Explanation Result used: Estimate of the Remainder “Let n be a fixed positive integer. Suppose there exist a number M such that, for all c | f ( n + 1 ) ( c ) | ≤ M between a and x inclusive. The remainder in the nth-order Taylor polynomial for f centered at a satisfies, | R n ( x ) | = | f ( x ) − p n ( x ) | ≤ M | x − a | n + 1 ( n + 1 ) ! .” Calculation: Consider the function f ( x ) = ln ( 1 + x ) at a = 0 . The ( n + 1 ) th derivative of f ( x ) = ln ( 1 + x ) is f n + 1 ( x ) = ( − 1 ) n n ! ( 1 + x ) n + 1 . Since ln ( 1 + x ) attains the maximum value at x = − 0.15 and this maximum is less than ( 1.2 ) n + 1 ⋅ n ! and by using the Result stated above, | f ( n + 1 ) ( c ) | ≤ ( 1.2 ) n + 1 ⋅ n ! for c ∈ [ − 0.15 , 0 ] . That is, the value of M = ( 1.2 ) n + 1 ⋅ n ! . The remainder in the n th-order Taylor polynomial for f centered at 0 is computed as follows, | R n ( x ) | ≤ M | x − 0 | n + 1 ( n + 1 ) ! = M | x | n + 1 ( n + 1 ) ! Substitute M = ( 1.2 ) n + 1 ⋅ n ! and x = − 0.15 in the above, | R n ( − 0.15 ) | = ( ( 1.2 ) n + 1 ⋅ n ! ) | − 0.15 − 0 | n + 1 ( n + 1 ) ! = ( ( 1.2 ) n + 1 ⋅ n ! ) ( 0.15 ) n + 1 n ! ( n + 1 ) = ( 0

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

See similar textbooks

Videos

Question

Chapter 11.1, Problem 62E

To determine

To find: The minimum order of the Taylor polynomial.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.1, Problem 61E

Chapter 11.1, Problem 63E

Students have asked these similar questions

Image attached thanks

Numerical analysis

Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10-3? (The answer depends on your choice of a center.) cos (-0.25)

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

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 minimum order of the Taylor polynomial.

Solution Summary: The author explains how the minimum order of the Taylor polynomial is n=3.

Videos

To find: The minimum order of the Taylor polynomial.

Want to see the full answer?

Chapter 11 Solutions