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math102_m2_key

math102_m2_key - vim“ sat-mi s1 “oilizs Bogazigi...

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Unformatted text preview: vim“? -.- sat-mi. s1 - “oilizs Bogazigi University Department of Mathematics caICUIUS points each Date: May 7, 2008 Full Name : EI If“ f Time: 18:10-19:10 -‘ Math 102 Number . ' Student ID : 'Tha‘" | ,— Spring 2008 — Second Midterm Examination l IMPORTANT 1. Write your name. surname on top of each page. 2. The exam consists of 4 questions some of which have more than one part. 3. Read the questions carefully and write your answers neatly under the corresponding questions. 4. Show all your work. Correct answers without sufficient explanation might not get full credit. 5. Calculators are n_ot allowed. 1. Express and in terms of r and 3 if .T ‘-w=:.-:+2y+32 ,I=-—~ ,y=r2+lns ,2221'. s Sol 0 Wﬁ Gradua/ 9»« o. ‘57: Q; +9.2 3X Br ’35 3r" '2 f -; (A? 1L1» 'hihze 2. (a) Find the derivative of f(:.:, y, z) = 3:3 * my? — 2 at Pg[l,1,0) in the direction of A = 2i — 3j + 6k. Sch 2. (b) In what direction does f change most rapidly, what are the rates of change in these directions? I i I i a) M3515..- :: l (1i'16_+6L-)Fhﬂr “All ’r and w; Mm Vﬁéxwﬁ): 43:3-31) —2xy/;4> 6°; VF(414)0): <Z;—2l)-4>_ Luna Dru¥(”{"2°) “'5 ? ea;— , :. i .é.__ C a, 7+9L 3‘ :7 b) ‘ .F Mcrqugg W034 th VWCU’4103—‘1-(‘2/‘7-2'4) and me F (4m) = wme 6950 =2 N VHMN) (I z.» 0 F dccrtwu my} (7915“; ' [M m WOHon .—~VF(4I7/OJ .1: (—7.,1/4> one} D— Vic'{‘ﬂ‘7/0) ‘0 (1,410) a m n- = W- nun—- H‘— (4:430) 3. Find the absolute maximum and minimum values of ab; MM ﬂay) = 2+2\$+2y—:r2 -y2 on the triangular plate in the ﬁrst quadrant bounded by the lines at = U, y = U, y = 9 — 3:. £3913 gm“ r: Don-P (Mal M-p-lﬁk Ts [fC(."o.r-(¢J”mo/ bcowda/ f album ﬁr: absohk max“ ﬁnal MIK- mﬂmW-W ch 4’11!— ' 33.18."- (rr/F‘daf [99M 1?: (Xis‘j)=(414> t 7‘ M M; trt'stIK 090ml" M '44». 1M WM!“ . \$0 we Chock (014)} (Of-'3); (W9) I3) we have. 3:3ﬁx MA x6 [0,3] an! W‘L [OWL . wlxl = 7, + ’Lx + 1(9' ’0 —.x"’- (9-2013: .—22<L+l‘5x —(;II I _ So we cheat (9/ 19/) _.... W 3‘ :1". ---’-«X r: ., .—- , 3. ‘3 w 2. A> ( 3 “(a p '> x “i (m) ,. cola) 1 _ ,- \$28.1» "Baits 4-32' “29 4. (a) Evaluate the following double integral I: n 41, dydm. 4. (1)) Write the integral f fR(y — 23:2) dA , as iterated integrals I ﬂy —- 22:2) dydm and I ﬂy v 22:2) dxdy with suitable limits of integration, where R is the square with vertices (D, 1), (1,0), (0,—1), (—1,0). Do not evaluate them. - :haa- éib. a) w; rcwni‘f— q ...
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math102_m2_key - vim“ sat-mi s1 “oilizs Bogazigi...

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