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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: $$\int 3xe^{−x}\, dx.$$

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