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Unformatted text preview: Engineering Mathematics (ESE 31?) Exam 2 March 3, 2010 This exam contains 7 multiplechoice problems worth two poﬁnts each, 6 {mewfaise problems worth one
point each, 3 shortanswer problems worth one point each, and 3 freeresponse probiems worth 17 points aitogether, for an exam total of 40 points. Part I. MultigieChoice (two points each) Ciearly £13 in the oval on your answer card which corresponds to the only correct response. 0 3 2 7'
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1. The vector V m 0 is an eigenvector ofthe matrix A 2 ' 3 2 1 —2 4
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,2; Eliminate the parameter from the following parsnietric equations, and identify the curve. mm?» ymsinhm? z=cosh2t (A) w = 3 9'2 ~ 32 m ellipse (including the possibility of a circie)
(B) :2: = z2 —~ y2 m ellipse (incluciing the possibility of a circie)
(C) a: = 1,42 —— 2:2 an ellipse (including the possibility of a circle)
(D) :1: 2 Z2 — y? m 4 ellipse (including the possibility of a circle)
(E) x = 3 y? —— 22 m hyperbole (F) i=3) 22~y2m1 hyperbeKE ”WWWmJWHmce A (G) a: = 3 y? — 22 m 4 hyperboi;
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(B) v”? <C> V35 ago : 222E 352%?) E :3 g: 9?; n 224:): E may) E (E) x/Ié «gifEiEEEEQZE 3:32 W <2“: E“ Consider the surface 2: m 65‘ —— 6339’. Find a normal vector to this surface at the point (E11 10, in 2, 2).
(Note that the opposite of a normal vector is also a normal vector. For exampie, if you come up
with . 10, w 18, ml], which is not in the list of answers, then you should choose {w 10, 18, If.) (A) I—10,18,1] §K2z (the: 41’ 2: we a? E» $33 .2: a? &J 03., 1051833
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(16, 3532, 4). Work carefuliy. The integration shouié be reasonable. é, (A) 5 Mfg” E13” 7’ i’?‘ ”it"; )5 ,. , ”” , , :7 (B)10 ?%é%} 2 i2???) 2% 2) E} (C) 15 i W
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(o) 2:: (H) 30 As you work through the true~false and short—answer probiems, you may assume that everything is “niee.”
Speciﬁcally, all vector and scalar functions are continuous with continuous partial derivatives, and all curves are piecewise smooth. Part Ht TrueFalse (one point each) Mark “A” on your answer card if the statement is true; mark “B” if it is false. 10. 11. 12. 13. 4 w» ——§ >
For any vectors 3' and b (with b % 0 ), the component of "21’ in the direction of b is greater than or equal to G. Let ?(t) be a vector function Then, for any value of t 3133(3) Emma t—) — 0 (You may assume thatr ,Ht) M v 4(t) is never zero so that the tangential and normal components of acceleration exist ) ’2 ' m?) *
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J $25 ﬂew eyecjw? ﬁﬂﬁgw M Myefjmﬂ go (one ﬁreaﬁyum a Viv/W; The vector function 7.? (ac, y, z) = [0, 2:2, 2372;] is a potential for the scalar function f (x, y, z) = yzg.
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Mae; ﬁWﬁ/t t r w m Meal/Z52}? fit": if , Let ‘39:, y, 2) be a vector function. Then grad(div "3) = V2(V). Mat em, /;n it? We pie: I: ago one; U The vector ﬁeld ‘1? m [2, y,:1:} is rotational 2 we ““37 i
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<3 3% it ”jg at) tat, k WWWW Let “P? (3, 31,2 2) be a vector ﬁmction let A and B he points and let C; and C2 be paths from A to B.
If the arc iength ofCﬁ is greater than the arc iength of Cg, then f Fv a! r > f F d r Egg/Jae, ”1e. fleets} {imam :41;me mid: Augie wt; «Wee e e out, \f’m Wymﬁi} 6%., WM a” jﬁyfﬂtﬁéw Sf” algae Part ill. Short Answer (one point each)
The answer to each of these is right or wrong: no work is requires, and no partial credit will be given. 14. Find a vector which is normal to the plane Ex ~i 3g + 42 m 5.
E 2 t 5 e 3 15. Correctly ﬁll in the blank with a two—word answer: In order to ﬁnd the derivative of a composite function, one must use the 16. Let ?'(t) be a vector function, and let a and b be real numbers with a < b. As usual, subdivide the
interval la, b] into 72 equal time increments, Where a = to, b m tn, and. At m t, — tel. Express the following limit as a deﬁnite integral. in order for you to receive credit, your notation must be gerfect.
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segre>mw‘t,jrgejge Part IV. Free Response (point values as shown) Follow directions careﬁrlly, and Show all the steps needed to arrive at your solution. (5) 17. Let V m [sinxcoshy,cosxsinhyﬂyﬂz]. Find (div VXW, 0, 1). (Be sure to work out the values of the trigonometric and hyperbolic trigonometric functions, so that
your final answer is a number. Simplify this number in any reasonable ways, but leave it exact.) . y )4 f, g
gig5;“? : not}: melt; we oesx {3,51% w, 2553?; a: E owemmaorsemieesweat M :22 e’eiWZ (6) 18‘ In a certain large Ieeture ciass of 290 students, the professor observes that 90% of the students “who
do attend on a given ciass day wiil attend again the next eiass day, and 70% of those who do net
attend on a given ciess day will attend the next class day. (a) F me the stochastic matrix for this Markov process. if: iris: Metal); eétvmf we. eéyj gig“: if? eﬂw ghee 65% z?)
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gig”t : aim} “it” egg}! (b) If ail students attend on the ﬁrst day of class, how many win attend 0:1 the seconci day? ”We; [5:] ante] its t3} (0) How many wiII then attend on the third day?
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{ft 7 { jg; 3”? we iiiaimﬁ, } L; i g 3 5, 253 i 2% J (d) Use an eigenvalue! eigenvactor approach to ﬁnd the limit of the number of students who wiii
attend on any given day. You will receive no credit for a guess~andcheek approach. .4: Circle your answer in parts (13), (e), and (d) above. (6) 19. Let ﬁg, 3;, z) E Ey,z, :cE, and let C be the curve which is parameirized by the vector function
?(t) 2 E32, cos 2i, 0} from t x 0 to t 2 E Find the value ofthe line integral fuﬁ'd?
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