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 \(x\), \(\frac{1}{x^2}\) is greater. For large \(x\), \(\frac{1}{x}\) is greater.

Step 1: Understand the Functions

We are comparing two functions: \( y = \frac{1}{x} \) and \( y = \frac{1}{x^2} \). Both functions are defined for \( x > 0 \). We need to plot these to understand their behavior and answer the questions about their intersections and relative value.

Step 2: Analyzing Intersections

To find where the two functions intersect, we set them equal to each other: \( \frac{1}{x} = \frac{1}{x^2} \). Solving this gives \( 1 = \frac{1}{x} \), meaning \( x = 1 \). Thus, they intersect at \( (1, 1) \).

Step 3: Plotting the Graphs

Plot the graphs of \( y = \frac{1}{x} \) and \( y = \frac{1}{x^2} \) on the same axis for \( x > 0 \). Notice how both functions approach zero as \( x \to \infty \).

Step 4: Comparing the Functions for Small Values of \(x\)

For \( x < 1 \) but still positive, \( \frac{1}{x^2} > \frac{1}{x} \) because squaring a number less than 1 (e.g., 0.5) makes it smaller (e.g., 0.25), thus the reciprocal becomes larger. Hence, \( y = \frac{1}{x^2} \) is greater.

Step 5: Comparing the Functions for Large Values of \(x\)

For large \( x > 1 \), the value of \( \frac{1}{x^2} < \frac{1}{x} \) because as \( x \) increases, the square of \( x \) increases even more, making \( \frac{1}{x^2} \) smaller than \( \frac{1}{x} \). Hence, the function \( y = \frac{1}{x} \) is greater.

Key Concepts

Function Comparison

When we compare functions such as \( y = \frac{1}{x} \) and \( y = \frac{1}{x^2} \), we are essentially analyzing their behavior and how they relate to each other over the domain \( x > 0 \). This type of comparison is crucial for understanding which of the functions yields larger values for different ranges of \( x \).

• For values of \( x \) approaching zero (small values just above zero), the behavior of these functions diverges significantly.
• Meanwhile, at larger values of \( x \), the rate at which each function approaches zero is different due to their distinct compositions.

Applying this concept practically involves flattening the function graphs side by side. This visual aid provides an immediate comparative look at the functions' properties at different \( x \) intervals.

Intersection Points

Finding the intersection points of two functions is an essential aspect of graphical analysis. For our functions \( y = \frac{1}{x} \) and \( y = \frac{1}{x^2} \), the intersection occurs where the outputs of the functions are the same given the same input. The solution to the equation \( \frac{1}{x} = \frac{1}{x^2} \) is found by simplifying to \( 1 = \frac{1}{x} \), leading to \( x = 1 \). Thus, the two functions intersect at the point \( (1, 1) \).

• This point is crucial because the behavior of the functions changes around this intersection.
• The functions are equal at this point, providing a specific \( x \) value where the transition in their relative size occurs.

Behavior of Functions

Understanding the behavior of functions over their domains is imperative, especially when it comes to comparing functions like \( y = \frac{1}{x} \) and \( y = \frac{1}{x^2} \). Both functions exhibit unique characteristics based on the value of \( x \).- **For small values of \( x \) (close to zero):** - The function \( y = \frac{1}{x^2} \) grows faster because squaring a fraction less than one gives a smaller number, thus a larger reciprocal. Consequently, \( y = \frac{1}{x^2} \) is greater than \( y = \frac{1}{x} \).- **For large values of \( x \) (greater than one):** - The situation reverses. The function \( y = \frac{1}{x} \) becomes larger because \( x^2 \) grows faster than \( x \), leading to \( y = \frac{1}{x^2} \) being smaller.Analyzing this behavior not only strengthens a fundamental grasp of Calculus concepts but also aids in portraying the gradual nuances as \( x \) changes.