Problem 37 - Calculus for Biology and Medicine

Question

Compare \(y=\frac{1}{x}\) and \(y=\frac{1}{x^{2}}\) for \(x>0\) by graphing the two functions. Where do the curves intersect? Which function is greater for small values of \(x ?\) for large values of \(x\) ?

Step-by-Step Solution

Answer

The curves intersect at \( (1, 1) \). For small \( x < 1 \), \( y = \frac{1}{x^2} \) is greater; for large \( x > 1 \), \( y = \frac{1}{x} \) is greater.

Step 1: Understand the Functions

First, understand the nature of the functions given: \( y = \frac{1}{x} \) and \( y = \frac{1}{x^2} \). Both functions are defined for \( x > 0 \) and describe curves that we aim to compare by graphing.

Step 2: Graph the Functions

Plot both functions for values of \( x > 0 \). The curve \( y = \frac{1}{x} \) is a hyperbola that decreases as \( x \) increases, approaching the x-axis but never touching it. The curve \( y = \frac{1}{x^2} \) is similar but decreases more rapidly as \( x \) increases.

Step 3: Intersection of the Curves

Set the functions equal to find the intersection point: \( \frac{1}{x} = \frac{1}{x^2} \). Multiply through by \( x^2 \) to obtain \( x = 1 \). Thus, the curves intersect at \( (1, 1) \).

Step 4: Compare for Small Values of x

Analyzing for small values (e.g., \( x = 0.5 \)): \( y = \frac{1}{x} = 2 \) and \( y = \frac{1}{x^2} = 4 \). Hence, \( \frac{1}{x^2} > \frac{1}{x} \) when \( x < 1 \).

Step 5: Compare for Large Values of x

Analyzing for large values (e.g., \( x = 2 \)): \( y = \frac{1}{x} = 0.5 \) and \( y = \frac{1}{x^2} = 0.25 \). Thus, \( \frac{1}{x} > \frac{1}{x^2} \) when \( x > 1 \).

Key Concepts

Intersection Points

In graphing functions, finding intersection points is crucial to understanding how two curves relate to each other. When two functions intersect, they share both an x-value and a y-value at the point of their intersection.

To determine where the functions \( y = \frac{1}{x} \) and \( y = \frac{1}{x^2} \) intersect, we set them equal to each other: \[ \frac{1}{x} = \frac{1}{x^2} \]By cross-multiplying, we get:\[ x = 1 \]This tells us that the x-coordinate of the intersection is 1. Substituting back into either function, we find the y-coordinate is also 1. As a result, the intersection point of these curves is (1, 1).

Understanding and finding intersection points helps in assessing when and where one function's values exceed another, marking an essential component of graphical analysis.

Function Comparison

When comparing functions graphically, we examine how one function behaves in relation to another over a specific domain. In this case, we are looking at the domain where \( x > 0 \). This involves observing which function's output is greater or smaller for various ranges of \( x \).

For small values of \( x \) (e.g., \( x = 0.5 \)), we calculate:

• \( y = \frac{1}{x} = 2 \)
• \( y = \frac{1}{x^2} = 4 \)

Here, \( y = \frac{1}{x^2} \) is greater than \( y = \frac{1}{x} \), showing that the latter decreases more slowly than the former.

For large values of \( x \) (e.g., \( x = 2 \)), the situation reverses:

• \( y = \frac{1}{x} = 0.5 \)
• \( y = \frac{1}{x^2} = 0.25 \)

Now, \( y = \frac{1}{x} \) is greater because it diminishes at a slower rate. Comparing functions graphically allows us to intuitively see which function dominates a particular x-range and to hypothesize about their behavior without explicit calculation for every point.

Hyperbola

The concept of a hyperbola is central to understanding how the function \( y = \frac{1}{x} \) behaves. A hyperbola is a type of curve in mathematics defined by its symmetry about the coordinate axes. It embodies two opposing bending curves that never meet, known as asymptotes.

The function \( y = \frac{1}{x} \) results in a hyperbola. As \( x \) increases, the curve approaches the x-axis but does not cross it, symbolizing perpetual decrease. Unlike a parabola, which opens either up or down, the hyperbola associated with \( y = \frac{1}{x} \) opens right and downwards when restricted to \( x > 0 \).

The hyperbolic nature means that as \( x \) becomes large, the function value significantly lowers, which is why during function comparison, we note \( y = \frac{1}{x} \) remains larger over a sufficiently broad domain than \( y = \frac{1}{x^2} \). Understanding the hyperbola helps in predicting and describing the behavior of rational functions, enriching our comprehension of their graphical analysis.