MakeupExam3_Solutions - Exam 3 Math 408D Name Spring 2009...

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Unformatted text preview: Exam 3 Math 408D Name Spring 2009 TA Discussion Time: TTH You must show sufficient work in order to receive full credit for a problem. Do your work on the paper provided. Please write legibly and label the problems clearly. Circle your answers when appropriate. N0 calculators allowed. 1.(14 pts) Find parametric equations for the tangent line to the curve traced out by the vector function m) = e~t§+ cos(t)j’+ (t2 + 4);? at the point (1,1,4). 2. (14 pts) Find an equation for the tangent plane to the surface 2 = 49:2 —xy at the point (1, —1, 5) 3. (14 pts) The radius of a right circular cone is decreasing at the rate of 2 cm/ sec and the height is increasing at the rate of 4 cm/ sec. At what rate is the volume changing at the instant when the height is 10 cm and the radius is 6 cm? (The volume of a cone with radius 7“ and height h is V = §7rr2h.) 4. (14 points) The temperature (in degrees Celsius) at a point ($,y,z) is given by T(m, y, z) = sin(:c + y)ez. (Assume distance in 3—space is measured in meters.) (a) A particle at the origin moves in the direction of the point (1, 0, 2). Does the temperature increase or decrease in this direction? At what rate? (b) Find a unit vector in the direction the particle should move in order for it to see the most rapid increase in temperature (again assuming that the particle starts moving from the origin.) 1 8 5. (16 points) Find all critical points of f (ac, y) = my — g — — and identify y each as a local max, a local min, or a saddle point. 6. (14 pts) Minimize f(:v, y, z) : 4x + 23/ + 4z subject to 3:2 + 3/2 + 32 = 36. 7. (14 pts) Show that the limit below does not exist: 1' 3’2 1m 2 (saw—40,0) 5'3 — y Bonus (5 pts): Show that every plane that is tangent to the cone 3:2 +312 2 22 passes through the origin. M‘W‘XD I“va WW 3 SM'IIM \. {fit Cast)t1+q7 ) ’/ gar It): Z-c't, ~s/th)at> Flt) PW +Lefomf 0,1,4) Lab.» {3:0} 30 a, $0M».an MW 9m {L4 {"3 F4103: 4-|)0/0>, PMWLQQIMW Q{X"l>”‘ (Ni-l)—(g—'§):O m QX-ta—z :5- - elm)no)(.23 #1317). L4 VT— (X¢3,2 ) :<Co§[¥+‘3) {E} 093(X+3)~C%) S~L(x+j)cz> M VT(O'0'03 3 <1) ‘J 0> <43 A w mm m m Mom. 4% (0,0,0) Mum) ‘ -‘ I Q! ,1 IS u;-V-_S:-<,)OIZ> I 30 A W ("m M W6» (3 VTK0,0,0)-E : (\l),o>~ZI/fi)ofl/V§_>: 1/“? SofL’ WWOM w;qu Jam 4) [/yg abywoé PM W. 1L4 [WWW WW UM “4) W a} (0,020) / so aWMWmMo’WlJ1NF'S fiéllilo>' g 6 . (¢ = _ __ LYfi) x: ,)L( E: firer wwal 4—0 M Md Pom/43 f _ J. * gfxl 1(23’ ¥+§~L S‘Avlw E; j+'/XL:0j (gt—wt 3/ :7 7C )H/‘jlco 7mg :0 ny‘BL 3:7 7C*3X4:O x(/+z(x5):a Mm be ,m JLWVM (3 m MM, go m7: Qw‘t CUVVJSPM "ll—04 M'aJ/Jomf, x3 lb X33 “’5? 1x3 3 ‘ ‘49. :10 So D<X¢¢3\ ~/ l DUé, H) :Lfi—K—S‘Z'L»; ; 444=§>0 Mal {a C‘lz/wB >0 , $0 £GVZ,'LI>\32 W~ ¥(¥ltj,%): 4X4Za+4g g()‘t‘3,£3: Xz+bz+ gL—BL V¥(\u3,23= 44,2»)4> Vé()‘,jla)=(4x133’3’i> UJJ Wacox 4—D JIM-6‘ 9M 093,2) SWLxM V'FCXH},Z) : )Vé/XUDIEB W x2443va g1: 34 ‘ SM" 442%). 7:7 X>CZ aura (11w Jib/W :1“: (33» =7 bk: ‘1 M ’Q‘WJWM =(92)> - L I L 2") Z) Z. )(lca’z X+b+z=li Mwm-g/wo 414/543: 4~4+2'3+‘4"‘ 3 3C 'p(“7'1"/"’) : ‘34 MM w W W {,9 MM a #4532231 1’3 flow 7‘44 M.m~ w/Lq. [3 #30. ...
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