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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.R.4
Capitolo 11, Problema 11.R.4

Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = cos⁻¹ x, n = 2, a = 1/2

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1
Identify the function and the point of expansion: here, the function is \(f(x) = \cos^{-1} x\) and the center is \(a = \frac{1}{2}\).
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 \(f^{(k)}(a)\) denotes the \(k\)th derivative of \(f\) evaluated at \(x = a\).
Calculate the function value at \(a\): \[f(a) = \cos^{-1} \left( \frac{1}{2} \right).\]
Find the first and second derivatives of \(f(x)\): - First derivative: \[f'(x) = \frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}},\] - Second derivative: \[f''(x) = \frac{d}{dx} f'(x) = \frac{x}{(1 - x^2)^{3/2}}.\]
Evaluate the first and second derivatives at \(x = a = \frac{1}{2}\): \[f'(a) = -\frac{1}{\sqrt{1 - (\frac{1}{2})^2}}, \quad f''(a) = \frac{\frac{1}{2}}{(1 - (\frac{1}{2})^2)^{3/2}}.\] Then, substitute these values into the Taylor polynomial formula up to order 2: \[T_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2.\]

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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 Inverse Trigonometric Functions

To find the Taylor polynomial of ƒ(x) = cos⁻¹(x), it is essential to know the derivatives of the inverse cosine function. These derivatives involve expressions with square roots and powers, which are used to compute the coefficients of the polynomial.
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Derivatives of Other Inverse Trigonometric Functions

Centering the Polynomial at a Specific Point

Centering the Taylor polynomial at a = 1/2 means all derivatives are evaluated at x = 1/2. This shifts the polynomial's approximation to be most accurate near this point, affecting the polynomial's terms and their values.
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Taylor Polynomials
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