a. Explanation Given: The function f ( x ) = ( 1 + x ) − 3 . Calculation: The n th-order Tayler polynomial centered at a = 0 is as follows, p n ( x ) = f ( 0 ) + f ′ ( 0 ) 1 ! ( x ) + f ″ ( 0 ) 2 ! ( x ) 2 + ⋯ + f n ( 0 ) n ! ( x ) n (1) The value of f ( x ) = cos ( x ) and its derivatives at x = 0 as shown in the below table. n f n ( x ) f n ( 0 ) 0 f ( x ) = ( 1 + x ) − 3 f ( 0 ) = 1 1 f ′ ( x ) = − 3 ( 1 + x ) − 4 f ′ ( x ) = − 3 2 f ″ ( x ) = 12 ( 1 + x ) − 5 f ″ ( 0 ) = 12 Substitute 0 for n in equation (1) and obtain p 0 ( x ) b. To determine To sketch: The Taylor polynomial and the function.

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Chapter 9.1, Problem 21E

To determine

To find: The nth order Taylor polynomial centered at 0 and n=0,1,2.

To determine

To sketch: The Taylor polynomial and the function.

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Chapter 9.1, Problem 20E

Chapter 9.1, Problem 22E

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What is p1 and p2? Also what would the graph look like?

a. Find the nth-order Taylor polynomials of the given function centered at 0, for n = 0, 1, and 2. b. Graph the Taylor polynomials and the function. f(x) = In (1-8x) a. Po(x) = ₁ P₁ (x) = ₁P₂(X) = b. Graph the Taylor polynomials and the function. Choose the correct graph below. O A. f(x) 2- -0.5 P(x) Ау 2 Po(x) 015 P(x) X Q N O B. P(x) -0.5 P1) 4 2 Ay f(x) Po(x) 05 X O C. -1 -0.5 (X) 2 2 P₁(x) 05 R₂(x) X

Find the remainder term R, in the nth order Taylor polynomial centered at a for the given function. Express the result for a general value of n. 5x. f(x) = e-3%, a = 0 Choose the correct answer below. (-5)"e - 5c (n + 1)! O A. R,(x) = -x"' for some c between x and 0. (-5)+150 O B. R, (X) = -x"*' for some c between x and 0. (n + 1)! (-5)"e -50 (n + 1)! O C. R,(x) = -x" for some c between x and 0. - 50 (- 5)"+1e O D. R,(x) = for some c between x and 0. (n + 1)!

Chapter 9 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 9.1 - Suppose you use a second-order Taylor polynomial...Ch. 9.1 - Does the accuracy of an approximation given by a...Ch. 9.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 9.1 - Prob. 4E Ch. 9.1 - How is the remainder Rn(x) in a Taylor polynomial...Ch. 9.1 - Explain how to estimate the remainder in an...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...

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The n th order Taylor polynomial centered at 0 and n = 0 , 1 , 2 .

Solution Summary: The author explains the nth-order Taylor polynomial centered at a=0 and its derivatives.

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To find: The nth order Taylor polynomial centered at 0 and n=0,1,2.

To sketch: The Taylor polynomial and the function.

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Chapter 9 Solutions