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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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1
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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Taylor Polynomials

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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Finding Area Between Curves on a Given Interval

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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Determining Error and Relative Error
Related Practice
Textbook Question

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.


f(x) = sin x, a = 0

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Textbook Question

How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?

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Textbook Question

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series


(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ⋯, for −1 < x < 1.


1/(3 + 4x)²

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Textbook Question

Tangent line is p₁ Let f be differentiable at x=a


a. Find the equation of the line tangent to the curve y=f(x) at (a, f(a)).


b. Verify that the Taylor polynomial p_1 centered at a describes the tangent line found in part (a).

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Textbook Question

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

tan ⁻¹ (1/2)

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Textbook Question

{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.


e⁰ᐧ²⁵, n=4

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