Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to estimate \(f^{\prime}(a)\) for the given value of \(a\). $$f(x)=x \cos x ; a=\frac{\pi}{4}$$

Step-by-Step Solution

Answer

The derivative \(f^{\prime}(a)\) is approximately 1.

Step 1: Understand the Derivative Concept

The derivative of a function at a particular point gives the slope of the tangent line to the function at that point. In this exercise, you need to find the derivative of the function \(f(x) = x \cos x\) at \(a = \frac{\pi}{4}\).

Step 2: Use the Definition of Derivative

The derivative of a function \(f\) at a point \(a\) can be estimated using the limit definition of derivative: \( f^{\prime}(a) \approx \frac{f(a+h) - f(a)}{h} \) for a small \(h\). Here, set \(h = 0.001\) or a similarly small value for precision.

Step 3: Calculate \(f(a)\)

Find \(f(a)\) where \(a = \frac{\pi}{4}\): \[ f\left(\frac{\pi}{4}\right) = \frac{\pi}{4} \cos\left(\frac{\pi}{4}\right) \]Since \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\), calculate\[ f\left(\frac{\pi}{4}\right) = \frac{\pi}{4} \times \frac{\sqrt{2}}{2} \approx 0.555 \]

Step 4: Calculate \(f(a+h)\) for a Small \(h\)

Choose a small \(h\), for instance, 0.001, and calculate:\[ f\left(\frac{\pi}{4} + 0.001\right) = \left(\frac{\pi}{4} + 0.001\right) \cos\left(\frac{\pi}{4} + 0.001\right) \]Use the calculator to find this value. You'll find that it approximately equals 0.556.

Step 5: Compute \(f^{\prime}(a)\) Using the Difference Quotient

Substitute the values into the difference quotient to find the derivative:\[ f^{\prime}\left(\frac{\pi}{4}\right) \approx \frac{0.556 - 0.555}{0.001} = 1 \]

Key Concepts

Limit Definition of Derivative

Understanding the limit definition of a derivative is crucial in calculus. This definition helps us understand how a function behaves at a specific point, by estimating the slope of the tangent line. The formula is:

• \( f^{\prime}(a) \approx \frac{f(a+h) - f(a)}{h} \)

This equation tells us to calculate the function's value at a slightly larger point, subtract the original function's value, and divide by the small difference \(h\), to estimate the derivative.
This method works because it mimics the process of taking a limit as \(h\) approaches zero.
Though we use a small nonzero value for \(h\) in practice, this gives us a close approximation. Setting \(h = 0.001\) is often precise. This is how we estimate the slope, which is synonymous with the derivative.
Using this approach, you can find how quickly the function is changing at a particular point, which is essential for deeper analysis in calculus.

Tangent Line

The tangent line is an essential concept in understanding derivatives. When we talk about finding the derivative at a given point, we are essentially finding the slope of this tangent line.
The tangent line touches the curve of the function at exactly one point without crossing it.
This line shows the immediate rate of change of the function at that point. By calculating the derivative, we are determining this slope.

• In our exercise, the function \(f(x) = x \cos x\) at \(a = \frac{\pi}{4}\) gives us a tangent line with a slope estimated to be 1.

This approximation shows how quickly the value of the function is increasing or decreasing exactly at that point. By understanding tangent lines, you gain insight into the local linear approximation of the function.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, play a crucial role in calculus and, by extension, in understanding derivatives.

• The function \(f(x) = x \cos x\) combines both an algebraic term and a trigonometric term.

The cosine function, \(\cos(x)\), oscillates between -1 and 1, leading to a wave-like behavior. This behavior affects the slope and curvature of the function.
Calculating derivatives of trigonometric functions often involves using identities like \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\), which simplify calculations.
In our example, knowing the trigonometric value \(\cos\left(\frac{\pi}{4}\right)\) helps us estimate \(f(\frac{\pi}{4})\).

• This foundational knowledge enables you to examine how trigonometric functions change and how their slopes are determined at specific points. This understanding is pivotal in numerous applications across mathematics and engineering.