Question

In Exercises 31–38, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. $$ f(\theta)=4 \sin \theta-\theta \quad(0,0) $$

Short answer

The slope of the function \(f(\theta) = 4\sin(\theta) - \theta \) at the point (0,0) is 3.

Step-by-step solution

Find the derivative

The first step is to find the derivative of the given function. The function is \(f(\theta) = 4\sin(\theta) - \theta\) . Let's differentiate this function with respect to \( \theta \). Using the chain rule, the derivative of \( \sin(x) \) is \( \cos(x) \) and the derivative of \(x\) is 1. Hence, the derivative of the function \(f'(\theta)\) becomes \(4\cos(\theta) - 1\).

Find the slope at the given point

The next step is to find the slope of the graph of the function at the given point by substituting the \( \theta \) coordinate of the point into the derivative. The provided point is (0,0). When we substitute 0 into the derivative \(f'(\theta) = 4\cos(\theta) - 1\), we obtain \(f'(0) = 4\cos(0) - 1 = 3\)

Confirm results using graphing utility

In this final, use a graphing utility to graph the function and its derivative. At \( \theta = 0 \), the slope of the tangent line to the function should match the calculated value from step 2. In other words,the slope of the tangent line to the curve \( f(\theta) = 4\sin(\theta) - \theta \) at \( \theta = 0 \) should be 3.

Interpretation of Results

The results confirm that the slope (or rate of change) of \(f(\theta) = 4\sin(\theta) - \theta \) at \(\theta = 0 \) is given by 3. This means that at this particular point, the function is increasing at a rate of 3 units per unit increase in \(\theta\).

Key concepts

Understanding the Chain Rule

The chain rule is an essential concept in calculus for finding the derivative of composite functions. When dealing with a function like \( f(\theta) = 4\sin(\theta) - \theta \), we apply differentiation rules individually to each part. The chain rule tells us that if you have a composite function \( g(h(x)) \), the derivative is \( g'(h(x)) \cdot h'(x) \). In this scenario, we treat \( \sin(\theta) \) as a function. Its derivative, \( \cos(\theta) \), is determined by the rules of trigonometric differentiation. The rest of the function, \( -\theta \), has a straightforward derivative of \(-1\). By following these steps, the derivative \( f'(\theta) = 4\cos(\theta) - 1 \) arises. Understanding and applying the chain rule correctly is crucial for effectively finding derivatives of more complex expressions.

Exploring the Slope of a Function

The slope of a function at a particular point is a critical way to understand its behavior. The slope is essentially the derivative of the function, which tells us how the function is changing at any given point. In the context of the exercise \( f(\theta) = 4\sin(\theta) - \theta \), we wanted to find the slope where \( \theta = 0 \). By substituting \( \theta = 0 \) into the derivative \( 4\cos(\theta) - 1 \), we calculated that the slope is \( 3 \). This result means that at the point \( (0,0) \), the function's graph goes upwards with a steepness of 3. The slope gives insight into whether the function is increasing or decreasing at that point, helping you visualize its behavior.

Utilizing Graphing Utilities

Graphing utilities are powerful tools for visualizing mathematical functions and confirming analytical results. By graphing \( f(\theta) = 4\sin(\theta) - \theta \) and its derivative \( 4\cos(\theta) - 1 \), you can observe how the function behaves around specific points. This visualization helps verify the calculated slope at \( \theta = 0 \), where you should see the tangent line with a slope of 3. Graphing calculators or software like Desmos are excellent for these purposes, making it easier to understand the function's behavior across different intervals. The graphical representation complements the analytical process, creating a holistic understanding of the function's dynamics.

Concept of Rate of Change

The rate of change is essentially the speed at which a function's value is changing with respect to its input. It is synonymous with the slope of the function at a specific point. In our example, a slope of 3 at \( \theta = 0 \) indicates that for every unit increase in \( \theta \), the function \( f(\theta) \) rises by 3 units. This concept helps in understanding how fast or slow the function is moving as its input changes. Rates of change are vital across fields like physics, economics, and biology for interpreting dynamic phenomena. They help predict future behavior based on current trends by examining how one variable affects another.