Question

Use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=4 \\ y &=\ln x \end{aligned}\right. $$

Short answer

The solutions are approximately (1.1, 0.10) and (1.47, 0.39).

Step-by-step solution

- Graph the First Equation

The first equation is a circle: \(x^2 + y^2 = 4\). Use a graphing utility to plot this circle. It has a center at the origin \((0,0)\) and a radius of 2.

- Graph the Second Equation

The second equation is a natural logarithm function: \(y = \ln x\). Use the same graphing utility to plot this curve. It passes through the point \((1,0)\) and increases to infinity as \(x\) increases.

- Find the Intersection Points

Look for the points where the circle and the logarithmic curve intersect on the graphing utility. These points of intersection are the solutions to the system of equations. Make sure to round each coordinate to two decimal places.

- Verify the Solutions

Check the coordinates of each intersection point to ensure they satisfy both equations. Plug these coordinates into both \(x^2 + y^2 = 4\) and \(y = \ln x\) to verify.

Key concepts

graphing utility

Graphing utilities are powerful tools to visualize mathematical functions and find precise solutions. They can be software applications on computers or apps on smartphones. When using a graphing utility to solve a system of equations, you enter each equation separately.
The graphing utility will then display curves or shapes representing each equation on a coordinate plane. This helps you easily see where the graphs intersect, providing a visual solution to the equations.
For our example:

• Enter the circle equation: \(x^2 + y^2 = 4\).
• Enter the natural logarithm function: \(y = \ln x\).
• Observe the graph to identify the intersection points.
  By using the graphing utility's zoom and trace features, you can pinpoint the coordinates of these intersection points accurately.

natural logarithm function

The natural logarithm function, represented as \(y = \ln x\), is a logarithm with the base of the mathematical constant e (approximately 2.718). This function is only defined for positive values of x.
Key characteristics of the natural logarithm function include:

• It passes through the point (1,0) because \( \ln1 = 0\).
• It increases slowly and never touches the y-axis or crosses it.
• It rises to infinity as x increases.
  Graphing this function helps you understand its behavior and how it intersects with other graphs, like a circle in this exercise. You can input this function into your graphing utility and see its characteristic curve.

intersection points

In the context of solving systems of equations graphically, intersection points are where the graphs of the equations meet. These points provide the solution to the system.
In our problem:

• The circle \(x^2 + y^2 = 4\) intersects with the natural logarithm function \(y = \ln x\).
• We look for these intersection points on the graph.
  Using a graphing utility, you can visually inspect where the curve of the circle and the line of the log function meet. After identifying these points, you can obtain their coordinates and verify their accuracy by plugging them back into the original equations.

circle

A circle in mathematics is defined as the set of all points that are a fixed distance from a central point. This distance is called the radius. The standard equation of a circle is \(x^2 + y^2 = r^2\), where \(r\) is the radius.
For the circle in our exercise with the equation \(x^2 + y^2 = 4\):

• The center is at the origin (0,0).
• The radius is 2, since \( \sqrt{4} = 2\).
  When using a graphing utility, you simply enter this equation to see the circle displayed on the graph. This visualization makes it clear how the circle interacts with other functions, such as the natural logarithm function in this exercise.