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265_Final_Spring_09_Solutions

# 265_Final_Spring_09_Solutions - Math 265 Name& Section...

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Unformatted text preview: Math 265 Name & Section: Final Exam S2009 Instructor: Answer each question completely. Show all work. No credit for mere answers with no work shown. Show the steps of calculations and give exact answers. State the reasons that justify conclusions. 1. A particle moves in space with-position vector r(t) = cos(2t)i + 3tj + Sin(2t)k. a) Show that the velocity vector v(t) and the acceleration vector a(t) have constant length. b) Show that V(t) and a(t) are orthogonal for each t. c) Find the distance the particle moves between t = 0 and t = 27r. ( . a : ‘0': N: #254201th tzcebzzalﬁ \l x) \I i r \ 2 r ‘3 H v (HM : («2440th 4 3Z (lave) 71 ad) : V<Ce:(’m : 4%(26)? — a'sghaQK A H firm—1 ‘ ‘ '9 r 7 :: — ll owl-l ll : Manage—DZ Tesla-2602 = ultra (ZH-Um [Zeﬂ bl “(th ’OKC’V) ‘: (Sis-(429,3 5953 W4 “ 4 @1525 l) 0, ~q'sin(2€l> =8314<zecaczo -+ a *63iaCZt/G) (Cl—ll >> W¥ML>BCMl {75(96ch (2:77 2: Q r 20‘ , A ‘ ,— C elkleth : SD INCH/(4+ — 3% (513 6H = 113 -Z n 2. Find an equation of the plane containing the point (2, 1, 3) and the line g) m=1+3t,y=—2—t,z=3t. (IQ/‘1“ ' L K {kl/33> (1,33>><<3,-<,a> : (2g 3 : [/(gl4/37 ' I U520) ‘c 33\4‘ llglfliléjl ’ ’ ’ As 9 35 , 22 c *5) ’lﬁlk 3. Find an equation of the tangent plane to the surface 3:2 +asy +y2 + 22‘: 16 at the point (1,2,3). Lﬁ———-’—‘ \$<Xﬁ31233 v? :<ZX+3’ leZJ " " O " QWSV. I mama) : <ng/(O7 (in; no‘ Mn \$243804 93an gawk Oe<><~t5~13\$>oﬂ5lé7 _ : mm Dray—WU?” s f 1329 1-7 4. Find all of the critical points of the function f(a:, y) = 3:3 — 93:11 + ys, and classify each one as local maximum, local minimum, or saddle point. 5. Find the mass and the center of mass of a thin plate with density function 6(x, y) = y occupying the triangle with vertices (0, 0), (1, 1), and (—1, 1). 6. Convert the integral below from cylindrical coordinates to an equivalent integral in a) Cartesian, b) spherical coordinates. DO NOT EVALUATE. 7r 1 m (rslmsll PAZOerIS IQ\$W910KV am I = / / TZSingdz dr d6. l but 0R. {LSD CXD‘r‘MN—B Z:"%»(‘ 1 ’k 4 ma or 21: 3&159 <P CM- X2? 53ml (959 D 2 f §{“é§fka 2 2 60) CL [DrkﬁbeW‘ﬂ’ .‘IW-r K F \XWN’M afﬁdawces 7. Let S be thE pherel centered F = (3:322 + 2yz2)i + (323223 — yz2)j + (3m2y2 + 23)k, and let n be the outward pointing unit normal vector on the boundary of S. Calculate ffF-ndS.-'—' SS§V~FAV 3 2 1 , ’i 2 5.21 Vs : 5.x(3X214ZJLé/fi 4%D(§XIZ§_DEZ)A§%<gxl:lz+z > ; g2 C£ Jrgg 217W”, 2 a171,, 2 ,AAMJAQ H W) of) op r O ngzzv-F ‘W 271 g 49:? 1 r. jzwjﬁ 9- Ffzwsz‘bémdg _ )0 L5!” 46 O D F :o __7__V_A_) w ’31 Seal“? 3 r (fowl —S,,{CAU Agnew» Zita”? 3 ': Magi) 8. Consider the vector liéld F(ac, y) = (3:5 — 31,22,453 — a) Calculate curl F and div F. b) Use Stokes’ Theorem to calculate [C F - T ds, Where C is the triangular path from (2, 0, 0) to (0,3,0) to (0,0,2) and back to (2,0, 0-). . ‘ r K 2,393 2% 4r} ' ‘ *2 34o -\ : @ (31-3)+%(2z§*%%["lx—Z—d Pg : UN'\¥"Y\ : S (,ZFL{'(7 a<éé<gi ( 3/- Cr / R If. 2 . l» U 3' \ C ’[Z-FZJAA : '8‘ ,I 03:3,?“ 2 E S “(0 Lil / -2 24:3 — r w E7: AA F M3327“: WOW : 21 mtlmml : A, <33?) ...
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265_Final_Spring_09_Solutions - Math 265 Name& Section...

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