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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 MuitiQEeChoice 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
Find 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“: [1i 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—éyE—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 eomputa’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
Hat40W
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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 Spring '08
 Hastings
 Linear Algebra, Derivative, vector function

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