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(a) Calculate the quartic (degree 4) Taylor polynomial T4(x) for f(x) = x - ln (x) with center a = 1 directly from the definition of a Taylor polynomial. Show the derivatives and their evaluations.

(a) Calculate the quartic (degree 4) Taylor polynomial T4(x) for f(x) = x - ln (x) with center a = 1 directly from the definition of a Taylor polynomial. Show the derivatives and their evaluations.

Calculus: Early Transcendentals
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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(a) Calculate the quartic (degree 4) Taylor polynomial T4(x) for f(x) = x · ln (x) with center a = 1 directly from the definition of a Taylor polynomial. Show the derivatives and their evaluations. (b) Use the polynomial T₁ (x) that you found in part (a) to write down a sum of terms that gives an approximation to the value of f(1.3) = (1.3) In(1.3). Give an answer with 6 decimal places. (c) Use Taylor's Inequality to find an upper bound on the error |R₂(x)] when T₁(x) is used to approximate ƒ(1.3) = (1.3) In(1.3). Part of your work will be to find a suitable value for "M." Give an answer with 6 decimal places. (d) Use a calculator to estimate the value of ƒ(1.3) = (1.3) · In(1.3) to 6 decimal places, and find the actual error [R₂(x)| resulting from using your estimate in part (b). How does this actual error compare to the upper bound on the error that you found in part (c)? Explain.
Transcribed Image Text:(a) Calculate the quartic (degree 4) Taylor polynomial T4(x) for f(x) = x · ln (x) with center a = 1 directly from the definition of a Taylor polynomial. Show the derivatives and their evaluations. (b) Use the polynomial T₁ (x) that you found in part (a) to write down a sum of terms that gives an approximation to the value of f(1.3) = (1.3) In(1.3). Give an answer with 6 decimal places. (c) Use Taylor's Inequality to find an upper bound on the error |R₂(x)] when T₁(x) is used to approximate ƒ(1.3) = (1.3) In(1.3). Part of your work will be to find a suitable value for "M." Give an answer with 6 decimal places. (d) Use a calculator to estimate the value of ƒ(1.3) = (1.3) · In(1.3) to 6 decimal places, and find the actual error [R₂(x)| resulting from using your estimate in part (b). How does this actual error compare to the upper bound on the error that you found in part (c)? Explain.
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