Math 230 Calculus II Summer 2004
FINAL EXAM
13 problems, 10 points each
Provide complete explanations of all solutions
Evaluate the integral
Evaluate the integral:
Evaluate
Let
(a) Use the graph of
to estimate
(b) Estimate
and compare it to
.
The region bounded by the graphs of
and
is revolved about the
-axis.
Find the surface area of the solid
generated.
Find (by integration) the length of a circle of radius
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):
(a) State the definition of the sum of a series. (b) Use (a) to show that the series converges.
Test the following series for convergence (including absolute/conditional):
Find the radius and the interval of convergence of the series
Find the power series representation of the function and its interval of convergence.
Find the Taylor polynomial
of order
centered at
of the function
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:
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