Math 230 Calculus II Summer 2004

FINAL EXAM

13 problems, 10 points each

Provide complete explanations of all solutions

1. Evaluate the integral   2. Evaluate the integral:     3. Evaluate  4. Let  (a) Use the graph of  to estimate  (b) Estimate  and compare it to  . 5. The region bounded by the graphs of  and  is revolved about the  -axis. Find the surface area of the solid generated. 6. Find (by integration) the length of a circle of radius  7. Find a parametric or polar representation of a curve similar to the one below, a spiral wrapping around a circle - from the inside! (no proof necessary):  8. (a) State the definition of the sum of a series. (b) Use (a) to show that the series  converges.

9. Test the following series for convergence (including absolute/conditional):   10. Find the radius and the interval of convergence of the series  11. Find the power series representation of the function  and its interval of convergence.

12. Find the Taylor polynomial  of order  centered at  of the function   13. Indicate which the following statements below is true or false (no proof necessary):

1. If the series  and  diverges then the series  also diverges.

2. The series  converges.

3. If  and  converges, then  also converges.

4. The series  converges.

5. The sequence  converges to  Bonus Problem (10 points). Test for convergence:  This document created by Scientific WorkPlace 4.1.