Problem 15 - Calculus - AP Edition

Question

Confirm that the stated formula is the local linear approximation of \(f\) at \(x_{0}=1,\) where \(\Delta x=x-1\). $$\tan ^{-1} x ; \tan ^{-1}(1+\Delta x) \approx \frac{\pi}{4}+\frac{1}{2} \Delta x$$

Step-by-Step Solution

Answer

The formula is confirmed to be the local linear approximation.

Step 1: Understand Local Linear Approximation Formula

The local linear approximation formula is given by:\[ L(x) = f(x_0) + f'(x_0) \cdot (x - x_0) \]Here, \( f(x) = \tan^{-1}x \) and \( x_0 = 1 \). We need to find \( f(1) \) and the derivative \( f'(1) \).

Step 2: Calculate Function Value at \(x_0\)

Calculate \( f(1) \) using the function \( f(x) = \tan^{-1}x \):\[ f(1) = \tan^{-1}(1) = \frac{\pi}{4} \]

Step 3: Compute Derivative of Function

First, find the derivative of \( f(x) = \tan^{-1}x \). The derivative is:\[ f'(x) = \frac{1}{1 + x^2} \]

Step 4: Evaluate Derivative at \(x_0\)

Now, evaluate this derivative at \( x_0 = 1 \):\[ f'(1) = \frac{1}{1 + 1^2} = \frac{1}{2} \]

Step 5: Construct Local Linear Approximation

Substitute \( f(1) = \frac{\pi}{4} \) and \( f'(1) = \frac{1}{2} \) into the local linear approximation formula:\[ L(x) = \frac{\pi}{4} + \frac{1}{2} \cdot (x - 1) \]

Step 6: Substitute \(\Delta x\) and Simplify

Substitute \( \Delta x = x - 1 \) into the approximation:\[ \tan^{-1}(1+\Delta x) \approx \frac{\pi}{4} + \frac{1}{2}\Delta x \]

Step 7: Present the Final Conclusion

By reconstructing the linear approximation with the calculated values, we confirm that the given formula, \( \tan^{-1}(1 + \Delta x) \approx \frac{\pi}{4} + \frac{1}{2} \Delta x \), is the local linear approximation at \( x_0 = 1 \).

Key Concepts

Derivative Calculation

Calculating the derivative of a function is a crucial step in understanding its rate of change. For the function \( f(x) = \tan^{-1}(x) \), known as the arctangent function, the derivative is not immediately obvious. To determine the derivative of this inverse trigonometric function, we use a specific formula

• The derivative of \( \tan^{-1}(x) \) is \( f'(x) = \frac{1}{1 + x^2} \).

This formula stems from the relationship between the tangent function and its inverse. Here, \( x \) represents any input value for the function, and the calculation of the derivative provides the slope of the tangent line to the function at a given point.
At \( x = 1 \), we evaluate this derivative:

• Substitute \( x = 1 \) into the derivative: \( f'(1) = \frac{1}{1 + 1^2} = \frac{1}{2} \).

This simple calculation shows us the slope of the function \( \tan^{-1}(x) \) at that specific point. Understanding how to find the derivative helps in constructing the local linear approximation formula.

Arctangent Function

The arctangent function, expressed as \( \tan^{-1}(x) \), is the inverse of the tangent function. It is used to find an angle whose tangent is a given number.

• One of its unique features is that it assigns an angle in radians between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
• For example, \( \tan^{-1}(1) \) equals \(\frac{\pi}{4}\), because the tangent of \(\frac{\pi}{4}\) is 1.

The arctangent function is widely used in various mathematical and engineering fields, particularly in situations where angles need to be calculated from ratio measurements.
In this exercise, understanding the arctangent function enables us to apply the appropriate linearization to the function at \( x = 1 \), confirming its local behavior near this value. By knowing the function value at \( x = 1 \), we can make predictions about the function's behavior around \( x = 1 \), which is foundational in calculus and approximations.

Taylor Series Expansion

The Taylor series is a powerful mathematical tool used to approximate functions with polynomials. It represents a function as an infinite sum of its derivatives at a single point. The local linear approximation is actually the first-order expansion of the Taylor series.

• This first-order approximation captures the function's value and slope at a point \( x_0 \).
• For the function \( \tan^{-1}(x) \) at \( x_0 = 1 \), the Taylor series starts as \( \frac{\pi}{4} + \frac{1}{2}(x-1) \).

This expression is known as the linear approximation, which provides an estimate for \( \tan^{-1}(1 + \Delta x) \), where \( \Delta x = x - 1 \).
The Taylor series can continue with higher-order terms, but the local linear approximation focuses on the simplest, first-order term. This makes it a convenient and practical tool for estimating function values near a specific point, particularly when complex calculations are unnecessary. By understanding Taylor series expansion, we gain insight into how functions behave locally, making approximations both simpler and more effective.