Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 11 - Power Series

Problema 11.1.12

Capitolo 11, Problema 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.

Guida verificata passo dopo passo

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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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.

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Linearization

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.

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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 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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Estimating the Area Under a Curve Using Left Endpoints

Pratica correlata

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Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (tan ⁻¹ x − x)/x³"

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Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)

∑ₖ₌₀∞(√x − 2)ᵏ

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Combining power series Use the power series representation

f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,

to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.

f(3x) = ln (1 − 3x)

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Differential equations

a. Find a power series for the solution of the following differential equations, subject to the given initial condition

b. Identify the function represented by the power series.

y′(t) − 3y = 10, y(0) = 2

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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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Radius of convergence Find the radius of convergence for the following power series.

∑ₖ₌₁∞ (1−cos (1/2ᵏ)) xᵏ

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