# buseneki120-2 - M E T U Department of Mathematics Code 3...

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Unformatted text preview: M E T U Department of Mathematics Code 3 Math 120 Acad. Year =2005—2006 Semester : Spring Coordinator: Muhiddz'n Uguz Date : April.29th.2006 Time 1 09:30 Duration i 120 minutes Last Name ! Name 3 Student No- C Department : Section Signature : 6 QUESTIONS ON 6 PAGES TOTAL 60 POINTS SHOW YOUR WORK Question 1 (8 points) Find the set of all points on the ellipsoid 2:2 +23;2 + "-12.v2 — 12 = 0 at which the tangent plane to the ellipsoid is parallel to the plane y + z t 120. .Le’: Wow} =y1+231+h%1-l3- Wm TIFMAJJ) :5 (\ormoA 3m k“ lth\ swgau. F20 0“ (1.3.3:). WCWG'A' VF -\o be poru\\c\ ’rQ 'Rzm'hﬁ. K7174 numb???) 2 >~ (and ‘3 2RD x=o omA 9:22: 4‘3: 3 {\I Q 8%: 1“ Guzman”: r41 821w?" :\1 2°”:\ ;:I-'\ .53 ’3‘ 7‘ Q’ (30min ox: (011,”) oné (O.—2‘-\) Question 2 (4+4+6=14 points) Let ﬂu) be a diﬂ'erentiable function with f(1) = 1, f'(l) = 2 and g(a:,y) = If(%). a)Compute gx(1, 1). 95min gift) + x 513; kg 013;; CMLLQ :- 300 + 1. §‘[\\ 1. ('13 Ji- g.- A. '\' 2 (’1') 5 ’3 b)Find a vector 3 in the nay—plane such that h(sr:,y) = 2:122 — 63y -+- 33/2 increases most rapidly when (22,31) moves along 7:? starting at (1,1) . M3x(03 mm) a M“ (qt .3) gmaqmm ease) - ~‘ 3 V v s \wk“ My ((059) :. WW \I at Olex,~51 (Aidan \\<l\f\\\\ 9:0 ﬁe Qzﬂ WM! Hence, told Maxtmm VA“. MM‘JS‘ \QV“ Q1 5° (.056: L ;¢/ A m ~ 0 Thus q ; E712)...— ;. 5 (4‘0) “&,[+x,‘u “qktiﬁlu 2. \A '- -5K1-“a ﬂ . . 7T 2 . . c)Fmd sm(0.002(— —— 0.01) ) approx1mately usmg the tangent plane 2 (differential) approximation at a suitable point. Hint: Consider the function f (9:, y) = sin(m(g — m2). 1”) 2 o o A 1 - o.°\\ '- ~ ‘ w am) a \$an Fem" ) «m wane?- L 1. g ' .. 1 1. 1' ‘- gvé—a mm 3W} :2 amen) 4%) 5 .2x (1,20 cm(M?;'m‘)=ru {3mm 9 0 8:1 C ox) 1m £00110) *IQJO‘O) (0.092).? 43(OIO) [Q 8( 0'®"’) N O ,- O -\- (EV’ (0.0513 'i’ " z =— (1531(0ng Question 3 (10 points) a)Find all critical points of f (my!) = 22:3 — ﬁzy + 3y2. 'L Y '7 szl :3 xso at x:\ 0 p) X 3x: 6x564“ :3! 53:: —6X+-‘3 // ' n- ’- n X‘ 0 =3 4:150 3 (0,0) @ LIA.) an. 36M U‘J‘WJ 90%“ X; l '5) GA: 1 S b)Classify all critical points of f (2:, y) = 2:33 — 65cy+ 3312 as local maximum, local minimum or a saddle point using second derivative test. 1. CU? L ' (2 (63—- (-6) (1,43, afwljr’lxa ~ ‘3 1312—36 >0 Question 4 (8 Boints) Using Lagrange Multipliers Method, ﬁnd a point of the surface 3112 = 2 that is clasaqt ti) the origin. 1 1. L 0‘90"“ ‘53; 0 90%" (ngcﬁ ’m We, magmas Nagasq x r; 1% mg 'L - - '. M. Hts": «in'wnvm H; MA “M3 i8 x1 avg-11"} \S (“\an 1““ , “*- heeA «‘so gtné; '1 1 1— «1mm vomit a; Sauna}: x13 +% 1-2:0 5u\01‘3(c-\ *0 %(~f_~j‘%): X‘QE . I «7370. 1 “3‘ pages; 03, «We. Suwéfatc x3l'27— era’s‘n; Chad’s DMS Question 5 (10 points) ‘1, X a)Evaluate/leylsin(z2)dzdy :- SWCX1) 6“ 5 I S 5*“(7‘136‘6 d x R O 0 K _ ‘ 1.) A! _. 5 %5m(\$ \Q ...._.. ‘ 0 gr 5 5‘x5m(i~133‘ o “‘XL K:0#)Q:Q "We £5..ch “Baud A! x A \ g. )1: S Stﬁud“ 5 Icosukg o ..\ ’ ‘Jzﬁoapcwoﬁi'h’W 1 1 b)After drawing the region for each integral, express / / f(m,y)dmdy + 0 xii-y 3 1 f f f (at, y)da:dy as one double integral. 1 I; Question 6 (10 points) Let R be the ﬁnite region, bounded by the line y = a: and the curve 3; = 1'2. Find the volume of the solid ovegL R bounded from below by the my—plane and the graph of f(.1:,y) = my. ...
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buseneki120-2 - M E T U Department of Mathematics Code 3...

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