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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.1.58
Chapter 11, Problem 11.1.58

{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.


√(1+x) ≈ 1 + x/2 on [−0.1,0.1]

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Identify the function to approximate: \(f(x) = \sqrt{1+x}\).
Recognize that the given approximation \(1 + \frac{x}{2}\) is the first-degree Taylor polynomial of \(f(x)\) centered at \(x=0\).
Recall the Lagrange form of the remainder (error) term for the Taylor polynomial of degree 1: \(R_1(x) = \frac{f^{\prime\prime}(c)}{2!} x^2\) for some \(c\) between 0 and \(x\).
Compute the second derivative of \(f(x)\): first, \(f'(x) = \frac{1}{2\sqrt{1+x}}\), then \(f''(x) = -\frac{1}{4(1+x)^{3/2}}\).
Find the maximum absolute value of \(f''(c)\) on the interval \([-0.1, 0.1]\) to bound the error, then use it in the remainder formula \(|R_1(x)| \leq \frac{\max|f''(c)|}{2} x^2\) to get the error bound.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Taylor Polynomial Approximation

A Taylor polynomial approximates a function near a point using derivatives at that point. For √(1+x), the linear approximation at x=0 is 1 + x/2, derived from the function's value and first derivative. This approximation is accurate close to the center point.
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Remainder Term (Lagrange Error Bound)

The remainder term estimates the error between the actual function and its Taylor polynomial. The Lagrange form uses the (n+1)th derivative evaluated at some point in the interval to bound the error, ensuring the approximation's accuracy within a specified range.
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Error Bound on a Closed Interval

To find the maximum error on [−0.1,0.1], evaluate the maximum absolute value of the relevant derivative on this interval. This maximum derivative value, combined with the remainder formula, provides a guaranteed upper bound on the approximation error.
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Related Practice
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