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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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.

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.

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.

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