W a 23%: : 3% AS require/X. 3’ (b) Find the equation of the tangent plane to the surface 2 = \$2 + 3331; _ y2 at the point (9:,y,z) = (1, 2,3). Vi‘l'ix {(\$30 : 7(1-6 3X3 “‘31 i N 7. M +W‘11 '3: :11-43? “ML (‘3'? : 35(‘27 TL“ glhn=2+é=8 ) 9;:(I,1)=3—L/=vll ﬁgl f0 H1 ‘iMa-ei Fiend. ‘l‘ 1': 16“,?» 07+ (’11,?) Page 4 of 4 ...
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