Problem 48 - Calculus for Biology and Medicine

Question

Use a graphing calculator to sketch the graphs of the functions. $$ y=x^{1 / 3}, x \geq 0 $$

Step-by-Step Solution

Answer

The graph of \( y = x^{1/3} \), \( x \geq 0 \), is a gradually increasing curve starting from (0,0).

Step 1: Understand the function

We are dealing with the function \( y = x^{1/3} \) for \( x \geq 0 \). This function represents the cube root of \( x \) with a domain restricted to non-negative numbers.

Step 2: Set up the graphing calculator

Turn on your graphing calculator. Clear any previously entered equations or data so that we have a clean slate to work with. This prevents any old data from interfering with our new graph.

Step 3: Input the function into the calculator

Enter the function \( y = x^{1/3} \) into the calculator. You may need to find the cube root button or key which might be labeled as \( x^{1/3} \) or similar symbols.

Step 4: Adjust the window settings

Adjust the calculator's window settings to best capture the graph from \( x = 0 \) to a reasonable value (like \( x = 10 \)), and from \( y = 0 \) to a slightly larger value (like \( y = 2 \)). This ensures that the relevant part of the graph is visible.

Step 5: Graph the function

Once the function is entered and the window is set, use the graph function on the calculator. Observe the graph; you should see a curve starting at the origin and gradually increasing.

Step 6: Analyze the sketch

Ensure that the graph starts at the point (0, 0) and rises slowly to the right, reflecting the increasing nature of the cube root for positive \( x \). Note the shape: slightly curved but always increasing in this range.

Key Concepts

Understanding the Cube Root Function

The cube root function, represented mathematically as \( y = x^{1/3} \), is a classic example of a root function. Unlike square root functions, cube roots can take any real number as input because taking the cube of any real number (negative or positive) results in another real number. However, in the exercise, we focus only on non-negative values of \( x \).

The cube root function has some unique characteristics:

• It's defined for all real numbers but our interest here is for non-negative \( x \).
• The function is defined even at \( x = 0 \), where \( y = 0 \).
• The graph of the function is a smooth, continuously increasing curve that starts at the origin and rises slowly, reflecting how cube roots increase slower than linear functions.

Knowing these properties helps when sketching the graph and interpreting its behavior.

Steps for Graphing Functions on a Calculator

Graphing functions like the cube root can be effortlessly achieved with graphing calculators. Here are the simple steps to follow when plotting a function:

• Power up your calculator and ensure it's cleared of any previous inputs.
• Locate the function input area, commonly labeled as "Y=" or similar, to enter the equation.
• If necessary, find the cube root function button, sometimes represented as \( x^{1/3} \), or compute it using the exponent of \( 1/3 \).

Once your function is entered, you can move on to selecting appropriate window settings. With these steps, even complex functions can be graphed smoothly.

Defining Mathematical Functions

Mathematical functions are the building blocks of calculus and many advanced math concepts. A function like \( y = x^{1/3} \) calculates the value of \( y \) for each corresponding \( x \) in the domain. A fundamental attribute of functions is their domain and range:

• Domain: This includes all possible input values. For our function, the domain is non-negative numbers \( x \geq 0 \).
• Range: The set of possible output values. Here, \( y = x^{1/3} \) will have a range starting from 0, increasing slowly as \( x \) increases.

Understanding functions in terms of their domains and ranges provides deeper insight into their graphs and how they behave.

Optimizing Calculator Window Settings

When graphing functions, especially with calculators, window settings are crucial. These settings decide what portion of the graph you can see and how detailed it is.

To adjust these settings appropriately:

• Set the \( x \)-range: For \( y = x^{1/3} \), you might start from \( x = 0 \) and go to \( x = 10 \) to see enough detail.
• Set the \( y \)-range: Consider starting from \( y = 0 \) to \( y = 2 \) to capture the sluggish growth of the function.
• Modify the scale if needed, to ensure the graph is neither too compressed nor stretched.

Setting your calculator's window correctly helps in visualizing the function accurately, making analysis much easier and efficient.