Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 11 - Power Series

Problema 11.RE.11a

Capitolo 11, Problema 11.RE.11a

ƒ(x) = eˣ, a = 0; e-0.08

a. Find the Taylor polynomials of order n = 1 and n = 2 for the given functions centered at the given point a.

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Identify the function and the center point: here, the function is \(f(x) = e^{x}\) and the center point is \(a = 0\).

Recall the general formula for the Taylor polynomial of order \(n\) 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 \(a\).

Calculate the derivatives of \(f(x) = e^{x}\) and evaluate them at \(a=0\): - \(f(x) = e^{x}\), so \(f(0) = e^{0} = 1\), - \(f'(x) = e^{x}\), so \(f'(0) = 1\), - \(f''(x) = e^{x}\), so \(f''(0) = 1\).

Write the Taylor polynomial of order \(n=1\) using the formula: \[T_1(x) = f(0) + f'(0)(x - 0) = 1 + 1 \cdot x = 1 + x.\]

Write the Taylor polynomial of order \(n=2\) using the formula: \[T_2(x) = f(0) + f'(0)(x - 0) + \frac{f''(0)}{2!}(x - 0)^2 = 1 + x + \frac{1}{2} x^2.\]

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Taylor Polynomials

Taylor polynomials approximate a function near a point using a finite sum of its derivatives at that point. The nth-order Taylor polynomial includes terms up to the nth derivative, providing increasingly accurate approximations as n increases.

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Taylor Polynomials

Derivatives of Exponential Functions

The function f(x) = e^x has the unique property that all its derivatives are equal to e^x. This simplifies finding Taylor polynomials since each derivative evaluated at a point a is e^a, making the polynomial terms straightforward to compute.

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Derivatives of General Exponential Functions

Centering the Polynomial at a Point

Centering a Taylor polynomial at a point a means the polynomial approximates the function near x = a. The polynomial uses (x - a) as the variable, ensuring the approximation is most accurate close to this center.

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Pratica correlata

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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 (x - 1)ᵏ/(k5ᵏ)

k = 1

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Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.

a. √(1 + 2x)

A. p₂(x)= 1 + 2x + 2x²

B. p₂(x) = 1 − 6x + 24x²

C. p₂(x) = 1 + x − x²/2

D. p₂(x) = 1 − 2x + 4x²

E. p₂(x) = 1 − x + (3/2)x²

F. p₂(x) = 1 − 2x + 2x²

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

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∞

Σ (x/9)³ᵏ

k = 0

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Taylor series and interval of convergence

a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x)=2/(1−x)³, a=0

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