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Unformatted text preview: MATH 104  FINAL EXAM
Friday, January 17, 2003, 8:30AM—11230AM
McCosh 50 Note: Average was approximately 60 percent. Considered hard but fair. 1. ( 8 points) Compute the following integrals: “0 / (Tia—an“
(h) / a31n(a:+1)d:c
2. (12 points) (a) Let R be the region bounded by the :caxis and the graph of y = 1/(513‘1 + 1) as :1; runs
from O to 00. Find the volume of the solid of revolution obtained by revolving R about
the yaxis. (b) Calculate the area of the surface obtained by revolving the graph of y = 6” between the
points (0, 1)‘ and (1,6) around the xaxis. ' 3. { 16 points) Determine whether the following integrals converge or diverge. Give your reasons. 00 da:
(a) ﬁ + m3
0» f5 dw (c) [01 L0 + m) dad x3
°° dac
d f
( ) 1 m Ina: _
4. { 16 points) Determine whether the following series converge or diverge. Give your reasons.
00 "2
a
( ) “5 + 1
oo ( 1)nn2
00) ED mg + 1
oo 2. n
(c) n ,3
“:0 n. co 1 33—2 "'
.1 ' = '
5 {2pomts)Let ﬁx) r§)n_1.2( 3 ) (a) For what values of as does the series converge? Give your reasons.
(b) Find f(50) 6. {12 points) 3  1 — 2 cos a;
(a) Use Taylor series to compute lim (i—T—L.
w—rU x(s1n a: — 1:) dt 1 + t4 centered at a: = 0. For what values of :1: does 1;
(b) Find the Taylor series of = f
o
it converge? 7. {12 points) For the questions below express your answers in the form a + ib where a and b are
real numbers. Simplify your expressions for a and b. . . 7+z' 43
(a) Simplify (3+ 41,) . (b) Solve Z4 = —8iz.
8. {12 points) Find all real solutions to the following differential equations. (a) :11" + 221’ + 10y = 0
(b) 2y” + y’ — 3y = 0 ...
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This note was uploaded on 09/26/2010 for the course MAT 104 taught by Professor Edwardnelson during the Fall '08 term at Princeton.
 Fall '08
 EdwardNelson
 Calculus

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