Problem 47 - Calculus for Biology and Medicine

Question

Use a graphing calculator to sketch the graphs of the functions. $$ y=x^{3 / 2}, x \geq 0 $$

Step-by-Step Solution

Answer

Graph the function \(y = x^{3/2}\) for \(x \geq 0\) using a graphing calculator. Observe the upward curvature from the origin.

Step 1: Understand the Function

The given function is \( y = x^{3/2} \) with the condition \( x \geq 0 \). This function is a power function where the exponent is a fraction, meaning we are looking at the square root of \( x \) raised to the power of 3.

Step 2: Set Up the Graphing Calculator

Power on your graphing calculator and clear any existing graphs. Set the graphing mode to function (usually denoted as 'Y=') and make sure it's in the standard mode suitable for graphing real numbers.

Step 3: Input the Function

In the 'Y=' tab of your graphing calculator, enter the function \( Y = X^{3/2} \). Double-check to ensure that it is input correctly.

Step 4: Adjust Viewing Window

Set the viewing window to focus on \( x \geq 0 \). A typical setting could be \( x \) ranging from 0 to 10 and \( y \) from 0 to 100, allowing you to comfortably view the curvature of \( y = x^{3/2} \).

Step 5: Graph the Function

Press the 'GRAPH' button on your calculator to display the graph of the function. Observe how the graph curves upward, starting from the origin (0,0) and moving outward as \( x \) increases.

Key Concepts

Power Functions

In mathematics, power functions are expressions where a variable, x, is raised to a constant power. The general form of a power function is \( y = x^n \). Here, \( n \) is a constant exponent, which can be any real number. This type of function is fundamental because it represents growth and change in many natural scenarios, like physics and economics.Power functions have different behaviors:

• When the exponent \( n \) is an integer, the graph is a polynomial curve.


• If \( n \) is positive, as \( x \) increases, \( y \) also increases.


• When \( n \) is negative, the curve approaches the x-axis as it moves away from the origin.

For the function \( y = x^{3/2} \), this is a power function with a fractional exponent. The function grows as \( x \) gets larger, creating an upward curve. It's essential to understand these growth patterns when analyzing the shape of graphs in mathematics.

Graphing Calculator

Graphing calculators are handy tools for visualizing mathematical functions. They allow you to input equations and see their graphical representation. This visual aid makes it easier to understand how changes in variables affect the function's graph.To use a graphing calculator effectively:

• Ensure it’s in the correct mode – for most functions, you want to use the 'Y=' mode.


• Input your function carefully to avoid mistakes.


• Adjust the window settings to ensure all important parts of the graph are visible.

For example, with the function \( y = x^{3/2} \), you’d want to see how the graph behaves as \( x \) increases. By setting your calculator to show \( x \) from 0 to 10 and \( y \) from 0 to 100, you can clearly observe the steep growth of the graph. Utilizing these devices efficiently helps deepen your understanding of function behavior, a crucial skill in math education.

Fractional Exponents

Fractional exponents can seem tricky initially, but they are just a different way of expressing roots and powers. The notation \( x^{a/b} \) essentially means the b-th root of \( x \) raised to the power of \( a \).Here's a quick breakdown:

• \( x^{1/n} \) implies the n-th root of \( x \).


• \( x^{m/n} \) indicates the n-th root of \( x^m \).

In the function \( y = x^{3/2} \), \( x^{3/2} \) can be viewed as the square root of \( x \), all to the power of 3. This results in a curve that steadily rises as \( x \) increases from zero. Understanding fractional exponents is crucial because they simplify working with roots and powers, enabling a more dynamic approach to solving and graphing complex functions.