Question

Use a graphing utility to solve each equation. Express your answer rounded to two decimal places. $$ e^{2 x}=x+2 $$

Short answer

x ≈ 0.69

Step-by-step solution

- Rewrite the equation

Rewrite the given equation for easier handling. The equation is: \[ e^{2x} = x + 2 \]

- Set up functions for graphing

Rewrite the equation as two separate functions to graph:\[ y_1 = e^{2x} \]\[ y_2 = x + 2 \]

- Graph the functions

Use a graphing utility to plot both functions. Graph each function on the same set of axes.

- Identify the intersection point

Look at the graph and identify where the two functions intersect. The x-coordinate of this intersection point is the solution.

- Round the solution

Round the x-coordinate of the intersection point to two decimal places to get the final answer.

Key concepts

Graphing Utility

A graphing utility is a powerful tool that helps visualize mathematical functions. It's often used in algebra and calculus to plot equations and find their intersections.

• Graphing calculators and software like Desmos or GeoGebra are popular graphing utilities.
• They allow you to input equations and plot their graphs on coordinate axes.
• For our exercise, we need to graph the functions \( y_1 = e^{2x} \) and \( y_2 = x + 2 \).

Using a graphing utility makes it easier to see where these functions cross and understand their behavior. Open your graphing tool, enter the equations, and watch how they plot the curves.

Intersection Point

The intersection point is where two graphs meet. This point gives us the solution to the equation.

• For our functions \( y_1 = e^{2x} \) and \( y_2 = x + 2 \), their intersection point shows where both equations hold true at the same x-value.
• Graph the functions and look where they cross on the graph.
• The x-coordinate of this intersection is the solution to \( e^{2x} = x + 2 \).

This point can be identified visually on the graph. Depending on your graphing utility, you might zoom in to see the exact point where they meet.

Rounding

Rounding numbers can make your answers simpler and easier to understand. In math problems, you're often asked to round to a certain number of decimal places.

• For our exercise, we need to round the x-coordinate of the intersection point to two decimal places.
• To round a number, identify the digit at the place you're rounding to, and look at the digit immediately to the right.
• If this digit is 5 or higher, round up. If it's 4 or lower, round down.

For example, if our intersection point x-coordinate is 0.676, it would round to 0.68. This way, our final answer is more concise and easier to use in further calculations or applications.