Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)
f(x) = 1 / √(1 − x²)

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Identify the point at which the Taylor polynomial is to be generated. Since the problem does not specify a value for \( a \), assume it is at \( a = 0 \) unless otherwise stated.

Recall the formula for the Taylor polynomial of order 1 (linearization) of a function \( f(x) \) at \( x = a \): \[ L(x) = f(a) + f'(a)(x - a) \]

Calculate \( f(a) \) by substituting \( x = a \) into the function \( f(x) = \frac{1}{\sqrt{1 - x^2}} \).

Find the first derivative \( f'(x) \) using the chain rule. For \( f(x) = (1 - x^2)^{-1/2} \), differentiate to get \( f'(x) = \frac{d}{dx} (1 - x^2)^{-1/2} \).

Evaluate \( f'(a) \) by substituting \( x = a \) into the derivative expression, then write the linearization \( L(x) = f(a) + f'(a)(x - a) \) explicitly.

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주요 개념

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Taylor Polynomial

A Taylor polynomial approximates a function near a point using derivatives at that point. The polynomial of order n uses derivatives up to the nth order to create a polynomial that closely matches the function's behavior near the chosen point.

Linearization (Taylor Polynomial of Order 1)

Linearization is the first-order Taylor polynomial, which approximates a function near a point using the function's value and its first derivative at that point. It provides a linear approximation useful for estimating function values close to the point.

Derivatives of the Function f(x) = 1 / √(1 − x²)

To find Taylor polynomials, you need the function's derivatives at the point of approximation. For f(x) = 1/√(1−x²), calculating the first and second derivatives involves applying the chain and power rules carefully due to the square root and the composite function.

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function f(x) at x = a is called the quadratic approximation of f at x = a. In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1)f(x) = 1 / √(1 − x²)