CALCULUS BC



CALCULUS BC

WORKSHEET ON TAYLOR POLYNOMIALS

Work the following on notebook paper. Use your calculator only on problems 9, 10, and 12. Show all work.

On problems 1 – 5, find a Maclaurin polynomial of degree n for each of the following.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

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On problems 6 – 8, find a Taylor polynomial of degree n centered at x = c for each of the following.

6. [pic]

7. [pic]

8. [pic]

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9. Use your answer from problem 1 to approximate [pic] to four decimal places.

10. Use your answer from problem 7 to approximate [pic] to four decimal places.

11. Suppose the function [pic] is approximate near x = 0 by a sixth-degree Taylor

polynomial [pic] Give the value of:

(a) [pic] (b) [pic] (c) [pic] (d) [pic] (e) [pic]

12. Suppose that g is a function which has continuous derivatives, and that

[pic]

(a) What is the Taylor polynomial of degree 2 for g near 5?

What is the Taylor polynomial of degree 3 for g near 5?

(b) Use the two polynomials that you found in part (a) to approximate [pic]

TURN->>>

For problems 13 – 16, suppose that [pic] is the second degree Taylor polynomial

for the function f about x = 0. What can you say about the signs of a, b, and c if f has the graph given below.

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17. Show how you can use the Taylor approximation [pic] for x near 0 to find

[pic]

18. Use the fourth-degree Taylor approximation [pic] for x near 0 to find

[pic]

19. Estimate the integral [pic] using a Taylor polynomial for sin t about t = 0 of

degree 5.

Answers to Worksheet on Taylor Polynomials

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. 0.6042

10. 0.1823

11. (a) 0 (b) 3 (c) [pic] (d) 0 (e) 3600

12. (a) [pic]

(b) 3.205

3.2055

13. a > 0, b > 0, c < 0

14. a > 0, b < 0, c < 0

15. a < 0, b > 0, c > 0

16. a < 0, b < 0, c > 0

17. 1

18. [pic]

19. [pic]

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