112
Final Math Exam
December 17, 2004
Read each problem carefully. Please show all your work for each problem! Use only those methods discussed thus far in class. No calculators! 1. (10 points) Compute: (a) lim 1 - cos , 0 22 (b) lim x2 e-2x ,
x
(c) lim (1 + 2x)1/x .
x0+
2. (10 points) Integrate:
2
(a)
0
dx , 16 - x2
(b)
x cos 2x dx,
(c)
sin3 x dx
3. (12 points) Integrate (evaluate the improper integrals correctly): (a) dx , 2 - 3x - 4 x (b) 2x + 1 dx, x2 + 4
(c)
1
dx . x2
4. (12 points) Find the area of the region in polar coordinates lying outside the curve r = 1 and inside r = 2 cos . 5. (10 points) Find the first two non-zero terms in the Maclaurin Series of the function f (x) = arctan x. 6. (10 points) whether Determine the following series converge or diverge. Find the sum of those that converge. Justify your answer!
(a)
n=1
n,
(b)
n=0
2n - 3n , 4n
(c)
n=0
1 . xn
7. (12 points) Determine whether the following positive term series converge or diverge. State clearly which test you use.
(a)
n=0
n , 4+1 n
(b)
n=1
2n n!
(c)
n=1
ln n n
8. (12 points) Determine whether the following series converge absolutely, conv...