Problem 50 - Calculus for Biology and Medicine

Question

Use a graphing calculator to sketch the graphs of the functions. $$ y=2 x^{-7 / 8}, x>0 $$

Step-by-Step Solution

Answer

The graph of \( y = 2x^{-7/8} \), \( x>0 \) shows a curve decreasing towards 0 as \( x \) increases.

Step 1: Understand the function

The function given is \( y = 2x^{-7/8} \). This can be rewritten as \( y = \frac{2}{x^{7/8}} \). It is a rational function where the variable \( x \) is raised to the power of \(-7/8\). We are looking to plot this function for positive values of \( x \).

Step 2: Set up the graphing calculator

In the graphing calculator, ensure the function mode is set to 'function' or 'FUNC'. Input the function as it is written: \( y = 2 / (x^{7/8}) \). Make sure to enter the power correctly using parentheses: \( x^{(7/8)} \).

Step 3: Choose a window range

Select an appropriate window range for the graph. Since it is specified that \( x > 0 \), set the x-value minimum slightly greater than 0 (e.g., 0.1) to avoid division by zero and the maximum to a feasible number (e.g., 10). Choose the y-range by considering expected values from calculations of a few points, such as when \( x = 1 \), \( x = 2 \), etc.

Step 4: Graph and observe the output

After inputting the function and setting the window, press the graph button. The graph should show an asymptote approaching the x-axis as \( x \) increases and increasing sharply as \( x \) approaches 0 from the positive side.

Step 5: Analyze the graph

Examine the graph to note the behavior of the function. Observe that as \( x \) increases, \( y \) decreases towards 0, indicating the function approaches the x-axis asymptotically. As \( x \) gets smaller (still positive), \( y \) increases rapidly.

Key Concepts

Rational Functions

Rational functions are expressions that can be written as the quotient of two polynomials. For example, the function \( y = \frac{2}{x^{7/8}} \) is a rational function because it involves dividing a constant, 2, by a power function of \( x \).

One important characteristic of rational functions is that they can have undefined points, usually associated with division by zero. In our example, this occurs when \( x \) is zero. However, since we're only looking at positive values of \( x \), this is not a concern here.

• Numerator: The top part of the fraction. In this example, it's just the constant value 2.
• Denominator: The bottom part, including the variable raised to a power, \( x^{7/8} \), which affects how the function behaves.

The behavior of the function near its undefined points or as variables tend towards infinity is a key aspect to evaluate. Rational functions often show trends or patterns that can be helpful in understanding their graphs.

Graphing Calculator

A graphing calculator is a powerful tool used to visualize mathematical functions. For our function \( y = \frac{2}{x^{7/8}} \), using a graphing calculator simplifies the process of understanding its graph.

Here’s how to use it effectively:

• **Mode Setting:** Ensure your calculator is in 'function' mode. This setting allows you to input and visualize functions directly.
• **Input the Function:** Enter the function precisely as written with attention to parentheses, especially when dealing with exponents like \( x^{(7/8)} \).
• **Adjust View Window:** To get a meaningful view of the graph, set a window where \( x \) is slightly above zero, e.g., from 0.1 to 10, avoiding undefined values. Consider checking a few calculations for specific \( x \) values to set logical y-range limits.

Once set up, the graph will show you the trends and behaviors of the function visually, aiding greatly in your comprehension.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never quite touches. For the function \( y = \frac{2}{x^{7/8}} \), the x-axis \( (y=0) \) acts as a horizontal asymptote.

Understanding asymptotes can help you predict the behavior of a function:

• **Horizontal Asymptotes:** These occur when the value of y approaches a constant as \( x \) becomes very large or very small. For our function, as \( x \to \infty \), \( y \to 0 \).
• **Vertical Asymptotes:** Although not present in the positive domain for this particular function, these occur where a function becomes undefined (typically when the denominator is zero).

Asymptotes help us understand the limits of a function's behavior, especially at the extremes of its domain, such as how our function flattens out as \( x \) increases or rises steeply as \( x \) approaches zero from the positive side.