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Unformatted text preview: MAT 296 FINAL
Spring 2002 NAME: INSTRUCTOR: This examination has 10 problems. It is your responsibility to make sure that all are
present. Show ALL work. Minimal credit will be given for answers without supporting work. A graphics calculator may be used on this examination. However, a symbolic calculator,
such as the TI89 or TI—92, may NOT be used. Do Not Write Below 1. 10. TOTAL (1) [15 pts] Determine each of the following: (a) lim e ’1
x—>01n(x+1) 3!!
(b) the limit of the sequence an =[l — n (2) [10 pts] Consider the region in the first quadrant bounded by y = x2 and y=x. (a) Draw a carefully labeled diagram and set up an integral to find the volume when this
region is rotated about the x—axis. Do NOT evaluate the integral. (b) Draw a carefully labeled diagram and set up an integral to find the volume when this
region is rotated about the yaxis. Do NOT evaluate the integral. (3) [15 pts] A tank has the shape of a paraboloid obtained by revolving the curve y = x2, 0 S x S 3, about the y—axis. x and y are measured in feet. The tank is initially full of water. Find the work needed to pump the water to a height 10 feet above the tank.
The water weighs 62.4 pounds per cubic foot. (4) [20 pts] Evaluate each of the following integrals. (a) l —1*—dx (b) l 2—2"—dx (x244); x —5x+6
(Hint: use partial fractions) (5) [20 pts] Evaluate each of the following integrals. (a) J cos3(x)dx (b) J xsin(x)dx (6) [20 pts] Determine if the following improper integrals converge and, if they do
converge, evaluate them. (a) .[14\/—1—dx (b) f:xze_‘3c1x x—1 (7) [10 pts] Does the series 2 1 2 converge or diverge? Give reasons for your
":2 n(1n n) answer! (8) [10 pts] Determine whether this series converges absolutely, converges conditionally, or diverges. Give reasons for your answer! 2—21) :3
n + n=1 (9) [15 pts] Find the interval of convergence of Zz—zx”. Does it converge absolutely,
n=l n converge conditionally or diverge at the endpoints of the interval of convergence? (10) [10 pts] Compute the first three nonzero terms in the Maclaurin expansion of
f(x) = x + e“ sinx ...
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This test prep was uploaded on 04/10/2008 for the course MAT 296 taught by Professor Zacharia during the Fall '07 term at Syracuse.
 Fall '07
 Zacharia
 Calculus

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