(c) Use Taylor's Inequality to find an upper bound on the error |R4(x)] when T₁(x) is used to approximate f(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.
(c) Use Taylor's Inequality to find an upper bound on the error |R4(x)] when T₁(x) is used to approximate f(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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please solve for part C
![(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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F397485a0-a213-449a-932e-f264adae2773%2F7e66fb54-f59b-427b-bb26-fc7f08104385%2Fmsghezl_processed.png&w=3840&q=75)
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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