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MATH 122: Sample 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 deﬁned parametrically by x = t and y = ln | cos(t)| for
0 ≤ t ≤ π /4.
(b) Find all points of horizontal tangency on the curve deﬁned parametrically by x =
1 + t3
1 + t3 √ 4x2 + 1
(t − 4)(t + 2) 2. (a) Evaluate [15 pts]
[10 pts] 1
dt converges or diverges. If it converges deter+ 4t + 8
mine the value to which it converges.
[15 pts] 3. (a) Determine if the integral ∞ t2 (b) Use the error formula for Simpson’s Rule to ﬁnd the smallest number of subintervals
necessary to approximate ln(2) =
dx with an error less than 10−6 . Phrase your
answer as “let n be the least number of (even?) subintervals greater than . . . ”
[10 pts] dy
4. (a) Find the unique solution of the diﬀerential equation 2xy dx = x + 1 such that y (2) = 1. [12 pts] (b) A conical tank with a radius of 1 foot and a height of 2 feet is completely full of water.
Water weighs 62.5 pounds per cubic foot. Find the work necessary to pump all the water
out to a height of 3 feet above the top of the tank.
[13 pts] ∞
5. (a) Determine whether the sequence
converges or diverges. If it is convern
gent, determine to what value it converges.
(b) Find the center of mass of the region bounded by f (x) = x + 1 and g (x) = x + 1.
[15 pts] 6. (a) Determine whether the series
(b) Determine whether the series ∞
n2 e−n/2 converges or diverges. State all tests that you n=1 [10 pts] ∞
(−1)n ln(n) n=2 verges. State all tests that you use. n converges absolutely, conditionally or di[15 pts] ∞
7. Let f (x) =
n=1 (a) Find the interval of convergence for f (x). State all tests that you use.
(b) Use the given power series to ﬁnd an approximation of
f (x)dx that has an error
less than 0.01. 0 [12 pts] 8. (a) Let f (x) = x cos(t3 )dt. Find the Taylor series for f (x) about 0. [10 pts] 0 (b) A bucket containing water is raised vertically at the rate of 2 feet per second. Water
is leaking out of the container at the rate of 1 pound per second. If the bucket weighs
1 pound and initially contains 20 pounds of water, determine the amount of work W
required to raise the bucket until its empty.
[15 pts] Formulae
• 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) =
1− x = n! ∞
(2n + 1)! n=0 ∞
(−1)n n=0 ∞
n=0 xn (2n)! x2n KT
(b − a)3 where KT = the maximum of |f (x)|
(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 21-122 taught by Professor Winter during the Fall '08 term at Carnegie Mellon.
- Fall '08