Problem 41 - Thomas Calculus

Question

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 \(x=a.\) find the (a) linearization (Taylor polynomial of order 1) and (b) quadratic approximation of \(f\) at \(x=0\). \(f(x)=\ln (\cos x)\)

Step-by-Step Solution

Answer

(a) Linearization is 0; (b) quadratic approximation is \(-\frac{1}{2}x^2\).

Step 1: Define the Function and Derivatives

We start with the function given, which is \( f(x) = \ln(\cos x) \). The first derivative of \( f(x) \) is \( f'(x) = -\tan x \). The second derivative is \( f''(x) = -\sec^2 x \). We will evaluate these at \( x = 0 \).

Step 2: Evaluate Function and Derivatives at x=0

Calculate \( f(0) \): \( f(0) = \ln(\cos 0) = \ln(1) = 0 \). \( f'(0) = -\tan 0 = 0 \). \( f''(0) = -\sec^2 0 = -1 \).

Step 3: Determine the Linearization

The linearization, or Taylor polynomial of order 1, is given by \( L(x) = f(0) + f'(0)(x - 0) \). Since \( f(0) = 0 \) and \( f'(0) = 0 \), the linearization is \( L(x) = 0 + 0 \cdot x = 0 \).

Step 4: Determine the Quadratic Approximation

The quadratic approximation or Taylor polynomial of order 2 is \( Q(x) = f(0) + f'(0)(x - 0) + \frac{f''(0)}{2}(x - 0)^2 \). Since \( f(0) = 0 \), \( f'(0) = 0 \), and \( f''(0) = -1 \), we have \( Q(x) = 0 + 0 \cdot x - \frac{1}{2}x^2 = -\frac{1}{2}x^2 \).

Key Concepts

Quadratic Approximation

Quadratic approximation is a critical concept in calculus, particularly when dealing with Taylor polynomials. It provides a way to estimate the value of a function using a polynomial of degree 2, making it useful for analyzing the behavior of functions near a specific point. In particular, the quadratic approximation is the Taylor polynomial of order 2.

When using this method, we aim to approximate a function around a point, say \( x = a \). For a given twice-differentiable function \( f(x) \), the quadratic approximation consists of the function's value, its first derivative, and its second derivative at that point. These values determine the coefficients of the polynomial:

\[ Q(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 \]

In our exercise, the point \( x = 0 \) is used. For the function \( f(x) = \ln(\cos x) \), this gives us a quadratic approximation of \( -\frac{1}{2} x^2 \). This means that near \( x = 0 \), we can use this simpler polynomial to approximate \( \ln(\cos x) \), which can significantly simplify analysis and estimation.

Linearization

Linearization is a way of approximating a function with a straight line, which is known as the linear approximation or the first-order Taylor polynomial. This method is beneficial because it simplifies the analysis of functions, turning complex functions into linear ones that are easier to handle.

linear approximation at a point \( x = a \) uses the value of the function and its first derivative, giving us the equation:

• \( L(x) = f(a) + f'(a)(x-a) \)

This gives us the tangent line to the function at \( x = a \). It�s especially useful when dealing with small values of \( x-a \), where higher-order terms have little effect.

In the given exercise, linearization of \( f(x) = \ln(\cos x) \) at \( x = 0 \) results in a function \( L(x) = 0 \). Though it might seem overly simple, in contexts where the function changes very slowly, such as near this point, the linearization can give a surprisingly accurate estimate of the function’s actual value.

Twice-Differentiable Function

A twice-differentiable function is a crucial concept in calculus because it allows us to explore the curvature of functions, which is essential for constructing Taylor polynomials of second order or higher.

For a function to be twice-differentiable, it must have both a continuous first derivative and a continuous second derivative. This means,

• First, the function itself must be smooth;
• Second, its rate of change (first derivative) must be smooth.

For Taylor expansions, having these derivatives ensures that the polynomial is an accurate representation of the function near the expansion point, \( x = a \).

In our exercise, \( f(x) = \ln(\cos x) \) is twice-differentiable around \( x = 0 \), allowing us to compute the first and second derivatives: \( f'(x) = -\tan x \) and \( f''(x) = -\sec^2 x \). This illustrates not only the function's behavior but also how the quadratic approximation is derived from these derivatives.