Find the root of equation by using bisection method f(x)/ x. e ^ x - 1

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.

Discover how to use the bisection method to find the root of the equation f(x) = (e^x - 1) / x. This detailed explanation covers numerical methods, the Intermediate Value Theorem, and the behavior of exponential functions, suitable for students at the undergraduate level.

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