Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises 9-16, use the four-step process to find the slope of the tangent line to the graph of the given function at any point. \(f(x)=13\)

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of the function \(f(x) = 13\) at any point is 0, as its derivative \(f'(x) = 0\).

Step 1: Compute the derivative of the function

To find the slope of the tangent line, we first need to compute the derivative of the given function. \(f(x) = 13\) is a constant function, the derivative will be as follows: \[ f'(x) = \frac{d}{dx} (13) = 0 \]

Step 2: Determine the slope of the tangent line

The derivative of the function represents the slope of the tangent line to the graph of the function at any given point. Since we have calculated \(f'(x) = 0\), this tells us that the slope of the tangent line is 0 at any point on the graph. Therefore, the slope of the tangent line to the graph of the function \(f(x) = 13\) at any point is 0.

Key Concepts

Derivative of a Function

The concept of the derivative is foundational in understanding how things change. It gives us the rate at which one quantity changes with respect to another. In the realm of calculus, the derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.

When we find the derivative of a function, symbolized as \( f'(x) \) or \( \frac{d}{dx}f(x) \), we're determining this instantaneous rate of change. If you imagine the function graphed on a coordinate plane, the derivative at a given value of \( x \) tells you how steeply the curve is rising or falling at that location.

For linear functions, the derivative is a constant, since such functions have a constant slope. In more complex functions, the derivative can vary across the domain, and this is where much of calculus's analytical power comes into play.

Constant Function

A constant function is one where the value of the function does not change no matter what the input is. This is represented as a horizontal line on a graph. For any value of \( x \), a constant function \( f(x) = k \) will always return the same result \( k \), a constant.

The importance of a constant function in calculus lies in its derivative. As seen in the given exercise where \( f(x) = 13 \), regardless of the value of \( x \), the function's output remains unchanged. Thus, the derivative or the rate of change is zero. This zero rate signifies that there is no slope—no incline or decline—to the tangent line at any point on the graph of a constant function.

Four-Step Process

Understanding the 'four-step process' is essential for tackling problems involving the determination of slopes of tangent lines. This systematic approach leads students through the series of operations required to find a derivative and then apply it to find slopes of tangent lines.

The steps typically involved in this process are:

• Identify the function and rewrite it if necessary to make the differentiation process easier.
• Compute the derivative of the function with respect to \( x \), which gives the general slope of the tangent line.
• Substitute the specific \( x \)-value into the derivative, if required, to get the slope at that particular point.
• Use the slope and the point to write the equation of the tangent line, if asked to do so.

By following this structured method, students can methodically work through the problem and understand each step, thereby enabling them to replicate this process across different types of functions.