For exercise 3 from the image, we are tasked with solving the following:

Given the function $f(x) = x^3 + 4x^2 - 10$, we need to:

a. Find an approximation $x^*$ of the real root of the equation $f(x) = 0$ using the bisection method with an error less than $10^{-2}$.

Steps to Solve Using Bisection Method:

1. Function Definition: $f(x) = x^3 + 4x^2 - 10$

2. Initial Interval: To apply the bisection method, we first need an interval [$a_1$, $b_1$] where the function changes sign, i.e., $f(a_1) \cdot f(b_1) < 0$.

3. Choose the Initial Interval: From analyzing the function:

   • $f(1) = 1^3 + 4(1)^2 - 10 = -5$
   • $f(2) = 2^3 + 4(2)^2 - 10 = 18$

   Therefore, the interval $[1, 2]$ is valid because $f(1) \cdot f(2) < 0$.

4. Apply Bisection Method: The general formula for the midpoint is: $x_{\text{mid}} = \frac{a + b}{2}$ Then, evaluate $f(x_{\text{mid}})$. Depending on the sign of $f(x_{\text{mid}})$, update the interval:

   • If $f(a) \cdot f(x_{\text{mid}}) < 0$, then the new interval is $[a, x_{\text{mid}}]$.
   • Otherwise, the new interval is $[x_{\text{mid}}, b]$.

5. Stopping Criterion: Continue this process until the interval size $|b - a|$ is less than the desired error $\varepsilon = 10^{-2}$.

Implementing in Excel:

1. Setup the Function: In Excel, create columns for $a$, $b$, $x_{\text{mid}}$, $f(a)$, $f(b)$, and $f(x_{\text{mid}})$.

2. Calculate $x_{\text{mid}}$: Use the formula $\text{mid} = (a + b)/2$.

3. Determine the Sign: In the next column, check the sign of $f(a) \cdot f(\text{mid})$:

   • If it's negative, set the new $b = \text{mid}$.
   • Otherwise, set $a = \text{mid}$.

4. Error Calculation: After each iteration, compute $|b - a|$. Stop the iterations when this value is less than $10^{-2}$.

5. Convergence: The midpoint when $|b - a| < 10^{-2}$ will be the approximate root $x^*$.

I will create an Excel sheet template that applies the bisection method for this function and problem, allowing you to visualize the process and get the solution directly.The bisection method solution has been implemented in an Excel file. You can download it using the link below:

Bisection Method Solution - Excel File

This file contains the step-by-step iterations showing how the interval is refined, the midpoint, the function value at the midpoint, and the error at each step until the error is less than $10^{-2}$.

Would you like further details on how the method works or any specific part of the solution? Let me know if you have more questions or need adjustments.

Related Questions:

1. How does the bisection method ensure convergence to a root?
2. What are the advantages and limitations of the bisection method?
3. How would the number of iterations change if the tolerance was $10^{-4}$?
4. Can the bisection method be used for all types of functions?
5. How does the error bound $|b - a|$ relate to the tolerance?

Tip: The bisection method is a reliable root-finding technique, but it requires that the function changes sign over the initial interval. Always check for this condition before applying the method.