#### 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.

2. Evaluate the integral Show all work.

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

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?

(a) Given

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

6. Evaluate

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

8. Evaluate the integralShow all work.

9. Evaluate the integralShow all work.

10. Evaluate the integralShow all work.

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

12. Evaluate the integralShow all work.

13. Evaluate the integralShow all work.

14. Evaluate the integral Show all work.

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

16. Evaluate the integralShow all work

17. Evaluate the integralShow all work.

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.

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

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

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.

(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!

(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!

23. Find the exact length of the curve

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

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

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

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

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

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

30.

31. $a_n = sin (2pn)$

32. Find the sum

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

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

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

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

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

39. Find the first three nonzero terms of the Taylor series expansion of $f(x) = ln(x)$ about $x = e$.
40. Find the Maclaurin series for $f(x) = e^x + e^{2x}$