Problem 90 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Use a graphing utility to draw the graphs of \(y=\ln \left(1+x^{2}\right)\) and \(y=\frac{1}{x}\) on the same axes. Do these graphs intersect?

Step-by-Step Solution

Answer

Yes, the graphs intersect.

Step 1: Understand the Functions

Identify the two functions that need to be graphed: 1. The natural logarithm function: \( y = \ln \left(1 + x^{2}\right) \). 2. The hyperbola function: \( y = \frac{1}{x} \).

Step 2: Set Up the Graphing Utility

Open a graphing utility or software (like Desmos or GeoGebra). Set up the axes to display an appropriate range of values for both functions, considering potential intersections and behavior of both graphs.

Step 3: Graph the First Function

Input \( y = \ln \left(1 + x^{2}\right) \) into the graphing utility. Observe the curve and ensure it is plotted correctly.

Step 4: Graph the Second Function

Input \( y = \frac{1}{x} \) into the same graphing utility. Observe that this is a hyperbola with branches in the first and third quadrants.

Step 5: Analyze the Graphs

Examine the two plotted graphs. Look for points where the curves of \( y = \ln \left(1 + x^{2}\right) \) and \( y = \frac{1}{x} \) intersect, meaning where they share common coordinates.

Step 6: Determine Intersection Points

Identify if there are any intersection points between the two graphs. This can be done by inspecting the graph visually or using the intersection tool in the graphing utility, which will precisely show coordinates of any intersection points.

Key Concepts

natural logarithm

The natural logarithm, noted as \texttt{ln}, is a logarithm with base \texttt{e} (where \texttt{e} is approximately 2.71828). It is often used in mathematics due to its unique properties. The natural logarithm of a number gives the time needed to reach that number, starting from 1, and growing continuously at a rate of 100%.
In this exercise, we work with the natural logarithm function denoted as: \( y=\texttt{ln}(1 + x^2) \). Understanding this function is critical to graphing it correctly.
Some key properties of the natural logarithm function include:

• The domain is all real numbers where the argument (inside the ln) is greater than zero.
• The output or range is all real numbers.
• As x approaches infinity, ln(x) also approaches infinity.

When graphed, the function \( y=\texttt{ln}(1 + x^2) \) begins at 0 when \( x = 0 \) and increases very slowly as \( x \) moves away from zero because the logarithmic growth is quite slow.

hyperbola

A hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations. In our exercise, we use the hyperbolic function: \( y = \frac{1}{x} \) which is a classic example of a reciprocal function.
This hyperbola has distinct properties:

• It has two branches, one in the first quadrant and the other in the third quadrant.
• It has asymptotes that are the x and y axes, meaning the curve approaches these lines but never intersects them.
• Its domain includes all real numbers except zero.
• Its range also includes all real numbers except zero.

When you graph \( y = \frac{1}{x} \), you'll see that it approaches infinity as x approaches zero from the left and negative infinity as x approaches zero from the right. Similarly, it approaches zero as x becomes very large in magnitude (positive or negative).

graphing software

Graphing software is an essential tool for visualizing mathematical functions and understanding their behavior. Programs like Desmos, GeoGebra, and others provide intuitive interfaces for plotting equations and analyzing graphs.
To graph functions:

• Start by opening your chosen graphing software and entering your functions in the input fields provided.
• Ensure that the axes are scaled appropriately to accommodate the significant features of both functions.
• Use features like zooming and panning to closely examine the graphs.
• Utilize tools such as the intersection finder if available, to pinpoint where two functions cross.

In our exercise, entering \( y=\texttt{ln}(1 + x^2) \) and \( y=\frac{1}{x} \) into a graphing utility shows their curves on the same axes. This allows you to observe where these functions intersect, providing a visual understanding of their relationship.

intersection points

Intersection points are locations on the graph where two functions share the same coordinates. Identifying these points is crucial for understanding the interaction between the functions.
To find intersection points:

• Visually inspect the graph to see where the two curves meet.
• Alternatively, use the intersection tool in your graphing software, which will provide the exact coordinates of the intersection.
• Look closely at the shapes and behaviors of the curves near potential intersection regions.

In our problem, when both \( y=\texttt{ln}(1+ x^2) \) and \( y=\frac{1}{x} \) are plotted, you'll see that they may intersect at specific points. By using graphing utility tools, these points of intersection can be precisely determined and validated for a complete understanding.