a . Explanation Given: The function is f ( x ) = ( 1 + x ) − 4 . Result used: The n th-order Taylor polynomial centered at a is as follows, p n ( x ) = f 0 ( a ) 0 ! ( x − a ) 0 + f ′ ( a ) 1 ! ( x − a ) 1 + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f n ( a ) n ! ( x − a ) n . Calculation: The n th-order Taylor polynomial centered at 0 is as follows, p n ( x ) = f 0 ( 0 ) 0 ! x + f ′ ( 0 ) 1 ! ( x − 0 ) + f ″ ( 0 ) 2 ! x 2 + ⋯ + f n ( 0 ) n ! ( x − 0 ) n (1) The value of f ( x ) = ( 1 + x ) − 4 and its derivatives at x = 0 as shown in the below table. Substitute 0 for x in f ( x ) = ( 1 + x ) − 4 , f ( 0 ) = ( 1 + 0 ) − 4 = 1 Substitute 0 for n in equation (1), p 0 ( x ) = f ( 0 ) p 0 ( x ) = 1 Thus, the polynomial centered at 0 is p 0 ( x ) = 1 . Differentiate f ( x ) = ( 1 + x ) − 4 with respect to x , f ′ ( x ) = − 4 ( 1 + x ) − 5 p 1 ( x ) = 1 + f ′ ( 0 ) 1 ! x = 1 − 4 ( 1 + 0 ) − 5 1 x = 1 − 4 x Thus, the linear polynomial centered at 0 is p 1 ( x ) = 1 − 4 x b. To determine To sketch: For each Taylor polynomials give the interval on which the graph appears indistinguishable. c. To determine To find: The interval on which the above graph appears indistinguishable from the graph of f .

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

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

Publisher: PEARSON

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Chapter 11, Problem 66RE

To determine

To find: The value of Taylor polynomials p0,p1,p2,p3 centered at 0.

To determine

To sketch: For each Taylor polynomials give the interval on which the graph appears indistinguishable.

To determine

To find: The interval on which the above graph appears indistinguishable from the graph of f .

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Chapter 11, Problem 65RE

Chapter 12.1, Problem 1QC

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Chapter 11 Solutions

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 11.1 - Verify that p3 satisfies p3(k)(a)=f(k)(a), for k =...Ch. 11.1 - Prob. 2QC Ch. 11.1 - Prob. 3QC Ch. 11.1 - Write out the next two Taylor polynomials p4 and...Ch. 11.1 - Prob. 5QC Ch. 11.1 - Prob. 6QC Ch. 11.1 - Suppose you use a second-order Taylor polynomial...Ch. 11.1 - Does the accuracy of an approximation given by a...Ch. 11.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 11.1 - Prob. 4E

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The value of Taylor polynomials p 0 , p 1 , p 2 , p 3 centered at 0.

Solution Summary: The author calculates the nth-order Taylor polynomial centered at 0 by using p_n(x)=f

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To find: The value of Taylor polynomials p0,p1,p2,p3 centered at 0.

To sketch: For each Taylor polynomials give the interval on which the graph appears indistinguishable.

To find: The interval on which the above graph appears indistinguishable from the graph of f .

Want to see the full answer?

Chapter 11 Solutions