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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.R.6
Capitolo 11, Problema 11.R.6

Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = e^(sin x), n = 2, a = 0

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1
Identify the function and the point of expansion: here, the function is \(f(x) = e^{\sin x}\) and the center is \(a = 0\).
Recall the formula for the nth-order Taylor polynomial centered at \(a\): \[T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k,\] where \(f^{(k)}(a)\) is the \(k\)th derivative of \(f\) evaluated at \(x = a\).
Calculate the derivatives of \(f(x)\) up to order 2: - First, find \(f(0) = e^{\sin 0}\). - Then, find \(f'(x)\) using the chain rule and evaluate \(f'(0)\). - Next, find \(f''(x)\) by differentiating \(f'(x)\) and evaluate \(f''(0)\).
Substitute the values of \(f(0)\), \(f'(0)\), and \(f''(0)\) into the Taylor polynomial formula: \[T_2(x) = f(0) + f'(0)(x - 0) + \frac{f''(0)}{2!}(x - 0)^2.\]
Write the final expression for the 2nd-order Taylor polynomial centered at 0, which approximates \(f(x)\) near \(x=0\).

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Taylor Polynomials

A Taylor polynomial approximates a function near a point a by using the function's derivatives at that point. The nth-order Taylor polynomial includes terms up to the nth derivative, providing a polynomial that closely matches the function's behavior near a.
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Taylor Polynomials

Derivatives of Composite Functions

To find the Taylor polynomial of e^(sin x), you must compute derivatives of a composite function. This involves applying the chain rule repeatedly to differentiate e^(sin x), since it is a composition of the exponential and sine functions.
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Evaluate Composite Functions - Special Cases

Evaluating Derivatives at the Center Point

After finding the derivatives, evaluate each at the center point a=0. These values are used as coefficients in the Taylor polynomial, ensuring the polynomial matches the function and its derivatives at that point.
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Evaluate Logarithms
Pratica correlata
Domanda del libro di testo

Approximating ln 2 Consider the following three ways to approximate

ln 2.

e. Using four terms of the series, which of the three series derived in parts (a)–(d) gives the best approximation to ln 2? Can you explain why?

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Domanda del libro di testo

Approximating ln 2 Consider the following three ways to approximate

ln 2.

b. Use the Taylor series for ln (1 - x) centered at 0 and the identity ln 2 = -ln 1/2. Write the resulting infinite series.

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Domanda del libro di testo

Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)


ƒ(x) = eˣ; bound R₃(x), for |x| < 1

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Domanda del libro di testo

Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.


ƒ(x) = ln (1 - 4x)

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Domanda del libro di testo

ƒ(x) = eˣ, a = 0; e-0.08


b. Use the Taylor polynomials to approximate the given expression. Make a table showing the approximations and the absolute error in these approximations using a calculator for the exact function value.

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Domanda del libro di testo

Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.



x +x³/3 +x⁵/5 +x⁷/7 + ...

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