Question

Find the slope of the graph of the function at the indicated point. Use the derivative feature of a graphing utility to confirm your results. $$ f(\theta)=4 \sin \theta-\theta $$ $$ (0,0) $$

Short answer

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

Step-by-step solution

Find the Derivative of the Function

The derivative of a function measures how the function changes as its input changes. In this case, we need to find the derivative of \( f(\theta)=4 \sin \theta - \theta \). We apply the power rule and the chain rule to differentiate the sine function and the linear term separately. The derivative of \( 4 \sin \theta \) is \( 4 \cos \theta \) and the derivative of \( -\theta \) is -1. So the derivative of the function \( f'(\theta) \) is \( 4 \cos \theta - 1 \).

Evaluate the Derivative at the Given Point

Now that we have the derivative \( f'(\theta)=4 \cos \theta - 1 \), we can evaluate it at the point \( \theta = 0 \) to find the slope at that point. Plugging in \( \theta = 0 \) into the derivative, we get \( f'(0)=4 \cos (0) - 1 \). Given that \( \cos(0) = 1 \), the derivative at 0 simplifies to \( f'(0)=4(1) - 1 = 3 \).

Interpret the Result

The derivative at the point (0,0) is 3. This means that the slope of the graph of the function at the point (0,0) is 3.

Key concepts

Derivative of a Function

Understanding the derivative of a function is key to analyzing how the function behaves. In essence, the derivative represents the rate of change of a function's output with respect to its input. For example, if you have a function that defines the distance travelled over time, the derivative of this function gives you the velocity, or how quickly the distance is changing as time advances.

To find the derivative, you often use rules of differentiation such as the power rule, which states that the derivative of \( x^n \) is \( nx^{n-1} \), or the product rule for the derivative of two multiplied functions. Sometimes, when dealing with more complex expressions, such as trigonometric functions, specific rules such as the derivative of \( \text{sin}(x) \) being \( \text{cos}(x) \) come into play. In the given exercise, the derivative of \( f(\theta)=4 \text{sin} \theta - \theta \) involves applying these rules to find the expression for the instantaneous rate of change at any point on its graph.

Chain Rule in Calculus

Applying the Chain Rule

The chain rule is one of the most powerful tools in calculus, especially when it comes to finding the derivative of composite functions. A composite function is a function made by combining two or more functions. The chain rule allows you to differentiate the composite function by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

The classic formula for the chain rule is \( (f(g(x)))' = f'(g(x)) \times g'(x) \). When you're faced with a function like \( \text{sin}(x^2) \), the chain rule tells you that you will first take the derivative of the outer function, \( \text{sin}(x) \), then multiply it by the derivative of the inner function, \( x^2 \text{.} \) Understanding and applying the chain rule is essential when trying to find the slope of complex functions, as it simplifies the process of differentiation.

Graphing Utilities in Calculus

Maximizing Technology for Understanding

Graphing utilities have become a staple in studying calculus because they provide an immediate visual representation of functions and their characteristics. These utilities, whether they are software or graphing calculators, allow users to quickly graph a function, find its derivative, and assess various properties like roots, maxima, and minima.

One of the most beneficial features of graphing utilities is their ability to compute derivatives at specific points. Instead of performing differentiation by hand, which might be prone to errors, graphing utilities can give you an exact slope of the tangent line at any chosen point on the graph. It is in this way that students can confirm the solutions they get analytically, just like in our example where the graphing utility would confirm the slope as 3 at the point (0,0). The integration of graphing utilities in learning calculus empowers students to visualize and understand the implications of derivatives on function graphs more clearly.