3, Exam Math 1220 Fall 2005
Name
Instructions: Show all of your work. (But only if you want credit!) You may not use symbolic manipulation capabilities on your calculator: Only numerical work is allowed. 1. Solve the mixing problem dQ/dt = (200 Q)/10, Q(0) = 100, using either of our two methods. (Recall for integrating factors that if = exp( p(x)) dx, then y = 1 ( (x)q(x) dt + C).) (12 pts)
2. Tell whether the sequence converges or diverges. If it converges, nd its limit. (6 pts each) (a) {3 (1)n }
(b) {n3 e2n }
3. Tell whether the sequence is (eventually) strictly increasing, decreasing, or neither. Remember the tests: (i) Comparing an+1 an or (ii) dan /dn to 0, or comparing an+1 /an to 1. (Pts: a:8; b:6) (a) { 20n } n!
(b) {an }, if
d 2 (an ) = dn (2n + 3)2
4. (12 pts) Use the formula the for nth degree Taylor polynomial pn given by pn (x) = f (k) (x0 )(x x0 )k . k! k=0
n
to get a 4th order polynomial approximation for f (x) = ln(x 1) about x0 = 2.
5. Find the sum of the series (if it exists). In part (b), rst nd the nth term in the sequence of partial sums to show convergence or divergence. (Pts: a:8; b:10) 5 2n+1 (a) 3n k=1
(b)
k=1
ln(
k+1 ) (Expand rst.) k+2
6. Test the following series for convergence or divergence. Clearly demonstrate ...