S06_Final_Exam-D.Sarason

S06_Final_Exam-D.Sarason - UHIZHIZUUIJ 15:12...

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Unformatted text preview: UHIZHIZUUIJ 15:12 23112113422411: LILJH Mn'fl'ulN LLHHHHV l-‘n'fl'ullsilz Ulr’lfi‘j Mathematics 1.6B May 16 2006 Sarason ’ FINAL EXAMINATION . Name (Printed): Signature: SID Number: El Torn Doreey El Zak Mesyan El David Penneys El Arun Shanna GSI (check one): Section Number or Time: Put your name on every page. Closed book except for tWo crib sheets. No Calculators. Table of Natural Logarithms (to Four SHOW YOUR WORK. Cross out anything you have .DecirnalS) written that you do not wish the grader to consider. If you continue the answer to a question on the back of the page, put a note to that efieot on the front of the page. Make sure 1.0986 the grader can easily spot your final answer(s) to each question, for example by boxing or circling answers Where 4 1.3863 1.6094 appropriate. a 1.7918 la! n The points for each problem are in parentheses. Perfect score = 140. UHIZHIZUUIJ 15:12 bllfib4224lb LILJH Mn'fl'ulN LlHHIfluH‘r’ I-‘n'fl'ullsilz UZIUH Name 1. (15} Evaluate the integrals: (a) If: [A $13de 033;, where R ia the triangle with. vertices (0,0), (1,0), (1,1). m 2 #3 (13} I2 =3 / SSE—I (1:1: (c) I3 = / sin (ids: ‘ O o 8832832885 15:12 5188422415 LICE Mn’l‘qIN LIERflR‘r" PflGE 83.388 N ame ' 3 W ‘ 2. (15) Let Em, a) = [Rum — [:02 + (y — 45ij dy, ( 3 where R is the square with vertices 0,0 , (1,0), (0,1), (1, 1). For which («1,6) is Efa b) a minimum? ' , 84.388 8832832885 15:12 5188422415 LICE Mn’l‘qIN LIERflR‘r" PflGE Name 4 -——...__._,_________H__ 3. (20) The Pauvre Suceur Gambling Accessories Manufacturing Company has a contract to produce 960,000 decks of cards. For the plant where the cards are made, the production function fix, 3;) = 12,0001” 3111/3 gives the number of decks that can be produced with the utilization of .1: units of labor and 3; units of capital. Each unit of labor costs $1,000 and each unit of capital costs $4,000. (a) Write down the function 9(3, y) giving the cost to the company when it utilizes as units of labor and y units of capital. (b) Determine the values of :1: and 3;; that minimize the cost of producing 960,000 decks of cards. Use Lagrange’s method and take care not to confuse the objective and Constraint functions. (You will lose points if you do confuse them.) (c) Campare labor costs with capital costs for the minimizing values of :c and y. UHIZHIZUUIJ 15:12 bllfib4224lb LILJH Mn'fl'ulN LlHHIfluH‘r’ I-‘n'fl'ullsilz name Name ——+————w——_.___H ‘ 5 4. (20] (a) Find the general solution of the differential equation Byy’ = —(y2 -1)2- (b) Find the solution satisfying the initial condition y(1) = —2. (c) Find the solution satisfying the initial condition y(1) = —1. ' ' I-‘n'fl'ulsilz Ubflfi‘j UHIZHIZUUIJ 15:12 bllfib4224lb LILH Mn'fl'ulN LlHHIfluH‘r’ o. (20) Bianca Confusion takes out a $500,000 mortgage to buy a hovel near. the Berkeleyr campus. The yearly interest rate is 5%, compounded continuously, and yearly payments are $35,000, applied continuously. (a) Set up a differential equation satisfied by the unpaid amount P(t) of the mortgage at time t (with t measured in years). (b) Find the general solution of the differential equation! (0) Find the solution satisfying the initial condition P(0) = 500, 000. (d) Determine how long it will take Bianca to repay the loan in full. (You will need to use the logarithm table on the cover sheet.) 8832832885 15:12 5188422415 LICE Mn’l‘qIN LIERflR‘r" PflGE 8?.-"'8'E| Name u—.____H___u_________fi__ 7 6. (15) (a) Find the third Taylor polynomial. 113(22) at m = 1 for the function fix) = fl. (b) Use the result from (a) to estimate 1/ 1.2. ExpreSS your answer in decimal form. (c) [-Jee the remainder estimate to get a bound on the error in the approximation obtained m (13). Again, express your answer in decimal form. UHIZHIZUUIJ 15:12 bllfib4224lb LILJH Mn'fl'ulN LlHHIfluH‘r’ I-‘n'fl'ullsilz UHIUH Name ___u____________'___ 8 7. (15) For a contiguous random variable X with probability density function f = 31n23:, U 5 a: 5 E, compute the expected value E(X) and the variance Va.r(X). UHIZHIZUUIJ 15:12 bllfib4224lb LILJH Mn'fl'ulN LlHHIfluH‘r’ l-‘n'fl'uEJI: UHIUH Name 9 8. (10) Suppose the possible values of the discrete random variable X range over the nonnegatiVe integers, and the associated probabilities are given by p“ = Pr(X = n) = tin/7"” (n = 01112,..4). Compute Pr(X is even). 9. (10) (a) Derive the formula 5 I l: / mge"mzf2dm 2/ fig/grim + tie—“2’2 — lie—5W2. (b) Let X be a standard normal random Variable, 'i.e., a continuous random variable Whose density function is the function Me) = Abode/2, —oo :2: :r <: on. Use the #27:“ result from (a) to show that Var(X) == 1. ...
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This note was uploaded on 11/14/2011 for the course EDUCATION 190 taught by Professor Hull during the Winter '10 term at Berkeley.

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S06_Final_Exam-D.Sarason - UHIZHIZUUIJ 15:12...

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