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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.1.65c
Capitolo 11, Problema 11.1.65c

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


c. Only even powers of x appear in the nth−order Taylor polynomial for f(x)=√(1+x²) centered at 0.

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Recall that the nth-order Taylor polynomial of a function \(f(x)\) centered at 0 (Maclaurin polynomial) is given by: \[T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^k,\] where \(f^{(k)}(0)\) is the \(k\)th derivative of \(f\) evaluated at 0.
Consider the function \(f(x) = \sqrt{1 + x^2}\). Notice that \(f(x)\) is an even function because \(f(-x) = \sqrt{1 + (-x)^2} = \sqrt{1 + x^2} = f(x)\).
Since \(f(x)\) is even, its Taylor series expansion around 0 will contain only even powers of \(x\). This is a general property: the Taylor series of an even function centered at 0 contains only even powers, and the Taylor series of an odd function contains only odd powers.
To confirm this, you can compute the first few derivatives of \(f(x)\) at 0 and observe that all derivatives of odd order vanish at 0, i.e., \(f^{(1)}(0) = 0\), \(f^{(3)}(0) = 0\), etc., which means the coefficients of odd powers are zero.
Therefore, the nth-order Taylor polynomial for \(f(x) = \sqrt{1 + x^2}\) centered at 0 contains only even powers of \(x\), making the statement true.

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Taylor Polynomial

A Taylor polynomial approximates a function near a specific point using derivatives at that point. The nth-order Taylor polynomial includes terms up to the nth derivative, expressed as powers of (x - a), where a is the center. It provides a polynomial approximation that matches the function's behavior locally.
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Taylor Polynomials

Even and Odd Functions

An even function satisfies f(-x) = f(x), resulting in Taylor expansions with only even powers of x. An odd function satisfies f(-x) = -f(x), leading to only odd powers in its expansion. Understanding the symmetry of a function helps predict which powers appear in its Taylor series.
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Properties of Functions

Derivatives of Composite Functions

Calculating Taylor polynomials often requires finding derivatives of composite functions like f(x) = √(1 + x²). Using the chain rule and recognizing patterns in derivatives helps determine which terms vanish or remain, influencing the presence of even or odd powers in the polynomial.
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Evaluate Composite Functions - Special Cases
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c. Determine the interval of convergence of the series.


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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. If f has a Taylor series that converges only on (−2,2), then f(x²) has a Taylor series that also converges only on (−2,2).

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c. Determine the interval of convergence of the series.


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{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is

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c. Determine the interval of convergence of the series.


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{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals

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c. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).

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