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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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