MTH203fall05.Exam 2

# MTH203fall05.Exam 2 - American University of Sharjah...

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Unformatted text preview: American University of Sharjah Department of Computer Science, Mathematics, and Statistics MTH 203 - EXAM II Fall 2005 Name: ID# Secti0n# Instructor Name: Q1. (12%) Letﬂx,y) = x2 —’2xy +y2. Find the directional derivative of f at the point (—2,—1) in the direction from P(—2,—l) t0&(2,—3). "EM Z 2 x ’13 2: 2L} —/L i V\ ij axzx *7? fblr"; ‘TH “Tl : L ' VGQFQI") jz’2)?—> ' £2?er 29:952.: (V W < MID/aggronZAZIZ>‘<%§fdjé> : i —% 13%) WE l\ —/ X; (9 ' b) What is the maximum and minimum rate of change of f at the point (—2,_1)_ 4/ Mueraifchlwngegi(;Z/7>l:xiqg :25— /l/ wuim (- ; :’f<‘12>(:’2f2: QZ. (8%)Use differentials to approximate ‘/0.99 ea”. 79/X.w):J?e:5, 29/ I, 90‘ o 2 M, ,w 9 m 23M; 1%»)+~f(»o)/>[email protected] / x 430,0) :1 9 ﬂ ,. 9 5; _L ,2: J7 C @013— i g — ,5? gala) L f3 OaO'Z Z 1/ Data 6 1’1 + ’ (50> I 5 Q3:(18%) Find all critical values _of the function ﬂx, y)=x3 +y3 +3x2 — 3y2 and classify each as a local maximum, local minimum or saddle points f <7xlféXZO 3KO(—r?)=’0\$ X:O/X:——Z X : 3‘3/‘0—r/2):2a?’"ﬁ [7:019 1 / 2 f :- O 6 (ya) (6,7) plan) :CEMQ 3 >0/ 49x10)” 6? gig Q4.‘(16%) Use Lagrane Multiplier to find the maximum and minimum of the functionf(x,y) = x2 + 3y2 + 2y on the unit disk x2 +y2 5 1. - ([0177; ?(‘}{7v”)) @ Xl/mlfk ’ F. (9% xx:oox%wls: do; OUCIV‘CE 92 i) évw- (Dim (9,4) 2 A 70 - m @ v“ 7‘ “:9”? sum, A’ L 9) 10:, h {wig/g 3W3” LM " (02' 2 gum. War-Vt 19) @ ﬂ LOU/“L (gm/Ozi xhr J5! : ‘7 Yaﬁ HQWZVV <% ‘ L MW VD 015.13%) a) Change the order of integration for [1/4 I ﬁ/(xwwydx- (12%) b) Find the center of mass of the lamina bounded by x +y2 — 2y = 0 and x + 2y = 0. The density function p(x,y) = 1. (Hint: m = H p(x,y)dA, ‘M =j j y p(x,y)dA,My 2:} j x p(x,y)dA, 4’; 4m 4 2x—x2 06. (12%) Consider the integral ex2+y2dy¢lx 0 x (a) Sketch the region of integration (b) Change the above integral to polar coordinates. ( Do not evaluate the integral).) 1mg ' C“ ’l/ 1 v1 {0" DU? 6 {Av/Cl (9 "3 e vd( O D tar ii Q7. (12%) Set up a double integral that represents the volume of the solid bounded above by the parabolic cylinder 2 =x2 and and below by the region enclosed by the parabola y = 6 ~ x2 and the liney = x in the xy-plane. ( Do not evaluate the integral) ...
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## This note was uploaded on 10/20/2008 for the course MATHS MTH 203 taught by Professor None during the Spring '08 term at American University of Sharjah.

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MTH203fall05.Exam 2 - American University of Sharjah...

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