Domanda del libro di testo
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = sin 2x, n = 3, a = 0
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 11, Problema 11.R.47
Convergence Write the remainder term Rₙ(x) for the Taylor series for the following functions centered at the given point a. Then show that lim ₙ → ∞ |Rₙ(x)| = 0, for all x in the given interval.
ƒ(x) = sinh x + cosh x, a = 0, - ∞ < x < ∞
Guida verificata passo dopo passo1
Identify the function and the center of the Taylor series expansion. Here, the function is \(f(x) = \sinh x + \cosh x\) and the center is \(a = 0\).
Recall the Taylor series remainder term in Lagrange form:
\[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x - a)^{n+1}\]
where \(c\) is some number between \(a\) and \(x\).
Find the \((n+1)\)-th derivative of \(f(x)\). Since \(f(x) = \sinh x + \cosh x\), note that the derivatives cycle in a predictable way. Compute \(f^{(n+1)}(x)\) explicitly or recognize the pattern.
Express the remainder term \(R_n(x)\) using the derivative found, substituting \(a=0\):
\[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\]
with \(c\) between \(0\) and \(x\).
To show that \(\lim_{n \to \infty} |R_n(x)| = 0\) for all real \(x\), analyze the growth of the numerator \(|f^{(n+1)}(c)|\) and the factorial in the denominator. Use the fact that factorial growth dominates exponential growth to conclude the limit is zero.
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Taylor Series and Remainder Term
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point. The remainder term Rₙ(x) measures the error between the function and its nth-degree Taylor polynomial. It is often expressed using the Lagrange form, involving the (n+1)th derivative evaluated at some point between a and x.
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Taylor Series
Hyperbolic Functions and Their Derivatives
The functions sinh x and cosh x are hyperbolic sine and cosine, respectively, with derivatives that cycle predictably: d/dx(sinh x) = cosh x and d/dx(cosh x) = sinh x. Understanding these derivatives helps in finding the Taylor series terms and the remainder for the combined function f(x) = sinh x + cosh x.
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Percorso guidato
5:50Asymptotes of Hyperbolas
Limit of the Remainder Term and Convergence
To prove convergence of the Taylor series, we show that the remainder term Rₙ(x) approaches zero as n approaches infinity for all x in the interval. This involves bounding the remainder term and using properties of the function and its derivatives to demonstrate that the error vanishes, ensuring the series converges to the function.
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06:32Alternating Series Remainder
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sin 20°
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ƒ(x) = 1/(4 + x²), a = 0
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Definite integrals by power series Use a Taylor series to approximate the following definite integrals. Retain as many terms as necessary to ensure the error is less than 10⁻³.
∫₀1 x cos x dx
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Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = cos x, a = π/2
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