Math 230 Calculus II Summer 2004

FINAL EXAM




13 problems, 10 points each

Provide complete explanations of all solutions

  1. Evaluate the integral MATH
    MATH

  2. Evaluate the integral: $\dint $ MATH
    MATH

  3. Evaluate MATH


  4. Let MATH (a) Use the graph of $y=f(x)$ to estimate $L_{3},M_{3},R_{6}.$ (b) Estimate $T_{3}$ and compare it to $I$.
    MATH

  5. The region bounded by the graphs of $y=\sqrt{x},y=0,$ and $x=1$ is revolved about the $x$-axis. Find the surface area of the solid generated.
    MATH

  6. Find (by integration) the length of a circle of radius $r.$


  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):
    final__19.png

    MATH

  8. (a) State the definition of the sum of a series. (b) Use (a) to show that the series MATH converges.


  9. Test the following series for convergence (including absolute/conditional): MATH
    MATH

  10. Find the radius and the interval of convergence of the series
    MATH

    MATH

  11. Find the power series representation of the function MATH and its interval of convergence.


  12. Find the Taylor polynomial $T_{2}(x)$ of order $2$ centered at $a=\pi $ of the function $f(x)=\sin ^{2}x.$
    MATH

  13. Indicate which the following statements below is true or false (no proof necessary):

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

2. The series MATH converges.

3. If $a_{n}>b_{n}>0$ and MATH converges, then MATH also converges.

4. The series MATH converges.

5. The sequence MATH converges to $0.$




Bonus Problem (10 points). Test for convergence: MATH

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