Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 11 - Power Series

Problem 11.1.12

Chapter 11, Problem 11.1.12

Use of Tech Linear and quadratic approximation

a. Find the linear approximating polynomial for the following functions centered at the given point a.

b. Find the quadratic approximating polynomial for the following functions centered at a.

c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.

f(x)=e⁻²ˣ, a=0; approximate e⁻⁰ᐧ².

Verified step by step guidance

Identify the function and the point of approximation: here, the function is \(f(x) = e^{-2x}\) and the center point is \(a = 0\).

Find the value of the function at \(a\): calculate \(f(0) = e^{-2 \cdot 0} = e^0 = 1\).

Compute the first derivative \(f'(x)\) and evaluate it at \(a\): since \(f(x) = e^{-2x}\), then \(f'(x) = -2 e^{-2x}\), so \(f'(0) = -2 e^0 = -2\).

For the linear approximation (first-degree Taylor polynomial) centered at \(a=0\), use the formula: \(L(x) = f(a) + f'(a)(x - a)\), which becomes \(L(x) = 1 - 2(x - 0) = 1 - 2x\).

For the quadratic approximation (second-degree Taylor polynomial), compute the second derivative \(f''(x)\) and evaluate at \(a\): \(f''(x) = 4 e^{-2x}\), so \(f''(0) = 4\). Then use the formula \(Q(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2\), which becomes \(Q(x) = 1 - 2x + 2x^2\). Finally, substitute \(x = 0.1\) (since \(e^{-0.2} = e^{-2 \cdot 0.1}\)) into both \(L(x)\) and \(Q(x)\) to approximate \(e^{-0.2}\).

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Key Concepts

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

Linear Approximation (Linearization)

Linear approximation uses the tangent line at a point to estimate the value of a function near that point. It is given by L(x) = f(a) + f'(a)(x - a), where f'(a) is the derivative at a. This method simplifies complex functions to linear polynomials for easier calculation of nearby values.

Quadratic Approximation (Second-Order Taylor Polynomial)

Quadratic approximation extends linearization by including the second derivative, providing a better estimate near the point a. It is expressed as Q(x) = f(a) + f'(a)(x - a) + (f''(a)/2)(x - a)^2, capturing curvature to improve accuracy over linear approximation.

Using Approximations to Estimate Function Values

Once linear and quadratic polynomials are found, they can approximate function values at points close to a. This avoids complex calculations of the original function, especially for transcendental functions like e^(-2x), by substituting x into the simpler polynomial forms.

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Related Practice

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∑ₖ₌₀∞(√x − 2)ᵏ

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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. Give the interval of convergence for the new series (Theorem 11.4 is useful). Use the Maclaurin series

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Textbook Question

How would you approximate e⁻⁰ᐧ⁶ using the Taylor series for eˣ?

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Is ∑ₖ₌₀ ∞ (5x − 20)ᵏ a power series? If so, find the center a of the power series and state a formula for the coefficients cₖ of the power series.

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Textbook Question

Exponential function In Section 11.3, we show that the power series for the exponential function centered at 0 is

eˣ = ∑ₖ₌₀∞ (xᵏ)/k!, for −∞ < x < ∞

Use the methods of this section to find the power series centered at 0 for the following functions. Give the interval of convergence for the resulting series.

f(x) = x²eˣ

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