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16B_final_two_previous_exams

16B_final_two_previous_exams - Ulflfi‘j lUIlZIZUUb 1'3...

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Unformatted text preview: ' ' ' Ulflfi‘j lUIlZIZUUb 1'3: 44 bllfib4224lb LILH Mn'fl'ulN LlHHIfluH‘r’ I-‘n'fl'uEJI: Mathematics 1613 December 14, 2005 Sarason ‘ FINAL EXAMINATION N eme (Printed): : Signature: ‘ SID Number: ‘ D Matt Geglierdi GSI (check one): D Jon Herel El James Kelley . ' Section Number or Time: Put your name on everyr page. Closed book except'for two crib sheets. No Calcu1etore. SHOW YOUR WORK. Cross out anything you have Written that you do not wish the grader to consider. The points for each question are in parentheses. Perfect score = 150. J.Uf12f2lfilfib 1'3: 44 bllfib4224lb LILJH Mn'fl'ulN LlHHIfluH‘r’ I-‘n'fl'ullsilz UZIUH Name 1. (10) What values of a and b minimize the functien E(e,bj=(t-—e+3)2+b2+(b+e—2)2? gv’iwe 2. (15) Let the eentinueus random variable X have density function f (1:) = a: E 1. Compute the expected value E(X) and the variance Van-(X). F'nfliGE 83.388 18.-"'12.-"'288Eu 15:44 5188422415 LICE MnfliIN LIERfliR‘r" Name 3 —“—"'—_——-~—————___ 3. (20) The number of pairs of shoes the Phlim Zee .Shoe‘Company can manufacture per week with the utilization of :5 units of labor and y units of capital is given by the production function fix, 3;) = stilts/W1“. Each unit of labor costs the company $200 and each unit of capital costs it $1,000. To manufacttire 1,600 pairs of shoes per week at minimum cost, how many units of labor and how many units of capital should the company utilize? What"'is the corresponding ratio of labor costs to capital costs? 18.-"'12.-"'288E- 15:44 5188422415 LICE MnflaIN LIERflR‘r" F'nflaGE 84.388 N arm? 4. (15) Perform the integrations. (a)f0w:csinmdm (b) famv1+mdw a . 2 . , ; ‘ (c) / f1; (3: u fiflmdy, Where R IS the trlangle w1t11 vertices (D, O), (1, O), (1, 1). 18.-"'12.-"'288Eu 15:44 5188422415 LICE MnfliIN LIERfliR‘r" F'nfliGE 85.388 Name ——_——i—v——_._.—_M‘—-___'_—' {LI—I 5. (20) For the difierentiai equation 2y’ = (e — y)3e‘.. (:1) Find the general solution. (b) Find the solution satisfying y(0) = e. (o) Find the solution satisfying MD) = e —— 1. F'nfliGE 8E-.-"'8'E| 18.-"'12.-"'288Eu 15:44 5188422415 LICE MnfliIN LIERfliR‘r" ‘ Name - E'- ——____.____________ 6, (20) The Dinky University Molecular Biology Scholarship Fund starts the year with assets of $1,000,000, invested so as to earn interest at the rate of 10% per year, coin- pounded continuously. The fund receives donations at the rate of $20 Assume that donations are received continuously at the preceding rate, and that ex— penditures (grants plus expenses) occur continuously at the rate of A dollars per year- (a) Set up a difierential equation satisfied by the DUMB Fund’s assets FOE) at time 25 (measured in years). i Us) Find the general solution of the differential equation. (0) Findfithe solution satisfying the initial Condition Pal) = 1, 000, 000. (d) What value must A have in order for the DUMB Fund’s assets to be $1,000,000 at the end of the year? Be clear about how you arrive at your answer. _ll:1.-"lZ.-"2Ulfib 1'3: 44 bllfib4224lb LILJH Mn'fl'ulN LlHHIfluH‘r’ I-‘n'fl'ullsilz U (HUB Nome _.___m__,___________ 7 "r. (15) (3.) Find the third Taylor polynomial 123(3) at a: = 0 for the function fies) = 111(1 «I- 2:), (13) Use the result from (a) to estimate In 1.1. (c) Use the estimate of the remainder Rgf. 1) W E613 on u er bound f t . * the estimate made in (b), PP ‘ or he error 111 ' I-‘n'fl'ullsilz UHIUH lUIlZIZUUb 1'3: 44 bllfib4224lb LILH Mn'fl'ulN LlHHnfliH‘r’ Name 8. (15) Let the random variable X denote the possible values of X are 2, 3, . . outcome of rolling a fair pair of dice. The . , 12, and their probabilities are given in the table below. (a) Compute the probabilities Pr(X a: 7), Pr(6 5 a: 3' 8), Pr(X is odd). (1:) Suppose someone gives you 2—to— the bet is for $1, you will Win $22 What is the expected value of 1 odds that you will not roll 6, T or 8. Thus, if if you roll 6, 7 or 8, otherwise you will lose $1. your winnings or losses, as the case may be? lUIlZIZUUb 1'3: 44 bllfib4224lb LILJH Mn'fl'ulN LlHHIfluH‘r’ I-‘n'fl'ullsilz UHIUH Name ———___.._________'_‘___ 9 9. (10) Suppose the poosible values of a discrete random ‘ ‘ variobl Y . - - negatlvo mtegors, with Pr(X = n) = Syn/4”“ ( B j FEDE‘B Over the T1011 n = 0,1,2,..-). Compute PI(X 3 3). 10. f 10) Find the Taylor series at a: =2 0 for the function f($)=f1—_m;)—2- F 02/27/2004 12:28 FAX 510 642 9454 .001 Name TA’s Name Section Math 16B ' Final Exam, December 11. 2003 Do Not Write Here R. Hartshorne Part I. Shorter questions. 5 points each. Show work and put answers in boxes. No partial credit. All answers must be in simplest-form. No calculators. You may leave expressions such as 71', 6, V2 in answers. a r I 1. and a (—12) and simplify. 8y $111 w — cos y 1 17 7r _1f ' ___ __ _ . 2 smt 3 and 2 <t< 2, find tant ‘ . 7-. I 3. Find the total area between the curve y = 1 $2 + 4 and the :c-axis. '3. 4. If y’ = 515 + Bty and y(0) = 1, find 3; = f(t). 02/27/2004 12:28 FAX 510 642 9454 .002 5. /x2sinx d3 = 7. Use the Taylor series for sins: to compute sin(0.3_.) t0 6 decimal places. 8. Find the 5th Taylor polynomial of y = tan 1'. Reduce fractions to lowest terms. _ ‘ E l: E 02/27/2004 12:28 FAX 510 642 9454 .003 9. Use two iterations of the Newton—Raphson algorithm, starting with $0 = 2, to find an approximation for x/i. Leave your answer as a fraction in lowest terms. Leave your answer as a fraction in lowest terms. 11. Find the rational number (as a fraction in lowest terms) whose decimal is 0.135135%. . . 12. Experiment: Pick a point at random in the half disk of radius 2. Let X be the distance from the center 0. Find the probability density function for —7. the random variable X. 10. Find the sum of the infinite series 1+1+1+l+i+1+ +i+ 2‘3 6 62 63 6n 02/27/2004 12:29 FAX 510 642 9454 .004 Part II. Longer problems. 10 points each. Show your work. Put answers in boxes. 1. Evaluate /2 v1 — $2 dx. 0 a) Make a trigonometric substitution, and write the new integral with new limits of integration. b) Evaluate the integral to find the answer. 2. Find the maximum value attained by the function y = 4sina: + 3 cosct 0n the interval 0 _<_ :c 5 7r. 02/27/2004 12:29 FAX 510 642 9454 .005 3. For each of the following, determine if the infinite series converges or diverges. State which method you use and show your work. 5 2 2 ) a g k+3 00 5 b) 2} H3 °° 5 0% 2+3 k 1 4. a) Find the Taylor series for f (as) = 1 + :r' b) Find the Taylor series for ln(1 + 2:). 111(1 + 332) c) Find the Taylor series for 2 as ...
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