Test1KeyYellow - MATH 251 Fall 2006 RAICH EXAM 1 —...

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Unformatted text preview: MATH 251, Fall 2006 RAICH EXAM 1 — VERSION A Name (printed): 0w Section #: “On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work.” 1 Signature: UIN : DIRECTIONS: 1. The use of a calculator, laptop or computer is prohibited. 2. Please present your solutions in the space provided. Show all your work neatly and concisely and clearly indicate your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. 3. There are questions on BOTH SIDES of the paper. Please be aware of this. DO NOT WRITE BELOW! 1Disputes about grades on this quiz must be handled the day the quiz is handed back and must be discussed before you leave the room. If the quiz leaves the room, it will not be re—evaluated (except for possible adding mistakes). 1 1. (a) (6 pts) For what values of c is the angle between the vectors (1, 2, 1) and (1, 0, 0) equal to %? Q: ————————-—~ 21:5—3 Ar\ -\ Co S = (\s—xz, ~ Gm , (0+0 = <3,-7,°\7 I ¥9\ %\ \43\—7.°17\: «imam *1 k“; E? l .3. —j_ ‘1 m<tm )Jrfimi—JTTR) (c) (8 pts) For P(0, 1,2), Q(2,4, 5), R(—1,0,1), and 5(6, —1,4), find the volume of the paral— lelepiped with adjacent edges PQ, PR, and PS. PR: <‘\i“\n '\7 PE 3(6), £2.27 PEXPR‘: a 3: \-§\ : <0 )- ~Z+~3\)“Z§:$\) —\ -1 q = (Oth \H <QW1‘7/v- 401..\\..\5\ = land +1 -A & x T Q QRXPS w\~\ 3 q \ = <~1~1\—K~1¥e\, 2H)» 0 =1 L : <~L'\y “Liva \J > \ 42‘s 3» L'Herfl: \~ ?—\2+M\ >44 2 g 3, . A q A : 2/(_2_7A:sk~1+b\ +39%} to :2 z -<g~\1 9% = L\ 2. (a) (8 pts) Find an equation of the line passing through (5, —2, 3) and parallel to the line m;3:y+l:_i3. #61th x-s‘q 2‘ 2-3 +3 W’ 0M \I=-2*‘r '5‘ Y‘“ ’ 2? 2:3’3‘l’ DY Vwr OK %’1 llvxe t6”? olL (b) (12 pts) Find an equation of the plane containing the points (4, 0,0), (6,0,0), and (1,2,1). \3 a R (Yd: <z\o,o7 +2 L: ‘7‘ 3. (a) (8 pts) Describe the level curves of the function f(m,y, z) = m2 + y2 — 22. You do not have to draw pictures. \bo szz+qz~22 wwbobh} 89v one sheet +3 \L=—o 21-: m2 come » arl \UO 2 lim my ( ) (0 0) 2 2 + 3314, if it exists, or show the limit does not exist. x111 _’ y (I: (b) (12 pts) Find y: W“ -\— 5 PM W W‘ a W aw» Leo 3517mm o\L VJPO 107W“ (:st : 203:3 a;\) L: \(5 Wm X Y2 +2, (and. (whfimm Z‘fl’EVH DNE 4. (a) (8 pts) For f(a:,y, z) : 69592 sin 2, find fxyz(x,y,z). 1 «Wm = V1 6)” 3% 3r2 1ny (MN = 2x, ex" 3M2 Ar 2m} QXYISME +3 QM LX\V)\E'\ = Ry QXYCDSZ "r QxYEgXYZosE +3 . 82 z . . (b) (8 pts) Flnd 5—5 and g—y 1f 2 : f(a:, y) and satlsfies avg/2 — yz + m3yz2 : m + y — z. me Xvi—v2 +><§VEZ~WE Ayah gix ‘ '2. 2 5 ~ (WAX 7’ V +3XW21~\ of +3><WZ +2XSY?§Z§( —‘-\“%'2§< +4 aw, = M ~‘a W224 H g); _ Vaaxwzhx +7- 3 - 5% = -v +Z><3yz H H X WM? -\ am 1 z a ' a ,9 65-2 =—_.l =-V*3XV2\ way 2 -2__% X32 Nagy- [bx (mat A? REA/QM +1 V X‘/ by ’« Ci»? +2 : 77/ a2/ A)an — 3Z- 5 2 — ‘7'— 3 .3 Y : __ RM; 2942 \ i ; xy Z+XZ \ y 5"" \/_ XEVZ _,\ ‘\l +Zx‘vz H d” gomubk f)k 5. (a) (6 pts) Find the directional derivative of f(:1:,y,z) : 22 + $393 at (2,1,3) in the direction of (—1, 1, 2). v1; = Q3, 3w“, 22> +2 mam = Q, (9,97 +\ a- <~\‘\\7«7 A. .L 1 u-——-——-—- ~. ' ~— 'M ‘L~\\\\_Z’»\ < V(I‘D/WEI 4 e e\ Oagkth's‘; 1 ‘ L‘éA—(L‘ ’ = “52*:2 = [7... +\ H b (b) (6 pts) What is the maximum rate of change of f at (2,1,3) and in What direction does it occur? Wm rate ogr Charge B \Vfrkztm“ % Merge, “Wk +Ll \ L (o VM'Zdig‘) I “8th ‘5 — s l W &, < BE ) {‘73 ) Ry) (c) (12 pts) Find the equation of the tangent plane to the surface 9:2 — 2y3 + z5 + wyz : 7 at the point (2, —1, «1). Hwfi = x2-1y3av23w2 W = {25042, ~ey3rx2.52%~xy> AH V ? kZ\'“\\-"t\ = '\’\)'C9 “2.; 5 :2,> = 45,-3.3) W 5(X~Z\ — ‘3 Wm AJSEM ‘»O M 0‘ 5X —ES\(¥3>2— = ~\ 0 +31 31;"3 : ...
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