Problem 105 - Brief Applied Calculus

Question

For any \(x\), the function \(\operatorname{INT}(x)\) is defined as the greatest integer less than or equal to \(x\). For example, \(\operatorname{INT}(3.7)=3\) and \(\operatorname{INT}(-4.2)=-5\) a. Use a graphing calculator to graph the function \(y_{1}=\operatorname{INT}(x) .\) (You may need to graph it in DOT mode to eliminate false connecting lines.)

Step-by-Step Solution

Answer

Graph \( y = \operatorname{INT}(x) \) using DOT mode for clarity.

Step 1: Understanding the Function

The function \( \operatorname{INT}(x) \) returns the greatest integer less than or equal to \( x \). For positive numbers, it rounds down to the nearest whole number. For negative numbers, it also truncates towards negative infinity.

Step 2: Setting Up the Graphing Calculator

Select DOT mode on your graphing calculator. This mode plots individual points rather than connecting them with lines. This is important for step functions like \( \operatorname{INT}(x) \), which are not continuous.

Step 3: Enter the Function

Enter \( y_1 = \operatorname{INT}(x) \) into the graphing function of the calculator. Ensure it's set to graph in DOT mode.

Step 4: Graph the Function

Graph the function by pressing the appropriate buttons to display the graph. In DOT mode, the function will appear as a series of horizontal segments, representing constant integer values between each pair of consecutive integers.

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool that helps us visualize mathematical functions, like the greatest integer function, effectively. To graph the function \( y_1 = \operatorname{INT}(x) \), it is crucial to understand how a graphing calculator can aid in this. By setting the calculator to DOT mode, it prevents unwanted lines from connecting the discrete points of the function, because the INT function is a step function. To begin, turn on your graphing calculator and navigate to the function editor. Here, you'll enter \( y_1 = \operatorname{INT}(x) \) as your function. Before graphing, switch your calculator to the DOT mode by adjusting the MODE settings. This mode plots individual points without connecting them, preventing misleading representations of the graph. Finally, execute the graphing function to display the step graph. You'll see a picture made up of horizontal segments indicating the integer values \( y = \operatorname{INT}(x) \) for every interval of \( x \).

Step Functions

Step functions like the greatest integer function \( \operatorname{INT}(x) \) are called so because of the way they look on a graph—like a set of stairs, with each step representing an integer value. This function starts with the integer at which \( x \) is rounded down. In more detail, when you graph \( y_1 = \operatorname{INT}(x) \), what happens is each input value, \( x \), produces a flat, horizontal segment at the integer level corresponding to \( \operatorname{INT}(x) \). For instance, any \( x \) value between 0 and just under 1 gives a \( y \) value of 0. Similarly, any \( x \) value between 1 and just under 2 gives \( y \) a value of 1. This behavior continues across all real numbers, leading to numerous small segments or steps on the graph. This unique step-like appearance is why such mathematical concepts are often termed as step functions. Exploring their graphs can greatly enhance comprehension of how these functions operate across different intervals of \( x \).

Graphing Discontinuous Functions

Discontinuous functions are functions that have breaks or gaps in their graphs. The greatest integer function \( \operatorname{INT}(x) \) is an example of such a function. For each interval \( x \) spans, there's no smooth transition from one integer value to the next—only jumps.When graphing discontinuous functions like \( \operatorname{INT}(x) \), the appearance will feature separate horizontal lines broken at each integer. This is because the function's value drops to the next lower integer abruptly, without any gradual slope.To visualize this accurately, using DOT mode on a graphing calculator is beneficial. This mode allows each step to be plotted correctly as distinct segments. As you scan from one piece of the graph to another, these breaks become very apparent, showcasing the discontinuous nature of such a mathematical expression. Recognizing this trait is instrumental in understanding how discontinuous functions differ from continuous ones in their graphical representation.