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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 simplestform. 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 :caxis. '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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 Spring '06
 Sarason

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