# Exam III Practice Problems: Calculus II – Dr. Staples Scope of Exam

```Exam III Practice Problems: Calculus II – Dr. Staples Scope of Exam: 8.5‐8.6, 9.1‐9.3, and 10.1‐10.2 Section 8.5 and 8.6 Ratio, Root and Comparison Tests, Alternating Series Test For the following series, determine whether they diverge or converge. Clearly STATE the REASON/TEST that you employed. Show the values of all computed limits. To save time, you may assume the function is decreasing when applying the Alternating Series Test. !
1. 3. ! !
!!! ! ! 2. !
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!!! ! ! !&quot;# ! !!
!!! ! ! 4. !
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!!! ! !! 5. ! ! !!
!!! ! ! 6. 7. 9.
! !
!!! !! 8. !
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!!! ! ! !! !
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!!! !! !
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!!! (! !&quot;#)! 10. 11. ! ! !&quot; !
!!! ! 12. !
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!!! !
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!!!( !! ) !
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!!
!!!(! +3 ) !
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13. !
!!! ! ! !! 14. !!!(1 + ! ) !
!!
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15. !
!!!(−1) ! 16. !!!(−1) !&quot; ! !
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!!!(−1) ! ! 18. !!!(−1) !&quot; ! ! !&quot;# !
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19. !
!!!(−1) !!!! 20. !!! ! ! For #15 to #20, now determine if the series converge absolutely, converge conditionally, or diverge. Section 9.1: Taylor Polynomials For the following functions find the 5th degree Taylor Polynomial centered at 0. 21. 𝑓 𝑥 = sin 𝑥 22. 𝑓 𝑥 = 𝑒 !! 23. 𝑓 𝑥 = ln(1 − 𝑥) 24. For the function 𝑓 𝑥 = cos 𝑥 , find the 4th degree Taylor Polynomial centered at π/6. 25. For the function 𝑓 𝑥 = 𝑥, find the 4th degree Taylor Polynomial centered at 4. Section 9.2: Power Series, Radius of Convergence, and Interval of Convergence For the following power series, find the radius and interval of convergence 26. 28. !
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27. !!!
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30. !
!!! ! ! !! Using information for geometric series and rules of manipulation of power series, find the power series and interval of convergence for the following functions. !!
31. 𝑓 𝑥 = !!! !
32. 𝑓 𝑥 = !!!! Using the power series for 𝑓 𝑥 find the power series and interval of convergence for the function 𝑔 𝑥 . !
33. 𝑓 𝑥 = !!! and 𝑔 𝑥 = ln 1 − 𝑥 !
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34. 𝑓 𝑥 = !!!! and 𝑔 𝑥 = (!!!!)! Section 9.3: Taylor Series For the following functions find the Taylor Series 35. 𝑓 𝑥 = sin 𝑥 about 𝑥 = 0. 36. 𝑓 𝑥 = ln 𝑥, about 𝑥 = 4. 37. 𝑓 𝑥 = 𝑒 !! about 𝑥 = 1. Section 10.1: Parametric Equations For the following problems, a) eliminate the parameter to obtain an equation in x and y b) graph the curve and indicate the positive orientation c) find the requested derivative and tangent line 38. 𝑥 = 𝑒 !! , 𝑦 = 𝑒 ! + 1, 𝑓𝑜𝑟 0 ≤ 𝑡 ≤ 25. Find the derivate dy/dx when t=1 and the equation of the tangent line to the curve there. 39. 𝑥 = 4 cos 𝑡 , 𝑦 = 3 sin 𝑡 , 𝑓𝑜𝑟 0 ≤ 𝑡 ≤ 2𝜋 Find the derivate dy/dx when t=𝜋/4 and the equation of the tangent line to the curve there. 40. Find a set of parametric equations for the circle centered at the origin with radius 6, generated counterclockwise. Section 10.2: Polar Coordinates !
41. Change the polar coordinates (2, ! ) to rectangular coordinates. 42. Express the point (‐4, 4 3) in polar coordinates with the angle 𝜃 between and 2π. 43. Graph the polar curve 𝑟 = 4 sin 𝜃. 44. Graph the polar curve r=2. 45. Graph the polar curve r= 2(1‐cos 𝜃). 46. Graph the polar curve 𝜃 = 2𝜋/3. Convert the following equations to Rectangular (x‐y) form. 47. 𝑟 sin 𝜃 = 3. 48. 𝑟 = 6 cos 𝜃. 49. 𝑟 = 4. 50. 𝑟 = cot 𝜃 csc 𝜃. ```