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Fall 06 Exam 2 solutions

# Fall 06 Exam 2 solutions - Engineering Mathematics(E35 3 E7...

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Unformatted text preview: Engineering Mathematics (E35 3 E7) Exam 2 October 18, 2006 This exam cantaias six 'ﬁeeﬁrespbﬁsé prbbiems Warth'Z’f' paints aktogether, fem: Shari-answer probiems worth. (me point each, and nine twenfalse problems werth one point each, far an exam total ofiii} points. Pan I. Free Respanse In each groblem in this section, fo-E'iew directions carefuliy, and Show ail the Stags needeii to arrive at the correct answer. The point vaiue for each problem is ShOWI‘k to its leﬁ. (4) 1. Find an equation of the plane centaining the points A 3 (~23, 3,1), B m (2, 0, 4), and C = (1, ~2, —1). x5 {3 XJQ'; 5" w§ jijgfi: +ZEEME§J<C W «2 "2 g 3 Wf/E.M}> 25:} I. m. ENFZ‘jW% 3W2 (3) 2. Find parametric equations far the circie given by 2:2 ~§- 2'2 + 42 = 12, y m 1. 22:2 2: 3-5; «W? :32.ng § ,3: Eli» {%‘*§"Z}2 ‘5‘— fie: “ﬁg :fi%&§% g “a: 2 w“: yﬁéﬁé .f :35 :2 éﬁéggé «75 f g :3 M2 “g'ézygﬁfﬁr‘ig (5) 3, Cansidez‘ the curve traced Out by the vecter function “38) x [2cm 2t, 2 + \$112121??? Find parametric equations or a vector equatian far the tangent ﬁne ta this curve a: the point where 3‘.’ 2 . (5) 4. Consider the scaiar function f (x, y, z) = 3136“ (a) Find V f at the paint (3, 2, 0). gr" “KE- 3 ﬁg] Vﬁgrggﬁg Wﬁﬁjxjﬁ 4):}? (b) Fimi the directienal ﬁeﬁvative of f at the point (3,2,0) in the directien of the vector “:3? = {5} «3,21. (4) 5. Consider the vector ﬁmction '3 2 {33; w \$051123, 555,2 in (3.) Find Iciiv 1? . ,v S . W “pg? “*2? ' (b) Fmd curl 1?. f ,4; j ” g} 3;} .ﬂs f” M? iwﬁi? _ M "r ‘ M W i? w E Coarg V“ £3}; a631, 53%? "L533; £33) E w § 3 “mix; "a @1432; :3 3 (6) 6. Find the work done by the force “I? m [:3, 5y, 2m] acting to displace a body from the point (3,0,1) to the point (1,1,8) aiong the curve traced out by the vector function ?(t) = [2,12, 8‘}. Part 11. Short Answer Each ef'the foﬂowing' is werth one'pomt. Ariswer brieﬂy. 7. Let f (as) be a function, (a usual function, with one éependent 21:26 0138 independent variabie). Answer 0118 or the other ofthe foﬁowéng, but not both. (a) Write out the limit deﬁnition of the derivaiive f’ (x), ,‘ fifxmawéx} “’ﬁK/XB m) : Aw , ‘ w AXﬁa £33K (b) Write out the limit éeﬁnition 0mg deﬁnite integrai fjﬂmmx. ~mmﬁbw a: I? I Ema : 3‘; aft/mm 8. The divergence ofthe gradient equais 9. Which of the four operators (gradient, divergence, curl, or Laplacian) has ﬂuid ﬂow as a ﬁmdamentai application? Qﬁnggﬁﬁm IQ. Which of the faur aperators (gradient, éivergence, curl, ear Lapfacian) has rotational motion as a ﬁméamentai appiication? .9: CM}; é“ WWWWWWW.‘M_ .,. Part {IL True-False Write on: the word “true” or “fake” ﬁt)? each ofti’ae foiiowing. Each is worth one point. it -—> 1 11. m k a...» j >< J} M”) 37\$ng- 21 “is m, —b ___, ——+ ma. 12. Foranytwovectors a and b,{a >< b)-b m0. * we“; «5? i. f; - i xmm axe ymamﬂv ’eefe 13. The parametric equations as = cos it, y m sin t, z- : 3 represent a heiix. em 14. Let ?(z) = [ﬁeﬂtet], and set dt = 0.2‘ Then, when t = 1, db: = 0.4. ﬁw gig: e ﬁﬁi{2}ﬂ}5&2j :63? 15. The direction of maximum increase of the scalar ﬁmction f (37, 31,2) = a: + ye at the point (1, 2, 3) is the direction of the vector [1, 3, 2]. “I m :7? a [1, egg} vﬁ’ﬁzjs) : {gem 16. The scaiar fieid f(x,y,z) mx+y+z is a potentiai of the vector ﬁeid v(x,y,z) m [1,1,1]. jms my 17. For every vector ﬁeld V, grad(curl ti?) m 8 . a W wiwji é We are; wwéw... I; ﬂier/z MM \ 5‘” If We ‘ W; 5:6, e.“ We? eke we?“ if 18‘ Every conservative vector ﬁeid gel-{rotationai g at“; m, g» ,wﬁéee like «i- f M . «e W . Wisﬁmﬂ r: e 19. The diﬁeremriai form ygdx —§- (2%: - z)-dy w 33:13 is exact. “V a a?“ \$95 “a. we "‘ "a 2"” ﬁg”; (35?: M ‘ _ w W “"1?” 4 é ragga: A??? -> ‘a WW «1 . "9 & # Hive gem igrwi;f’:§ wmeégx ﬁg J 5535 5%}; £53: Si? in WWW-muuwm—muwwwamu «um. ...
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