please #5, 9, 13, 17, 21, 25, 29

1. If f(x) = >no b,(x - 5)" for all x, write a formula for bg. 13. f(x) = cos x 14. f(x) = e-2x 2. The graph of f is shown. 15. f (x) = 2x4 - 3x2 + 3 16. f(x) = sin 3x 17. f(x) = 2" 18. f(x) = x cos x 19. f(x) = sinh x 20. f(x) = cosh x 21-30 Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that R,(x) -> 0.] Also find the associated radius of convergence. 21. f (x) = x5 + 2x3 + x, a= 2 (a) Explain why the series 1.1 + 0.7x2 + 2.2x' + . .. is not 22. f (x) = x6 - x+ + 2, a=-2 the Maclaurin series of f. (b) Explain why the series 23. f(x) = Inx, a =2 24. f (x) = 1/x, a= -3 1.6 - 0.8(x - 1) + 0.4(x - 1)2 - 0.1(x - 1)' + ... 25. f (x) = e2x, a= 3 26. f(x) = 1/x2, a=1 is not the Taylor series of f centered at 1. 27. f(x) = sin x, a = T 28. f(x) = cos x, a = T/2 (c) Explain why the series 29. f(x) = sin 2x, a = T 30. f (x ) = Vx, a = 16 2.8 + 0.5(x - 2) + 1.5(x - 2)2 - 0.1(x - 2)3+ ... is not the Taylor series of f centered at 2. 31. Prove that the series obtained in Exercise 13 represents cos x for all x. 3. If f (0) = (n + 1)! forn = 0, 1, 2, . . ., find the Maclaurin series for f and its radius of convergence. 32. Prove that the series obtained in Exercise 27 represents sin x for all x. 4. Find the Taylor series for f centered at 4 if 33. Prove that the series obtained in Exercise 19 represents sinh x f') (4) = - (-1)"n! for all x. 3" (n + 1) 34. Prove that the series obtained in Exercise 20 represents cosh x What is the radius of convergence of the Taylor series? for all x. 5-10 Use the definition of a Taylor series to find the first four 35-38 Use the binomial series to expand the given function as a nonzero terms of the series for f(x) centered at the given value of a. power series. State the radius of convergence. 5. f (x) = xe', a =0 6. f(x) = - 35. V1 - x 36. V8 + x 1 + x' a = 2 7. f (x) = Vx, a=8 8. f(x) = Inx, a=1 37. 38. (1 - x) 3/4 (2 + x)3 9. f(x) = sin x, a = T/6 10. f(x) = cos-x, a = 0 39-48 Use a Maclaurin series in Table 1 to obtain the Maclaurin 11-20 Find the Maclaurin series for f(x) using the definition of a series for the given function. Maclaurin series. [Assume that f has a power series expansion. 39. f(x) = arctan (x2) 40. f(x) = sin(Tx/4) Do not show that R,,(x) -> 0.] Also find the associated radius of convergence. 41. f(x) = x cos 2x 42. f ( x) = ex - e2x 11. f(x) = (1 - x)-2 12. f(x) = In(1 + x) 43. f(x) = x cos(x2) 44. f(x) = x2 In(1 + x3)