To find the root of the equation using the bisection method, we can follow these steps:

Given Equation:

$f(x) = \frac{e^x - 1}{x}$

Bisection Method:

The bisection method works by narrowing down the interval where the root lies. Here's how you can apply it:

1. Define the function $f(x) = \frac{e^x - 1}{x}$.

2. Choose an interval $[a, b]$ where $f(a)$ and $f(b)$ have opposite signs. This ensures that there is at least one root in the interval due to the Intermediate Value Theorem.

3. Compute the midpoint $c = \frac{a + b}{2}$.

4. Evaluate $f(c)$.

   • If $f(c) = 0$, then $c$ is the root.
   • If $f(a)$ and $f(c)$ have opposite signs, set $b = c$.
   • If $f(b)$ and $f(c)$ have opposite signs, set $a = c$.

5. Repeat the process until the interval is sufficiently small.

Implementation:

Let's implement this process with a concrete interval.

Step 1: Define the Function

$f(x) = \frac{e^x - 1}{x}$

Step 2: Choose an Interval

To find a suitable interval $[a, b]$, we must ensure that $f(a)$ and $f(b)$ have opposite signs.

Let's consider $a = -1$ and $b = 1$.

Step 3: Compute the Midpoint and Evaluate the Function

We will iterate until the interval is sufficiently small (say, the length of the interval is less than $\epsilon = 0.001$).

Let's start with these values and compute.

Step 4: Apply the Bisection Method

I'll perform the calculations now.The initial interval $[-1, 1]$ is not suitable because $f(a)$ and $f(b)$ do not have opposite signs. Let's adjust the interval to ensure that the function changes sign, which indicates the presence of a root.

I'll test another interval and proceed with the bisection method. Let's try the interval $[0.1, 1]$, since $f(x)$ could behave differently as we avoid the origin where the function is discontinuous.It seems that the interval $[0.1, 1]$ also doesn't work because $f(a)$ and $f(b)$ do not have opposite signs.

Let's explore another interval, specifically one where $x$ approaches zero but doesn't include zero, such as $[-0.5, 0.5]$. This will help us narrow down the location of the root. I'll proceed with this interval.It seems the chosen intervals still aren't suitable for finding a root using the bisection method because $f(a)$ and $f(b)$ do not have opposite signs. Given that the function $\frac{e^x - 1}{x}$ behaves continuously, we should identify a better interval by exploring the behavior of the function in smaller steps around zero.

To do this, I'll analyze the function at specific points and identify a valid interval where the sign changes.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, you can try testing different intervals manually, or I can help you with another method or problem.

If you have any questions or need further assistance, feel free to ask!

Here are 5 related questions you might consider:

1. How does the bisection method compare to other root-finding methods like Newton's method?
2. What is the significance of the interval's initial choice in the bisection method?
3. How does the function $\frac{e^x - 1}{x}$ behave as $x$ approaches 0?
4. What are the limitations of the bisection method for solving equations?
5. How can we improve the efficiency of the bisection method?

Tip: When using the bisection method, always ensure that the initial interval is chosen such that the function values at the endpoints have opposite signs. This guarantees that a root exists within that interval.