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# Calculus II -- Fall 2018 -- final exam

**MATH 230 -- Fall 2018 -- Final exam**

Name:_________________________ $\qquad$ 8 problems, 10 points each

- Write the problems in the given order, each problem on a separate page.
- Show enough work to justify your answers. Explanations matter more than answers!

$\bullet$ **1.** Sketch the parametric curve: $x=t^2-1,\ y=2t-3$.

$\bullet$ **2.** The velocity of the object at time t is given by $v(t)=1+3t^2$. If at time $t=1$ the object is at position $x=0$, where is it at time $t=3$?

$\bullet$ **3.** (a) Provide the definition of the sum of a series. (b) Give examples of convergent and divergent series.

$\bullet$ **4.** Apply the Integral Test to show that the series converges or diverges:
$$\sum \frac{1}{n^{1/3}}.$$

$\bullet$ **5.** The Fundamental Theorem of Calculus includes the formula $\int _a^bf(x)\, dx=F(b)-F(a)$. (a) State the whole theorem. (b) Provide definitions of the items appearing in the formula. (c) What is absolute convergence?

$\bullet$ **6.** 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 $K_{n+1}=1$.

$\bullet$ **7.** Find the interval of convergence of the power series:
$$\sum \frac{(x-2)^n}{n}.$$

$\bullet$ **8.** Integrate by parts:
$$\int 3xe^{−x}\, dx.$$