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Unformatted text preview: Final Examination — MATH 141 — May 12, 2007 Instructions: Answer each of the 9 numbered problems on a separate answer sheet. Each answer
sheet must have your name7 your TA’s name, and the problem number (=page number). Show all your
work for each problem clearly on the answer sheet for that problem. You must show enough written work
to justify your answers. Finally, please copy and sign the Honor Pledge on page 1. NO CALCULATORS OR OTHER ELECTRONIC DEVICES 1. (a) (15) The base of a solid is the right triangle in the anyplane with vertices (0, 0), (1,0), and (0,2).
The cross sections perpendicular to the my plane are square. Draw a picture of the base, and include a cross section. Then set up the integral for the volume V of the solid. DO NOT
EVALUATE THE INTEGRAL. (b) (10) Let = 2553/2 + 4, for O S :10 S 1 . Find the length L of the graph of f. 2. A tank has the shape of the surface generated by revolving the curve y = 18 — 2x2 for 0 S :c S 3
about the y axis, where units are in feet. The tank is ﬁlled with water to a level 1 foot from the top
of the tank. Note that the density of water is 62.5 pounds per cubic foot. (a) (5) Draw a picture of the situation, with appropriate labels. (b) (10) Set up the integral for the work W required to empty the water out the top until there is
a depth of 2 feet in the tank. DO NOT EVALUATE THE INTEGRAL. 3. (a) (5) Simplify the expression sec(tan“1 :62). (b) (10) Let f(:c) = 3m — 21;, for x > 0. Show that f has an inverse, and ﬁnd (f‘1)’(2). d .
(c) (10) Find the unique solution of the differential equation 3y2xﬁ — at +1 = 0 for which y(e) = 1. 4. (a) (10) Evaluate /exln(1+em)dx. V 2 1
(b) (10) Evaluate /—————4m 4+ dw.
x 6t
5. a 10 Evaluate /—————— dt. ) (t—4)(t+2)
6 1
(b) (10) Determine if / W dx is convergent, or divergent. If convergent, ﬁnd its value.
4 _
V2 — v2 —
6. (a) (10) Compute lirgi+ —~:$m———m, giving reasons.
x—> (b) (10) Compute lim {II—+00 1 .7}
<1 + —2) , giving reasons.
00 00
. . 1 1
7. (a) (10) Flnd the numerlcal value of 3:2 <n+ 1 —~ n+4).
°° 1
(b) (10) Determine Whether the series E (—1)’“ k In k converges absolutely, converges conditionally, 16:13
or diverges, giving reasons. (c) (10) Let f(3c) : x21n(1+a:). Find the fourth Taylor polynomial p4 at m = 1/2 (that is, p4(1/2)). 8. Let me) = icy—1)” 3—; 96%“. n=1 (a) (10) Find the radius of convergence R of f. 1/4
(b) (10) Use the given power series to ﬁnd an approximation of f (t) dt that has an error of 0
less than 0.01. Don’t simplify your answer. 9. (a) (10) Let 2 = ~16. Write the 4th roots 21,22, zg, 24 of z in the form we”. Then display these
roots in the complex plane, and identify each of them. (b) (15) Consider the rose deﬁned by r = sin(40). Sketch the entire rose in the plane (including
pertinent labeling), and ﬁnd the area A of one leaf of the rose. END OF EXAM —— GOOD LUCK! ...
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This note was uploaded on 02/05/2009 for the course MATH 141 taught by Professor Hamilton during the Fall '07 term at Maryland.
 Fall '07
 Hamilton
 Calculus

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