Problem 33 - Essential Calculus

Question

Linear and Quadratic Approximations In Exercises 33 and 34 use a computer algebra system to find the linear approximation \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and the quadratic approximation \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) of the function \(f\) at \(x=a\). Use a graphing utility to graph the function and its linear and quadratic approximations. \(f(x)=\tanh x, \quad a=0\)

Step-by-Step Solution

Answer

The linear and quadratic approximations of the function \(f(x) = \tanh x\) at \(x = 0\) are both \(x\).

Step 1: Find the value of the function at \( a \)

Evaluate the function at \( x = a \). Given that \( f(x) = \tanh x \), and \( a = 0 \), then \( f(a) = \tanh(0) = 0 \).

Step 2: Find the first derivative of the function

The derivative of \( \tanh x \) is \( \frac{d}{dx} \tanh x = \frac{1}{\cosh^2 x} \). Evaluate this at \( x = a \) gives \( f^{\prime}(a) = \frac{1}{\cosh^2 0} = 1 \).

Step 3: find the second derivative of the function

The derivative of \( \frac{1}{\cosh^2 x} \) is \( \frac{d}{dx} \frac{1}{\cosh^2 x} = -2 \tanh(x) \cdot \frac{1}{\cosh^2 x} \). Evaluate this at \( x = a \) gives \( f^{\prime \prime}(a) = -2 \tanh(0) \cdot \frac{1}{\cosh^2 0} = 0 \).

Step 4: Find the linear approximation

The formula for linear approximation is \(P_{1}(x) = f(a) + f^{\prime}(a)(x-a)\), which with the values already obtained is: \(P_{1}(x) = 0 + 1(x - 0) = x \). This is the equation for linear approximation.

Step 5: Find the quadratic approximation

The formula for quadratic approximation is \(P_{2}(x) = f(a) + f^{\prime}(a)(x-a) + \frac{1}{2} f^{\prime \prime}(a)(x-a)^2 \). With the values already obtained, this results in: \(P_{2}(x)= 0 + 1(x-0) + 0 = x \). This is the equation for quadratic approximation.

Key Concepts

Linear Approximation

In the realm of calculus, a linear approximation is often the simplest way to approximate a function near a specific point. The concept employs the tangent line at a given point to estimate the values of a function close to that point. The equation for the linear approximation of a function, also known as the first-order Taylor series, is given by:\[ P_{1}(x) = f(a) + f'(a)(x-a) \]where \( f(a) \) is the value of the function at \( x=a \), and \( f'(a) \) is the slope of the tangent line, or the derivative of the function at that point. This method assumes the function is sufficiently smooth and continuously differentiable around the point of approximation.

• It provides a good approximation when the changes in \( x \) are small.
• The closer \( x \) is to \( a \), the more accurate the approximation is.
• Linear approximations can be particularly useful for complex functions that are difficult to compute directly.

For our specific case of \( f(x) = \tanh x \) with \( a = 0 \), the linear approximation works out to be simply \( P_{1}(x) = x \). This is because \( f(0) = 0 \) and \( f'(0) = 1 \), leading to a linear function passing through the origin with a slope of 1.

Quadratic Approximation

Quadratic approximation is one step deeper than linear approximation. It includes the curvature of the function using the second derivative to provide a better fit for values near \( a \). The formula used here is the second-order Taylor series expansion:\[ P_{2}(x) = f(a) + f'(a)(x-a) + \frac{1}{2} f''(a)(x-a)^2 \]where \( f''(a) \) is the second derivative of \( f \) at \( a \). This aims to add accuracy by accounting for the concavity or convexity of the function.

• Incorporates the change in the slope of the tangent line through the second derivative.
• Approximates better than linear, especially for points further from \( a \).
• It is useful for analyzing the behavior of functions that aren't perfectly linear near \( a \).

For \( f(x) = \tanh x \) at \( a = 0 \), the second derivative \( f''(0) \) is \( 0 \). Hence, the quadratic approximation simplifies to \( P_{2}(x) = x \), which coincidentally aligns with the linear approximation.

Derivatives

Derivatives are core to understanding both linear and quadratic approximations. They represent the rate of change of a function's value with respect to changes in \( x \). Each derivative gives essential information:- The first derivative \( f'(x) \) indicates the slope or the angle of the tangent at a particular point on the function curve. It's crucial for linear approximation: \[ f'(a) = \frac{1}{\cosh^2 a} \] (for \( f(x) = \tanh x \)), evaluated at \( a=0 \) as 1.- The second derivative \( f''(x) \) provides insight into the curvature or concavity of the function: \[ f''(a) = -2 \tanh(a) \times \frac{1}{\cosh^2 a} \] evaluated at \( a=0 \), results in 0, in our exercise.These derivatives help construct approximations, capturing the linearity and curvature of functions:

• The first derivative is vital for the linear term in both approximations, giving a direct correlation of function and variable increase.
• The second derivative is only used in quadratic approximation, enhancing its precision when non-linearity significantly affects the function's shape.

Understanding derivatives allows us to model scenarios effectively using approximations, improving computational efficiency especially within small intervals.