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# Calculus II -- Fall 2014 -- final exam

Name:_________________________ $\qquad$ 12 problems, 10 points each

- Justify every step you make with as thorough an explanation as possible, in English.
- Unless requested, no decimal representation of the answers is necessary.
- Start every problem at the top of the page.

$\bullet$ **1.** The graph $y=f(x)$ of function $f$ is sketched below (it's not a parabola). Based on the graph, estimate the value of the derivative $f'$ of $f$ for $x=0$ and $x=5$. What can you say about $f' '$?

$\bullet$ **2.** Write the formula and illustrate with a sketch the left-end Riemann sum $L_4$ of the integral $\int _1^3 f(x)\ dx$ for the function plotted above.

$\bullet$ **3.** Find the $x$-coordinate of the center of mass of the region between $y=x^2$ and $y=x^3$.

$\bullet$ **4.** Evaluate: $\int_0^{1} \frac{1}{2x} dx.$

$\bullet$ **5.** Use substitution to evaluate the integral: $\int _0^{\pi} \sin x \ \cos^2 x \ dx.$

$\bullet$ **6.** (a) State the definition of absolute convergence. (b) Give an example of a series that converges but not absolutely.

$\bullet$ **7.** What degree Taylor polynomial one would need to approximate $\sin (-.01)$ within $.001$? Explain the formula: $E_n \le K_{n+1} \frac{|x-a|^{n+1}}{(n+1)!}$ and why you can choose $K_{n+1}=1$.

$\bullet$ **8.** (a) Plot these points in polar coordinates: $(r,\theta )=$ $(0,1)$, $(1,0)$, $(1,\pi)$, $(2,3\pi)$. (b) Sketch these three polar curves: $r=1,\ \theta=0,\ r=\theta$.

$\bullet$ **9.** Evaluate the limit: $\lim_{n\to \infty}\ln \left({\frac{n}{n+1}} \right)$.

$\bullet$ **10.** Plot this entire parametric curve: $x=\sin t,y=\cos 2t$.

$\bullet$ **11.** Find the interval of convergence of the series: $\sum \frac{(x-2)^n}{n}.$

$\bullet$ **12.** Explain how functions are represented by power series and how they both are differentiated. Demonstrate on $f(x)=e^x$.