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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.RE.11b
Capitolo 11, Problema 11.RE.11b

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.

Guida verificata passo dopo passo
1
Identify the function and the point of expansion: Here, the function is \(f(x) = e^{x}\) and the expansion point is \(a = 0\). This means we will use the Taylor series of \(e^{x}\) centered at 0, also known as the Maclaurin series.
Recall the Taylor polynomial formula for \(f(x)\) centered at \(a\): \[T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x - a)^k\] Since \(f(x) = e^{x}\), all derivatives \(f^{(k)}(x) = e^{x}\), so at \(a=0\), \(f^{(k)}(0) = 1\) for all \(k\).
Write the Taylor polynomials of various degrees for \(x = -0.08\): \[T_n(-0.08) = \sum_{k=0}^{n} \frac{(-0.08)^k}{k!}\] Calculate these partial sums for increasing values of \(n\) (e.g., \(n=1, 2, 3, 4, 5\)) to get successive approximations.
Calculate the exact value of \(e^{-0.08}\) using a calculator or software to use as a reference for error calculation.
Create a table listing each polynomial degree \(n\), the corresponding approximation \(T_n(-0.08)\), and the absolute error defined as: \[\text{Absolute Error} = |e^{-0.08} - T_n(-0.08)|\] This will show how the approximation improves as \(n\) increases.

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Taylor Polynomials

Taylor polynomials approximate a function near a point by using its derivatives at that point. For a function f(x) centered at a, the nth-degree Taylor polynomial sums terms involving derivatives of f at a, multiplied by powers of (x - a). This provides a polynomial approximation that becomes more accurate as n increases.
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Taylor Polynomials

Exponential Function and Its Derivatives

The exponential function e^x is unique because its derivative at any point is equal to the function itself. This property simplifies the Taylor polynomial for e^x, as all derivatives at a point a are e^a. Understanding this helps in constructing the polynomial terms efficiently.
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Derivatives of General Exponential Functions

Absolute Error in Approximations

Absolute error measures the difference between the exact value of a function and its approximation. Calculating this error helps evaluate the accuracy of Taylor polynomial approximations. It is found by subtracting the approximate value from the exact value and taking the absolute value.
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Determining Error and Relative Error
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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

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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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.


∞

Σ x⁴ᵏ/k²

k = 1

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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

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) = ln (1 - x); bound R₃(x), for |x| < 1/2

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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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