exam 2 spring 2009 - Engineering Mathematics (ESE 3 E?)...

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Unformatted text preview: Engineering Mathematics (ESE 3 E?) Exam 2 February 25, 2,009 This exam contains 9 muitipiencheice preblems worth twe points each, 9 true~false emblems werth one point each, 4 short—answer problems worth one point each, and 3 free—response problems worth 9 paints aitogefiler, for an exam total of 46 points. Part L MuitiQEe-Choice Clearly fifl in the eve} on your answer card which corresponds to the only correct response. Each is went: two points. 1. If the vector '5’ has magnitude 5 and the vector g has magnitude 2, an& if "a? and E; point in —O -'I opposite directions, then what are the values of "a" b and [3) X b g respectively? (A) “10, ~10 (B) 40,0 (C) —10, 10 (D) 0, ~10 (E) {1,0 (F) 0,10 (G) 10, mm (H) 10,0 (1) 10:10 (I) not eneugh information is given to determine both of these values Fin-d the area of the triangle determined by the three pointg (0, 1, 2), (0, 3, 4), and (2,13 1). (A) 2 (B) x/5 ((2) \/§ (D) 3 (E) x/E (F) 4 (G) v35 (H) fié (I) 6 0) fig 3. Mich one ofthe foliewing vector equatiens represents a helix? (A) "ngj m ggcoszz, 33m 2:,3} {:8} ?(t) u {360323, 331112;, 2t} C) “i? t : ’3cas 2t,3sin2t, cos 213‘: ( i J (D) ?(t) 2: §3cos 2t§3s£n 2t,cosh 225} (E) fit} m [ 300511 2%, asinh 2:, 3] (F) “1%) m g 3 cosh 2t, 3 sinh 2t, 2:} (G) ?(t) = 3 cosh 2t, 3 sin}; 22?, cos 2t} (H) “£303 2 [3 cosh 2t, 3 sinh 2t, cash 21?} 4. Find an equation of the line which is tangent to the curve determined by the vector equation ?(t_) m [13, 2:, film? w t] at the point Wheret z e. (A) 3(a)) w + em, 2 + 2810,11 (B) 1“: [1-i- 8w, 2 + 2610, e] (C) m {1 + 810,2 + 2611;? 1 "3;" 610] (D) 3W) x [1+ 610,2 + 2810, e ~§~ ew] (E) 3 {e + £12,262 + 220, w} (F) 2 [e ~é~ 18:26 + 2w, 621;} (G) 3(w):{e+w=2ew§~2w,e+w§ (H) :2 [6+ w,28+2w,e+ aw} 5. Find the tangential compcnent of acceleration 3mm for the following vector function at the point wheret m 0. 76(3) - (:3: e32 sin 37:} (B) {0:m%a“§$§ P 3 3“: EO$§§ §l g 9 9 £01231); (E) {019,0}? 3 3; (F) {23""9'5; (G) {23031 (H) {2:039} (I) 32,35—3} 9 (I) [13,—5] Find the length of the are traced out by the vector function "138) x: [$152, grim, 815] from the point (D, 0, 0) to the point ( 3 fig, 32). (Work carefuin and double—check each step so it “comes out m‘cciy.”) - (A) 4 (B) 8 (C) 12 (D) 16 (E) 20 (F) 24 (G) 28 (H) 32 (I) 36 (J) 40 "E .1. Find the direction of maximum increase of the fimction f y, 2:.) m yooshx ~— 85” from the point (EL ILE 2). (A) EMMA; (B) 9—2,me (c) {43134} (D) {—1158} (E) {4,0544 (F) Emma; (G) §“1a19”1:§ (H) HAO} Find an equation of the piane which is tangth to the surface 2: 2 3" — say? at the point (2, 1, 5). (A) x+4ym532~19 (B) w+4yw321 (C) x+£§y+zm11 (D) :c—E—éy-E—SzflBl (E) 233+y—5zm} (F) 2x+y—zm—19 (G) 2x+y+zm3l (H)2x+y+52m11 Let if, V, and (A? be vector fimctions in three—space, and consider the foilowing eomp-uta’zions. m; (I) ("6.39) x w (KI) grad(cur1'w7) (In) divfif x T?) (W) (div “x? ).(cur1“v*} Which of these computations make sense; in other words, which could realiy be done? (A) I oniy (B) B only (C) HI only (D) W only (E) I and H onEy (F) I and HI only (G) i and EV only (H) 11 and HI only (I) II and W only (3) EH and W oniy Part I}. Tme«FaIse Mark “A” on your answer card if the statement is true; mark “B” if it is faise. Each is worth one goint. ‘10. The value A = 0 can never be an. eigenvaiue. I}. The vector “i? = is an eigenvector for the matrix A m HHi—JHH Hat-40W QHOHCD F—‘QHQH OHQHCD MOI—10H ‘ «4 , 12. For any vector “it except the zero vector, I = 1. —+ -—+ 13. Foranytwevectors—zi and b, 3%) g «a» «up J 14. The vectors 1;, ’ "i" (in that order) form a rightwhanded triple. } 15. For any vectors 31+ and ('3 X Mg)”; m G. 16. For any scaiar function f and any vector function ii, %(f(t)?(t)) m f(t)-{i’f(t) + f’(t)m{r’ (t). 17. Consider the motion of a fluid in a region around a point P in three~spaee, and iet 3(m,y,z,t) 2*: p(x,y,z,t)?(x,y,z,t), where p is the density filfiCinfl and 75' is the veiocity fimetien. If the fluid is incompressible and div "if > 0 at R then P is a scarce. 18. The vests: fimetion V(x,y, x) :2 Eye, 2:23, xy] is retationai. Part BI, Show Answer Each of the foiiowing is worth one point. Each answer is either right or wrong: no work is required, and no pattiai Ciedit will be given. . . .45 .20 . 0 192 24G 19. The stochastic matrix [.55 .88] generates the Mairkov chain [Q60], {7,68} [?28]?.... . (You .1] do not need to verify any of this.) What is the limit of the Markov chain? Be sure to write your answer as a column vector. i The eigenvalue A. m 1 for this matrix leads to the famiiy of eigenvoctors given by i: 11 3’ 20. Fill in the folkowing blanks. (Both must be fined in correofly in order for credit to be given here.) is the change in y as you follow the curve, but is the change in y as you foilow the tangent lino. 21. Let w r: f (33-, y), where x 2 g(t) and y x Mt). Use the chain rule to mite an expression for fit}? (Your notation must be gerfeot and must be clearly readabio in order for credit to be given here.) 22. The soaiar function flag, y, z) m x + 3,62 + 2:3 is a potential for what vector function? Part 1%" Free Resganse FOEIGW directions carefully, and Show a1} the steps needed to arrive at your soiution. The paint value fer each preblem is ShOWfi to its left. (2) 23. Find all eigenvalues of the matrix A m . —1 W2 1 (5) 24. Let A m 3 MS 3 . The value A w -2 is an eigenvalue of A of muItipiicity two. (You —~1 2 “3 do not need to verify this.) Find two linearly independent eigenvectors of A corresponding to Amwil. (2) 25. Fimi divgzcoswfl,ezmflcesy}. ...
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This note was uploaded on 01/10/2011 for the course ESE 317 taught by Professor Hastings during the Spring '08 term at Washington University in St. Louis.

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exam 2 spring 2009 - Engineering Mathematics (ESE 3 E?)...

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