To solve the equation $3x - e^x = 0$ for $1 \leq x \leq 2$ using the Bisection method to an accuracy of $10^{-5}$, we need to follow these steps:

Steps for Bisection Method:

1. Define the function: Let $f(x) = 3x - e^x$.

2. Initial Interval: The interval given is $[1, 2]$. We need to ensure that the function has opposite signs at the endpoints.

3. Check Sign at Endpoints:

   • Calculate $f(1)$ and $f(2)$.
   • $f(1) = 3(1) - e^1 = 3 - 2.71828 = 0.28172$.
   • $f(2) = 3(2) - e^2 = 6 - 7.38906 = -1.38906$.

   Since $f(1) > 0$ and $f(2) < 0$, there is a root in the interval.

4. Bisection Iteration:

   • Calculate the midpoint $c = \frac{1 + 2}{2} = 1.5$.
   • Evaluate $f(c) = f(1.5) = 3(1.5) - e^{1.5}$.
   • Continue updating the interval depending on the sign of $f(c)$ compared to the endpoints, and repeat until the interval is sufficiently small.

Let's perform the iterations numerically using Python to find the solution accurate to $10^{-5}$.The root of the equation $3x - e^x = 0$ within the interval $[1, 2]$ using the Bisection method, accurate to within $10^{-5}$, is approximately:

$x \approx 1.51214$

The method took 16 iterations to converge to this result.

Questions to Consider:

1. What are the advantages of the Bisection method compared to other root-finding methods?
2. How would the convergence speed change if the interval were narrower initially?
3. What are the implications if $f(a)$ and $f(b)$ are not of opposite signs?
4. How does changing the tolerance affect the number of iterations?
5. Can you apply the Bisection method to functions with multiple roots within the same interval?

Tip:

Always ensure that the function changes signs at the initial interval endpoints when using the Bisection method, as this guarantees the existence of a root within the interval.