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 11.1.14

Chapter 11, Problem 11.1.14

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) = cos x, a = π/4; approximate cos (0.24π)

Verified step by step guidance

Step 1: Identify the function and the center point. Here, the function is \(f(x) = \cos x\) and the center point is \(a = \frac{\pi}{4}\).

Step 2: Find the value of the function and its derivatives at the center point \(a\). Calculate \(f(a) = \cos\left(\frac{\pi}{4}\right)\), the first derivative \(f'(x) = -\sin x\) and evaluate \(f'(a) = -\sin\left(\frac{\pi}{4}\right)\), and the second derivative \(f''(x) = -\cos x\) and evaluate \(f''(a) = -\cos\left(\frac{\pi}{4}\right)\).

Step 3: Write the linear approximating polynomial (the first-degree Taylor polynomial) centered at \(a\): \(L(x) = f(a) + f'(a)(x - a)\).

Step 4: Write the quadratic approximating polynomial (the second-degree Taylor polynomial) centered at \(a\): \(Q(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2\).

Step 5: Use the polynomials \(L(x)\) and \(Q(x)\) to approximate \(\cos(0.24\pi)\) by substituting \(x = 0.24\pi\) into each polynomial.

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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. It is given by L(x) = f(a) + f'(a)(x - a), where f'(a) is the derivative at a. This method provides a simple, first-order estimate of the function close to a.

Quadratic Approximation (Second-Order Taylor Polynomial)

Quadratic approximation improves accuracy by including the second derivative term. The polynomial is Q(x) = f(a) + f'(a)(x - a) + (f''(a)/2)(x - a)^2, capturing curvature near a. It provides a better estimate than linear approximation for values near the center point.

Using Approximations to Estimate Function Values

Once linear and quadratic polynomials are found, they can approximate function values at points near a. This avoids complex calculations of the original function, especially for transcendental functions like cosine. Comparing both approximations shows the improvement in accuracy.

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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

Textbook Question

Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that lim ₙ→∞ Rₙ(x)=0, for all x in the interval of convergence.

f(x) = e⁻ˣ, a = 0

Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₀∞ (2x)ᵏ/k!

Textbook Question

Evaluating an infinite series Write the Maclaurin series for f(x) = ln (1+x) and find the interval of convergence. Evaluate f(−1/2) to find the value of ∑ₖ₌₁∞ 1/(k 2ᵏ)

Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

−x²/1 + x⁴/2! −x⁶/3! + x⁸/4! − ⋯

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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