MATH 1620

Final - Practice 1 | Fall '15

1. Find the area bounded by the line $y = \sqrt{4 - x}$, the $x$-axis and the $y$-axis.

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2. Find the area bounded by the $x$-axis and the curve $y = \sin(x)$ on the interval $0 \le x \le \pi$.

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3. Use the disc/washer method to find the volume of the solid formed by rotating the regino enclosed by the lines $y = 1, x = 0$ and curve $y = x^3$ around the $x$-axis.

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4. Use the shell method to compute the volume of the region formed by rotating the triangle with vertices $(0, 0); (1, 1); (0, 1)$ around the line $x = 1$.

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5. Find the volume formed by rotating the region bounded by $y = 2x, x = 1$ and the $x$-axis around the $y$-axis.

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6. Find the volume formed by rotating the region bounded by $y = e^x, x = 1$, the $x$-axis and the $y$-axis around the $x$-axis.

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7. A $10$ meter chain with mass $100$ kg is suspended vertically from a platform. Use an integral to compute how much work is done lifting the chain onto the platform.

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8. A leaky bucket weights $100$ lb when full of water. Suppose water leaks at a rate of $1$ lb per second, and the bucket is lifted at a rate of $2$ ft per second. Write an integral computing the work required to lift the bucket $50$ ft, assuming it is full to start.

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9. If $1$ lb of force extends a spring $3$ inches beyond rest length, how much work would be done extending it $6$ inches beyond rest length? Give your answer in foot pounds. Must show appropriate integral and correct answer for full credit.

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10. Find the area of the surface formed by rotating the line $y = 2 - x, x = 0$ to $x = 1$ around the $x$-axis.

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11. Use integration to find the length of the curve $y = \sqrt{1 - x^2}, x = 0$ to $x = \frac{1}{2}$.

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12. Use the Pappus Theorem to find the volume of the solid formed by rotating the diamond-shaped region with corners $(1, 0); (0, 1); (-1, 0); (0, -1)$ around the line $x = 2$.

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13. Compute $\int x \sec^2(x) dx$

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14. Compute $\int_0^{\frac{\pi}{4}} \tan^2(x) dx$

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15. Compute $\int \sqrt{1 - x^2} dx$

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16. Compute $\int \frac{x^3 + 2x^2 + 1}{x^4 + x^2} dx$

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17. Compute $\int_0^1 xe^x \; dx$

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18. Compute $\int \frac{2x + 1}{x^2 + x} dx$

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19. Compute $\int x^3 \sqrt{x^2 + 1} dx$

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20. Find the limit of the sequence $a_n = (1 - \frac{1}{n})^n$. Does the sequence $b_n = (-1)^na_n$ converge or diverge? Give a reason for your answer.

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21. Write the repeating decimal $.\overline{5}$ (this means $.555555...$ no end to the $5
#39;s) as a geometric series. Use the geometric sum formula to find a rational number equal to this repeating decimal.

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22. Determine if the series converges or diverges. If it converges, find its sum. If it diverges, state why. $\sum_{n=0}^{\infty}(-1)^n \frac{3^n}{2^(n + 1)}$

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23. Apply the integral test to the series $\sum_{n=1}^{\infty} \frac{2}{n(n + 1)}$. The associated improper integral must be written and solved correctly. State conclusion obtained.

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24. Determine if the series converges or diverges. Give reasons for your answer. $\sum_{n=1}^{\infty} \frac{n^2 + n + 3}{2n^3 + 2n - 1}$.

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25. Determine if the series converges or diverges. Give reasons for your answer. $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^3}}$.

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26. Determine if the series converges or diverges. Give reasons for your answer. $\sum_{n=1}^{\infty} \frac{1}{2^n - 1}$.

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27. Determine if the series $\sum_{n=1}^{\infty} \sqrt[n]{n}$ converges or diverges.

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28. Determine if the series $\sum_{n=1}^{\infty} \frac{(-1)^n n^2}{n!}$ converges absolutely, converges conditionally or diverges.

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29. Determine if the series $\sum_{n=2} \frac{(-1)^n}{\ln n}$ converges absolutely, converges conditionally or diverges.

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30. Determine the radius and interval of convergence of the power series $\sum_{n=1}^{\infty} \frac{3^n(x - 1)^n}{n + 1}$.

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31. Determine the radius of convergence of the power series $\sum_{n=1}^{\infty} \frac{n^n x^n}$.

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32. Determine the radius of convergence of the power series $\sum_{n=1}^{\infty} \frac{2^nx^n}{n!}$.

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33. Suppose that the power series $\sum_{n=0}^{\infty} a_nx^n$ is convergent when $x = $ and divergent when $x = 6$. Is the series convergent when $x = 3$? When $x = -7$? When $x = 5$? Explain.

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34. Use the geometric sum formula to find a power series, with radius of convergence, that converges to $f(x) = \frac{x}{2 + x}$.

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35. Find the $2^{nd}$ degree Taylor polynomial of $f(x) = \sqrt[3]{x}$ expanded at $a = 8$.

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36. Use the McLaurin series for $e^x$ to find a power series converging to an antiderivative of $e^{-x^2}$.

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37. Let $\mathbf{u} = \mathbf{i} - \mathbf{j} + 2\mathbf{k}$ and $\mathbf{v} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}$. Find the vector projection of $\mathbf{u}$ onto $\mathbf{v}$ and the vector component of $\mathbf{u}$ orthogonal to $\mathbf{v}$.

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38. Find a paramteric form of the line through the points $P = (1, 2, -1)$ and $Q = (2, 1, 3)$.

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39. Find an equation of the plane containing the point $P = (1, 2, -1)$ and the line $l(t) = (1 + t, 1 - 3t, 2 + t), -\infty < t < \infty$.

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40. Find the point of intersection of the line $l(t) = (1 + t, 1 - 3t, 2 + t), -\infty < t < \infty$ with the plane $x + y + z = 1$.

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