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MATH 122: Final Exam Section:
Thursday, December 10, 2009 Show all work and justify your answers. Your solutions should read nicely and be legible. They
should not be composed of regurgitated fragments of your mind scattered throughout the page.
If you run out of room for a problem on the front, then continue onto the back. Remember no
calculators are allowed.
√
1. (a) Find the length of the curve given parametrically by x = sin−1 (t), y = ln( 1 − t2 ) for
0 ≤ t ≤ 1/2. (Simplify your answer completely.)
[13 pts]
(b) Sketch the curve x = 2(t − sin(t)), y = 2(1 − cos(t)) on 0 ≤ t ≤ 2π and the ﬁnd the area
between the curve and the xaxis.
[12 pts] 1
dx
4x − x2
3x2 + 7x + 7
(b) Evaluate
dx
(x + 2)(x2 + 1) 2. (a) Evaluate √ [12 pts]
[13 pts] 1
dx converges or diverges. If it converges
x2 + 6x + 13
−∞
determine the value to which it converges.
[15 pts]
1
2
(b) Approximate
e−x dx using the Trapezoidal Rule with n = 5 subintervals.
[10 pts] 3. (a) Determine if the integral 0 −5 4. (a) A building demolisher consists of a 2000pound ball attached to a crane by a 100foot
chain weighing 3 pounds per foot. At night the chain is wound up and the ball is secured
to a point 100 feet high. Find the work W done by gravity on the ball and the chain
when the ball is lowered from its nighttime position to its daytime position at ground
level.
[12 pts]
√ √
√ ∞
(b) Determine whether the sequence
n n + 1 − n n=0 converges or diverges. If it
converges, determine what value it converges to.
[13 pts] dy
5. (a) Find the unique solution of the diﬀerential equation 3y 2 x dx − x + 1 = 0 for which
y (e) = 1. Solve explicitly for y if possible. (You may assume x > 0.)
[10 pts] (b) Let f (x) = 1 + 2x − x2 and g (x) = x2 − 2x + 1, and let R be the region bounded by the
graphs of f and g . Find the center of gravity of the region R.
[15 pts] √
∞
n + n2 + 1
√
6. (a) Determine whether the series
converges or diverges. State all tests
n 4 − 3n 2 + 1
n=2
that you use.
[10 pts]
∞
nn
(b) Determine whether the series
(−1)n
converges absolutely, conditionally or din!
n=1
verges. State all tests that you use.
[15 pts] 7. (a) Find the interval of convergence for ∞
(−1)n 4n+1 n=1 (b) Approximate ∞
n=1 (−1)n as a sum of fractions.) 1
n2 2n π n+2 · n xn+3 . State all tests that you use. [15pts] with an error less than 0.01. (You may leave your answer
[10 pts] 8. (a) Find the Maclaurin series for tan−1 (x2 )dx and state its radius of convergence. Provide appropriate justiﬁcation for your answer.
∞
3
(b) Find the numerical value of
(−1)n+1 n .
4
n=1 [15 pts]
[10 pts] Formulae
T
• Error bound for Trapezoidal Rule: En ≤ on [a, b]. S
• Error bound for Simpson’s Rule: En ≤ on [a, b]. • Taylor Remainder Formula: rn (x) =
• ex = ∞
xn n=0 • sin(x) =
• cos(x) =
• n! ∞
(−1)n
x2n+1
(2n + 1)! n=0 ∞
(−1)n n=0
∞ (2n)!
1
=
xn
1−x
n=0 x2n KT
(b − a)3 where KT = the maximum of f (x)
12n2 KS
(b − a)5 where KS = the maximum of f 4 (x)
180n4 f (n+1) (tx )
(x − a)n+1
(n + 1)! ...
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This note was uploaded on 08/25/2011 for the course MATH 21122 taught by Professor Winter during the Fall '08 term at Carnegie Mellon.
 Fall '08
 WINTER
 Math

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