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) = ln(cos x)

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Identify the point at which the Taylor polynomial is to be generated, denoted as \(a\). This is the center of the approximation.

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

Calculate the value of the function at \(x = a\): \[f(a) = \ln(\cos a)\]

Find the first derivative of the function \(f(x) = \ln(\cos x)\). Use the chain rule: \[f'(x) = \frac{d}{dx} \ln(\cos x) = \frac{1}{\cos x} \cdot (-\sin x) = -\tan x\]

Evaluate the first derivative at \(x = a\): \[f'(a) = -\tan a\] Then substitute \(f(a)\) and \(f'(a)\) into the linearization formula to write the linear approximation.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 function that is tangent to the original function at the chosen point.

Derivatives of Composite Functions

To find the Taylor polynomial of f(x) = ln(cos x), you must compute derivatives involving the chain rule, since f is a composition of ln and cos. Understanding how to differentiate composite functions is essential for finding accurate polynomial approximations.

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) = ln(cos x)