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Calculus II -- Fall 2012 -- final exam

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  • Show enough work to justify your answers
  • Write the problems in the given order, each problem on a separate page.
  • Don't simplify unless very easy or absolutely necessary.

1. The volume of a solid is the integral of the areas of its cross-sections. Explain and justify using Riemann sums.

2. The region bounded by the graphs of $y=x^{2}+1,y=0,x=0,$ and $x=1$ is revolved about the $x$-axis. Find the volume area of the solid generated.

3. Evaluate $\int x \sin x dx$.

4. (a) State the definition of the sum of a series. (b) Use (a) to prove the Sum Rule.

5. Find the sum of the series $$\sum _{n=0}^{\infty} \frac{(-1)^n+2}{3^n}.$$

6. Test the following series for convergence (including absolute/conditional): $$\sum \frac{(-1)^{n-1}}{(1.1)^n}.$$

7. Find the radius and the interval of convergence of the series $$\sum \frac{2(x+1)^n}{n^2}.$$

8. Find the Taylor series centered at $a=1$ of the function $f(x)=x^4$.

9. Given a parametric curve $x=\sin t,y=t^2$. Find the line(s) tangent to the curve at the origin.

10. Find a parametric representation of a curve that looks like the figure eight or a flower (no proof necessary).