Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 11 - Power Series

Problema 11.1.14

Capitolo 11, Problema 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)

Guida verificata passo dopo passo

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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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.

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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.

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Taylor Polynomials

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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Pratica correlata

Domanda del libro di testo

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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Domanda del libro di testo

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

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Domanda del libro di testo

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

∑ₖ₌₀∞ (2x)ᵏ/k!

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Domanda del libro di testo

Any method

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.

b. Determine the radius of convergence of the series.

f(x) = (1 + x²)⁻²/³

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Domanda del libro di testo

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

ln (1 + x²)

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Domanda del libro di testo

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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