Consider f(x) = arcsin(x) on the interval -≤ x ≤1 Here are the first six derivatives of this function: f'(x) √1-22 f"(x) (1-x2)2 f"" (x): 2x2+1 (1-x2) f(4)(x) = 6x3+9x (1-x2) f(5) (x) = 24x+72x²+9 (1-x²) f(6)(x) = 120x³+600x³+225x (1-2) (a) Write down the fifth degree Taylor Polynomial for this function centered at x = 0 (b) Use your polynomial from (a) to approximate arcsin() (c) Determine the maximum error for approximating arcsin(x) on ½ ≤ x ≤ using the fifth degree Taylor Polynomial. Hint: For (c) it will be helpful to know that |f(6) (x)| is maximized on this interval at x =
Consider f(x) = arcsin(x) on the interval -≤ x ≤1 Here are the first six derivatives of this function: f'(x) √1-22 f"(x) (1-x2)2 f"" (x): 2x2+1 (1-x2) f(4)(x) = 6x3+9x (1-x2) f(5) (x) = 24x+72x²+9 (1-x²) f(6)(x) = 120x³+600x³+225x (1-2) (a) Write down the fifth degree Taylor Polynomial for this function centered at x = 0 (b) Use your polynomial from (a) to approximate arcsin() (c) Determine the maximum error for approximating arcsin(x) on ½ ≤ x ≤ using the fifth degree Taylor Polynomial. Hint: For (c) it will be helpful to know that |f(6) (x)| is maximized on this interval at x =
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Consider f(x) = arcsin(x) on the interval -≤ x ≤1
Here are the first six derivatives of this function:
f'(x)
√1-22
f"(x)
(1-x2)2
f"" (x):
2x2+1
(1-x2)
f(4)(x) = 6x3+9x
(1-x2)
f(5) (x) = 24x+72x²+9
(1-x²)
f(6)(x) = 120x³+600x³+225x
(1-2)
(a) Write down the fifth degree Taylor Polynomial for this function centered at x = 0
(b) Use your polynomial from (a) to approximate arcsin()
(c) Determine the maximum error for approximating arcsin(x) on ½ ≤ x ≤ using the fifth
degree Taylor Polynomial.
Hint: For (c) it will be helpful to know that |f(6) (x)| is maximized on this interval at x =
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