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]
____________________________________________________________________________
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]
______________________________________________________________________________
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.
______________________________________________________________________________
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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