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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.
 Winter '10
 Hull

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