a . Explanation Given: The function f ( x ) = x 4 , a = 16 . Result used: The n th-order Taylor polynomial centered at a is shown below, p n ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f n ( a ) n ! ( x − a ) n (1) Calculation: Substitute 16 for a in equation (1), p n ( x ) = f ( 16 ) + f ′ ( 16 ) ( x − 16 ) + f ″ ( 16 ) 2 ! ( x − 16 ) 2 + ⋯ + f n ( 16 ) n ! ( x − 16 ) n (2) Substitute e for x in f ( x ) = x 4 , f ( x ) = x 4 = 16 4 = 2 That is, f ( 16 ) = 2 . The derivative of f ( x ) = ( x ) 1 4 is computed as follows, f ′ ( x ) = 1 4 ( x ) 1 4 − 1 = 1 4 ( x ) − 3 4 Substitute 16 for a , f ′ ( 16 ) = 1 4 ( 16 ) − 3 4 = 1 4 ( 1 8 ) That is, f ′ ( 16 ) = 1 32 . The derivative of f ′ ( x ) = 1 4 ( x ) − 3 4 is computed as follows, f ′ ′ ( x ) = 1 4 ( − 3 4 ) ( x ) − 3 4 − 1 = − 3 16 ( x ) − 7 4 Substitute 16 for a , f ′ ′ ( 16 ) = − 3 16 ( 16 ) − 7 4 = − 3 16 ( 1 ( 16 ) 7 4 ) = − 3 16 ( 1 128 ) = 3 2048 That is, f ″ ( 16 ) = − 3 2048 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

See similar textbooks

Videos

Question

Chapter 9.1, Problem 36E

To determine

To find: The nth-order Taylor polynomial centered at the point a for n=0,1,and 2.

To determine

To sketch: The Taylor polynomial and the function.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.1, Problem 35E

Chapter 9.1, Problem 37E

Students have asked these similar questions

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

answer both parts

. Find a 3rd-degree Taylor polynomial for f(r) Simplify all coefficients, and keep all values exact. = sec z.

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...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The nth -order Taylor polynomial centered at the point a for n = 0 , 1 , and 2 .

Solution Summary: The author explains the nth-order Taylor polynomial centered at the point a=16 and n=1.

Videos

To find: The nth-order Taylor polynomial centered at the point a for n=0,1,and 2.

To sketch: The Taylor polynomial and the function.

Want to see the full answer?

Chapter 9 Solutions