MATH 105A

Final - Practice 1 | Winter '14 | Fider, Butenko

1. Estimate the area under $f(x) = sin(x)$ on the interval $[0, p]$ by computing the Riemann sum using three subintervals and left endpoints.

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2. Evaluate the integral Show all work.

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3. Write the integralas a limit of Riemann sums using right endpoints..

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4. A function for the basal metabolism rate, in kcal/h, of a young man is $R(t)$, where t is the time in hours measured from 5 : 00 AM. What does the integral $f_0^{24}$ $R(t)dt$ represent? What are the units?

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(a) Given

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(b) LetFind $h'(x)$.

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6. Evaluate

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7. Suppose $?_1^{e2} f(z)dz = 10$.Evaluate $?_ 0^1e^{2x}f(e^{2x})dx$

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8. Evaluate the integralShow all work.

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9. Evaluate the integralShow all work.

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10. Evaluate the integralShow all work.

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11. Evaluate the integral $?e^{2x} sin(px)dx$.Show all work.

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12. Evaluate the integralShow all work.

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13. Evaluate the integralShow all work.

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14. Evaluate the integral Show all work.

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Evaluate the integral $4?sec^{20}(x) tan^5(x)dx$. Show all work.

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16. Evaluate the integralShow all work

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17. Evaluate the integralShow all work.

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18. Determine whether the improper integral $?_0^8re^{-3r}dr$r is convergent or divergent. Evaluate the integral if convergent, or explain why it diverges. Show all work.

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19. Determine whether the improper integralis convergent or divergent. Evaluate the integral if convergent, or explain why it diverges. Show all work

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20. A particle moves along a line with velocity function $v(t) = cost$, where $v$ is measured in feet per hour. Find (a) the displacement and (b) the distance traveled by the particle during the time interval

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21-22. Let R be the region bounded by the graphs of $x = 2y^2$ and $x =4+ y^2$.
(a) Sketch the region R and find its area.

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(b) Set up an integral to compute the volume of the solid generated by revolving the region $R$ (from part (a)) about the $y$-axis. Do not evaluate the integral!

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(c) Set up an integral to compute the volume of the solid generated by revolving the region $R$ (from part (a)) about the line $x = 1$. Do not evaluate the integral!

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23. Find the exact length of the curve

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24. Find the average of the functionover the interval [2, 3]. Show all work.

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25. Using integration find the area of the triangle with vertices $A = (-2,-4),B = (1, 5), C = (10,-1) and sides AB :3x-y = -2,BC : 2x + 3y = 17, CA : x-4y = 14$.

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26. For the sequencedetermine if the sequence is (a) monotone, (b) bounded, and (c) what conclusion can you make based on (a) and (b)?

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27. Use the Squeeze Theorem to show that the sequenceconverges.

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28. Determine the general term formula for the sequenceUse the formula to find the $100^{th}$ term.

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For each of the following sequencesIf a limit doesn’t exist, expain why not. Show all work.
29. $a_n = (-2)^n$

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30.

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31. $a_n = sin (2pn)$

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32. Find the sum

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33. Use the Divergence Test to determine that the seriesis divergent. Show all work

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34. Use the Alternating Series Test to determine whether the seriesis convergent or divergent. Show all work

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35. Use the Direct or Limit Comparison Test to determine whether the seriesis convergent or divergent. Show all work.

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36. Use the Ratio or Root Test to determine whether the following series is convergent or divergent. Show all work.

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37. Use any Convergence Test to determine whether the following series is convergent or divergent. Show all work

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38. Find the interval of convergence of the power series

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39. Find the first three nonzero terms of the Taylor series expansion of $f(x) = ln(x)$ about $x = e$.

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40. Find the Maclaurin series for $f(x) = e^x + e^{2x}$

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