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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.1.56
Capitolo 11, Problema 11.1.56

{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.


tan x ≈ x on [−π/6, π/6]

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Identify the function and the approximation: here, the function is \(f(x) = \tan x\) and the approximation is \(f(x) \approx x\) on the interval \(\left[-\frac{\pi}{6}, \frac{\pi}{6}\right]\).
Recognize that the approximation \(\tan x \approx x\) is the first-degree Taylor polynomial of \(\tan x\) centered at 0, so the error can be estimated using the Lagrange remainder term for the Taylor series expansion.
Recall the Lagrange remainder formula for the Taylor polynomial of degree 1 centered at 0: the error \(R_1(x)\) satisfies \[R_1(x) = \frac{f^{(2)}(c)}{2!} x^2\] for some \(c\) between 0 and \(x\).
Calculate the second derivative of \(f(x) = \tan x\). First, \(f'(x) = \sec^2 x\), then \[f^{(2)}(x) = \frac{d}{dx} (\sec^2 x) = 2 \sec^2 x \tan x.\]
To find the error bound, determine the maximum value of \(|f^{(2)}(c)|\) on the interval \(\left[-\frac{\pi}{6}, \frac{\pi}{6}\right]\), then use it in the remainder formula: \[|R_1(x)| \leq \frac{\max_{c \in [-\frac{\pi}{6}, \frac{\pi}{6}]} |f^{(2)}(c)|}{2} x^2.\]

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Taylor Polynomial Approximation

A Taylor polynomial approximates a function near a point using derivatives at that point. For tan x near 0, the linear approximation is tan x ≈ x, which is the first-degree Taylor polynomial. Understanding this helps in estimating how close the approximation is to the actual function.
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Taylor Polynomials

Remainder Term (Lagrange Form)

The remainder term quantifies the error between the function and its Taylor polynomial approximation. The Lagrange form expresses this error using a higher-order derivative evaluated at some point in the interval, providing a bound on the maximum possible error.
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Percorso guidato
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Alternating Series Remainder

Error Bound on an Interval

To find an error bound on [−π/6, π/6], we evaluate the maximum value of the relevant derivative in that interval. This maximum derivative value, combined with the remainder formula, gives a guaranteed upper limit on the approximation error over the entire interval.
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Determining Error and Relative Error
Pratica correlata
Domanda del libro di testo

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) = e⁻³ˣ

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


∑ₖ₌₁∞ (3x + 2)ᵏ/k

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Inverse sine Given the power series

1/√(1 − x²) = 1 + (1/2)x² + (1 ⋅ 3)/(2 ⋅ 4) x⁴ + (1 ⋅ 3 ⋅ 5)/(2 ⋅ 4 ⋅ 6) x⁶ +⋯

for −1<x<1, find the power series for f(x) = sin ⁻¹ x centered at 0.

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Limits Evaluate the following limits using Taylor series.

lim ₓ→₁ (x 1)/(ln x)

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{Use of Tech} Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10⁻³ ? (The answer depends on your choice of a center.)

ln 0.85

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{Use of Tech} Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.


f(x) = ∜x with a=16; approximate ∜13.

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