Problem 21 - Calculus for Biology and Medicine

Question

Use a graphing calculator to graph \(f(x)=x^{2}, x \geq 0\), and \(g(x)=x^{4}, x \geq 0\), together. For which values of \(x\) is \(f(x)>g(x)\) and for which is \(f(x)

Step-by-Step Solution

Answer

\( f(x) > g(x) \) for \( 0 \leq x < 1 \); \( f(x) < g(x) \) for \( x > 1 \).

Step 1: Analyze the Functions

First, understand that you need to graph two functions: \( f(x) = x^2 \) and \( g(x) = x^4 \), both defined for \( x \geq 0 \). These are both types of power functions, with \( f(x) \) being a parabola and \( g(x) \) being a quartic curve. We will explore how their values compare across different \( x \) values.

Step 2: graph the functions

Use a graphing calculator or graphing software to plot both functions. Ensure your graph includes the x-axis (from \( x = 0 \) onwards) and the y-axis to accurately reflect where each function lies as \( x \) increases. You will plot \( y= x^2 \) and \( y=x^4 \) on the same graph.

Step 3: Examine the graph at specific points

By studying the graph, observe the points at which the graphs intersect and diverge. Specifically, note that for small values of \( x \), \( f(x) = x^2 \) may be above \( g(x) = x^4 \), and look for where changes occur.

Step 4: Determine intersection and comparison

The intersection of \( f(x) \) and \( g(x) \) occurs where \( x^2 = x^4 \). Solving this, we get \( x^2 = x^4 \Rightarrow x^4 - x^2 = 0 \Rightarrow x^2(x^2 - 1) = 0 \). This gives the solutions \( x = 0 \) and \( x = 1 \). These are the critical points for comparison.

Step 5: Compare functions before and after intersection

For values of \( x \) where \( 0 \leq x < 1 \), evaluate \( x^2 \) and \( x^4 \). Since \( x^4 \) grows faster, \( x^2 > x^4 \) before the intersection at \( x = 1 \). After \( x = 1 \), \( g(x) = x^4 \) increases more quickly than \( f(x) = x^2 \), thus \( f(x) < g(x) \) for \( x > 1 \).

Key Concepts

Intersection Points

The concept of intersection points in a graph is fundamental when comparing two functions. These points occur where the two functions have the same output for a given input, essentially where they cross each other on the graph. In our exercise with the functions \( f(x) = x^2 \) and \( g(x) = x^4 \), we set these equal to each other to find the intersection points. To find where \( f(x) = g(x) \), solve the equation: \[ x^2 = x^4 \]Rewriting this, we get: \[ x^4 - x^2 = 0 \]Factor this to find:\[ x^2(x^2 - 1) = 0 \] This leads us to the solutions: - \( x = 0 \)- \( x = 1 \)These are the intersection points, meaning that at \( x = 0 \) and \( x = 1 \), the values of both functions are the same. Between and beyond these points, the functions differ in their relative size, which is a key aspect to analyze in graphing.

Graphing Calculator

A graphing calculator is an essential tool when working with functions and analyzing their graphs. This tool allows us to visualize the relationships between functions by plotting them on a graph. For the given functions \( f(x) = x^2 \) and \( g(x) = x^4 \), using a graphing calculator lets us see both functions plotted from \( x = 0 \) onward. Here's how to make the most out of a graphing calculator in this scenario:

• Plot both functions simultaneously: Input \( y = x^2 \) and \( y = x^4 \) into the calculator to view how both graphs look when overlaid.
• Set appropriate window size: Ensure your calculator's window is wide enough along the x-axis to cover the region of interest, typically starting at \( x = 0 \). Also, adjust the y-axis to capture the curves as they grow.
• Zoom in and out: This feature helps to better visualize the intersection points and identify where \( f(x) > g(x) \) and where \( f(x) < g(x) \).

The graphing calculator highlights visually where the functions intersect and diverge, assisting in determining at what intervals one function overtakes the other in value.

Power Functions

Power functions are mathematical functions of the form \( f(x) = x^n \), where \( n \) is a real number. In the context of our exercise, both \( f(x) = x^2 \) and \( g(x) = x^4 \) are power functions, with powers of 2 and 4 respectively. Here are some important aspects:

• Shape of Graphs: Power functions tend to have characteristic shapes depending on their exponents. \( f(x) = x^2 \) is a parabola, which is symmetric about the y-axis, whereas \( g(x) = x^4 \) resembles a steeper version, as it rises more quickly.
• Growth Rate: The critical observation with power functions is their growth rate. As \( x \) increases, functions with higher powers such as \( x^4 \) grow faster than those with lower powers such as \( x^2 \).
• Comparison by Intervals: For the functions in this exercise, \( x^2 \) is greater than \( x^4 \) for small values of \( x \) (specifically when \( 0 \leq x < 1 \)), and \( x^4 \) overtakes \( x^2 \) for larger values (when \( x > 1 \)).

Understanding power functions helps in analyzing and predicting the behavior of polynomial equations over different intervals. This understanding is key when making comparisons and finding intersections.