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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 11.1.73b
Chapter 11, Problem 11.1.73b

{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero. 


b. Estimate f(0.2) and give a bound on the error in the approximation.


f(x) =√(1+x) ≈ 1 + x/2

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Identify the function and its approximation: The function given is \(f(x) = \sqrt{1+x}\), and the approximation near \(x=0\) is \(f(x) \approx 1 + \frac{x}{2}\).
Calculate the approximate value at \(x=0.2\) using the linear approximation: Substitute \(x=0.2\) into the approximation to get \(f(0.2) \approx 1 + \frac{0.2}{2}\).
Understand the error bound concept: The error in the approximation can be estimated using the remainder term from Taylor's theorem, which involves the second derivative of \(f(x)\).
Find the second derivative of \(f(x)\): First, compute \(f'(x) = \frac{1}{2\sqrt{1+x}}\), then find \(f''(x) = -\frac{1}{4(1+x)^{3/2}}\).
Use the error bound formula for the linear approximation: The error \(R_2\) satisfies \(|R_2| \leq \frac{M}{2} |x|^2\), where \(M\) is the maximum value of \(|f''(x)|\) on the interval between 0 and 0.2. Determine \(M\) and calculate the bound.

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Key Concepts

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

Linear Approximation (Tangent Line Approximation)

Linear approximation uses the tangent line at a point to estimate function values near that point. For f(x) near x=0, f(x) ≈ f(0) + f'(0)(x - 0). Here, √(1+x) is approximated by 1 + x/2, which is the tangent line at x=0.
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Slopes of Tangent Lines

Error Bound Using the Remainder Term in Taylor's Theorem

The error bound estimates how far the approximation is from the true value. Using Taylor's theorem, the remainder term involves higher derivatives evaluated at some point between 0 and x, providing a maximum possible error for the approximation.
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Taylor Series

Derivative of the Square Root Function

Understanding the derivative of f(x) = √(1+x) is essential for both approximation and error estimation. The first derivative is f'(x) = 1/(2√(1+x)), which determines the slope of the tangent line used in the linear approximation.
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Root Test
Related Practice
Textbook Question

Taylor series and interval of convergence


b. Write the power series using summation notation.


f(x) = cosh 3x, a = 0

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. The function f(x) = csc x has a Taylor series centered at π/2.

Textbook Question

Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.

b. Integrate the series to find a Taylor series for Si.

Textbook Question

Taylor series


b. Write the power series using summation notation.


f(x)=sin x, a = π/2

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. Let f(x)=x⁵−1 The Taylor polynomial for f of order 10 centered at 0 is f itself.

Textbook Question

Taylor series and interval of convergence


c. Determine the interval of convergence of the series.


f(x) = e⁻ˣ, a=0

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