M 408M - M408MF09Exam2 - Fall 2009 - Rodin

# M 408M - M408MF09Exam2 - Fall 2009 - Rodin - Exam 2 Math...

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Unformatted text preview: Exam 2 Math 408M Name Fall 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. Write your name on this sheet and turn it in with your work. Please write legibly and label the problems clearly. 1. (10 points) Find all values for :0 and y such that the vectors 17 = ;+ \$j+ I; and 117' = 2?- ;+ y]; are perpendicular and have the same magnitude. 2. (14 points) Consider points P1(l, —6,2), P2(0, —8, 1), and Q1(1,0,1), Q2(2,2,1), Q3(0, —5, 0). Determine where, if ever, the line through P1 and P2 intersects the plane through Q1, Q2, and Q3. If the line does not intersect the plane, ﬁnd the distance from the line to the plane. 3. (12 points) Let L1 be the line given by the vector equation F(t) = (1 + 2t, —6 + 6t,2 + 4t) and let L2 be the line given by the vector equation ﬂu) = (3u,4 + 911,1 + 6U). Determine whether the two lines are parallel, intersecting or skew lines. 4. (12 points) Let (‘1' and gbe vectors such that projg" = (1,—1,2). Determine each of the vectors below. Justify your answers. (a) proj25c'i (b) proj_5c'i (c) projg2d’ 5. (12 points) Find a vector equation for the curve of intersection of the cylinder y2 + z — 4 = 0 and the paraboloid 1’2 + 33/2 = z. 6. (12 points) Sketch level surface f (3:, y, z) = 0 for the function f (ac, y, z) = x2 + 422 ~ y. Show clearly the traces of the surface in the coordinate planes. 7. (12 points) Let F(t) = (2,sint,t), 0 S t 3 7r. Find the point on the curve where the curvature is largest. 8. Let 7"(t) = 2133+ cos 2t; + sin 2t]? (a)(10 points) Find the unit tangent vector and unit normal vector to the curve at the point (271', 1, 0). (b) (6 points)Find the tangential and normal components of acceleration at the point (271', 1, 0). Bonus (5 points): A particle moves on the curve F(t) = 3ti+ 253+ 61;, for 0 S t S 8. How long will it take the particle to cover (bth the length of the curve? (Assume length is measured in meters and time in seconds.) I 7. 3+‘H45H3 :ij 43= -4; =7 YSQ‘ I/k/z q/L/ E “WC. 0: ‘ l’ 4— ‘ 2_ rs W a )0 ‘ ‘2 L‘ l -t 9‘44“)!ny [AC‘JTH iljz: ﬁmx : l ( IS {FAD-“1t 1—! )”2) —I >3 2::a—t WWW OQLW b3 Q. IQL, Q3 I’m; (-ﬂ)-§l‘h {Le F133» c725 x «979\$ =<i,a,c>7‘< . '1 la 17% _ [’Z. ’Zgj-T/—{_n/ =/ 72ie/: L{—§-I 2. hi ~55 -) +I: I!) '5/ 9=XL 7X 7Aaoe. m )(% ﬁlm 36% #230 :3 cfoﬁj (0,03, 2 7AA“) “A. (QM/vb FMOULJ 40M rC/L Xzflmwx Mfg») X “Mac-“ f V5 x (/7. . - ”/0 ‘ ”s )_— r So HA W TMW WW ! TU; [tr/(UH :2 iii/t): I}: < 1)—%2L, cub ;[:llt)~ %ZQI-an2t) —Qé~1?t> L HT’Ic)H = {er— W = 3; SUM/sh W [OW WW (9”) I ﬁn): gwm) '7 pf”): 40,-),0> u) Q W \ \ (‘f\ V N 43 4.. .43 W (I OK) 1 (T (wwoq’th \$0M 51m: \$1 :13 ‘5 4 3/ 3Cl+t3 L~2 :13 (’4' t )3/2': IY/L Ht: : (Iv/L>773 t = (1;)2’3 -I ...
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