Maclaurin and Taylor Series

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Function / Series / Terms

BC Calculus 2013 Test Form A Name ______

Maclaurin and Taylor Series Non Calculator Section

#1-2, Use direction substitution into known power series to find the power series (sigma notation is all that is needed—you do not need first few terms). Simplify your answer. 4 points each

1.

2.

In #3 and 4, find the radius and the interval of convergence for the following series.

5 points each

3.

4.

5. Find a power series for . (4 points)

6. Using the Maclaurin series for , find the coefficient of the term in the expansion of . (6 points)

7. Let equal the Taylor polynomial

Find the following: (2 points each)

a) b) c)

d) Explain whether the curve e) Can you tell whether P(x) has a

is concave up or down at x = 1. local min or local max at x =1?

Why or why not?

8. Find using power series. (5 points)

9. Find using power series. (5 points)

Multiple Choice Section. 4 points each

10. The Maclaurin series for the function f is given by . What is the value of f(3)?

(a) -3 (b) (c) (d) (e) 4

11. What is the radius of convergence of the series ?

(a) (b) (c) 3 (d) (e)

12. For x >0, the power series converges to which of the following?

(a) cos x (b) sinx (c) (d) (e)

13. The power series converges at x = 5. Which of the following must be true?

(a) The series diverges at x = 0. (b) The series diverges at x = 1.

(c) The series converges at x = 1. (d) The series converges at x = 2.

(e) The series converges at x = 6.

14. Let f be a function having derivatives of all orders for x >0 such that f(3)= 2,

f ʹ(3) = -1, f ʺ (3) = 6 and . Which of the following is the third-degree Taylor polynomial for f about x = 3?

(a) (b)

(c) (d)

(e)

15. Let be the fifth-degree Taylor polynomial for the function f about x = 0. What is the value of ?

a. b. c. -4 d. -24 e. -30

16. What is the coefficient of in the Taylor Series for about x = 0?

a. -24 b. -4 c. 6 d. -8 e. 3

BC Calculus 2013 Test Form A Name ______

Maclaurin and Taylor Series Calculator Section

17. a. Find the 2nd order Maclaurin polynomial for . (5 pts)

b. Approximate using T2(x). (3 pts)

c. Find the maximum possible size of 4 points

18. a. Use a fourth order Taylor Polynomial, , for the Maclaurin series for to estimate . (4 points)

b. Find the maximum possible size of . (4 points)

19. Use the LaGrange Error bound to find the maximum possible size of .

(4 pts)

20. Suppose can be approximated by a 4th-degree

Taylor polynomial, , centered at . Given

the graph of at the right, calculate the Lagrange

error bound for f at the following x-values: 3 points each

a. x = 0.7

b. x = 1.5