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Unformatted text preview: Final Exam for MAT 296, Spring 2003 There are 10 problems in this exam worth a total of 200 points. To receive,
full or partial credit the correct work leading to the correct answer must be
written down. Graphics calculators may be used in this examination.
However, calculators capable of symbolic computations, such as TI89 or
TI92 may NOT be used. Instructor’s Name Your Signature:  (1) Compute the following limits.  . 1 — 005(12
(3) (11 pomts) (b) (11 points) lim(er + x)1/r .‘r—tO 1
(2) (13 points) Find the length of the are of the curve f(.7‘l = §(2 + $2)3/2 on the interval [0,3]. (3) (11 points) The region in the ﬁrst quadrant bounded by the lines y 2 O,
.r = 1 and the curve y = 3:3 is rotated about the line 2: = 1. Set up
an integral to evaluate the volume of the resulting solid. Include a
carefully labeled diagram. DO NOT EVALUATE THE INTEGRAL. (4) (11 points each) Determine whether the following series converge or diverge. Be sure to Justify your answers:
00 3"n
(a) Z(2n~—1)n! 11:1 (5) (11 points each) Evaluate the following integrals: 2
:E
d
(a)_/:1:2+4 I (C) /se(:4zdx (6) (13 points) A tank having the shape of an inverted right circular cone
of height 6 leet and base radius of 3 feet, is full o1 water. How much work is performed in pumping all the water in the tank over the tOp
edge? The density of water is 62.4 lli/It3. (7) (11 points each) Determine whether the following integrals are conver—
gent or divergent. If convergent. find their value. 4232+?
a d1.
<>/O ﬂ u" (X)
  . (—1)"n
16 t D t h th the E
(8) ( pom s) e ermine w e er series ":1 n3 + 2 vergent, conditionally convergent, or divergent. Be sure to justify your
answers. is absolutely con (9) (21 points) Find the radius and the interval of convergence of the power “3 n / n
, (~1) nus—‘4) ,
series E Does it converge absolutely, conu rge con
71:1 ditionally, or diverge at the endpoints of the interval of nonvorgence? (10) (16 points) Compute the first four nonzero terms in the MacLaurin
expansion of the function f(:z:) = x3 + 8“ cos 2:. ...
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 Fall '07
 Zacharia
 Calculus, Cone

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