Problem 44 - Brief Applied Calculus

Question

Use a graphing calculator to verify that the derivative of a linear function is a constant, as follows. Define \(y_{1}\) to be a linear function (such as \(\left.y_{1}=3 x-4\right)\) and then use NDERIV to define \(y_{2}\) to be the derivative of \(y_{1}\). Then graph the two functions together on an appropriate window and observe that the derivative \(y_{2}\) is a constant (graphed as a horizontal line, such as \(y_{2}=3\) ), verifying that the derivative of \(y_{1}=m x+b\) is \(y_{2}=m\).

Step-by-Step Solution

Answer

The derivative \( y_2 = 3 \) is constant, confirming the slope of \( y_1 \).

Step 1: Define the Linear Function

On your graphing calculator, designate the first function as \( y_1 \) and enter the linear equation \( y_1 = 3x - 4 \). This will be the function for which you want to find the derivative.

Step 2: Define the Derivative Function

Use the `NDERIV` function of the calculator to define \( y_2 \) as the derivative of \( y_1 \). Input \( y_2 = \text{ndr}(3x - 4, x, x) \) or use your calculator's specific method to compute derivatives. This represents the mathematical derivative of \( y_1 \).

Step 3: Set the Graphing Window

Adjust the viewing window on your graphing calculator to appropriately show both functions. Generally, set the X-axis from, say, \(-10\) to \(10\) and the Y-axis from, say, \(-5\) to \(5\) to ensure both functions can be viewed clearly.

Step 4: Graph Both Functions Together

Plot both \( y_1 \) and \( y_2 \) on the graphing calculator. You will observe \( y_1 = 3x - 4 \) as a straight line with slope 3 and \( y_2 \) as a horizontal line, which indicates a constant value.

Step 5: Interpret the Graph

After graphing, you should see that the line \( y_2 \) is horizontal, indicating it is a constant. This confirms that the derivative of a linear function is constant. Specifically, \( y_2 = 3 \), which is the slope \( m \) of the original function \( y_1 = 3x - 4 \).

Key Concepts

Linear Functions Derivatives

In calculus, the derivative tells us how a function changes as its input changes. For linear functions, derivatives have a simple and predictable pattern. A linear function is generally expressed as \( y_1 = mx + b \). Here, \( m \) is the slope, which represents how steep the line is, and \( b \) is the y-intercept, showing where the line crosses the y-axis. The derivative of a linear function, \( y_2 \), is always the slope \( m \). This is because the slope of a line is constant, meaning it doesn't change as you move along the line. So if you calculate the derivative of \( y_1 = 3x - 4 \), you find that \( y_2 = 3 \), a constant value. This constant derivative is reflected as a horizontal line when graphed, reinforcing that the derivative is the same at any point across the graph.

NDERIV Function

The NDERIV function on a graphing calculator is a tool that helps you find the derivative of a function at a given point. This is particularly useful for visualizing and verifying derivatives graphically. When you input a function into NDERIV, such as \( y_1 = 3x - 4 \), the calculator computes and plots its derivative. In this example, using NDERIV yields \( y_2 = 3 \). Notice that the command often involves specifying the function, variable, and point of evaluation in a way such as: 'ndr(function, variable, point)'. Different calculators might have varying methods of input, so it's a good idea to refer to your specific calculator's guide for details. The power of NDERIV lies in its ability to quickly and accurately show how functions behave and change, something fundamental in calculus studies.

Graphing Techniques in Calculus

Graphing is an essential skill in calculus as it visually communicates the functions and their characteristics. To effectively graph a function and its derivative on a graphing calculator, understanding the correct window settings is crucial. For our example, the function \( y_1 = 3x - 4 \) is a straight line with a slope of 3. Graphing its derivative \( y_2 = 3 \) requires you to set a window where both lines can be clearly displayed. A common setting might be to center the graph, using an X-axis ranging from \(-10\) to \(10\) and a Y-axis from \(-5\) to \(5\).
This window setting ensures you see the whole behavior of \( y_1 \) and the constant value of \( y_2 \). By plotting both, you observe the original line moving diagonally, while \( y_2 \) forms a flat, horizontal line. This visual demonstration emphasizes the concept that the slope (or derivative) of a linear function is constant, and graphing these functions helps reinforce the theories and calculations done analytically.