The image shows a workshop (taller) on the Bisection Method for numerical methods, with several exercises related to finding roots of equations using this method. Here is an overview of each exercise from the image:

1. Exercise 1:
   Find an approximation of the root of the equation $x - \frac{1}{2} \cos^2(x) = 0$ with an error order less than or equal to $2 \times 10^{-2}$ using the Bisection Method.

2. Exercise 2:
   Find an approximation $x^n$ of the only real root of the equation $x^3 + 3x^2 - 8 = 0$ using:
   a. The Bisection Method,
   with an error less than $10^{-2}$.

3. Exercise 3:
   Given the function $f(x) = x^3 + 4x^2 - 10$,
   a. Find an approximation $x^*$ of the only real root of the equation $f(x) = 0$ using the Bisection Method with an error less than $10^{-2}$.

4. Exercise 4:
   Use the Bisection Method to approximate the solution of the equation $x - 2^{-x} = 0$ in the interval $[0,1]$.
   Calculate the number of iterations and find the approximation with five decimal places.

5. Exercise 5:
   Show that $f(x) = x^3 + 4x^2 - 10 = 0$ has a root in $[1,2]$ and use the Bisection Method to determine an approximation for the root that is accurate to at least $10^{-4}$.

6. Exercise 6:
   Determine the number of iterations necessary to solve $f(x) = x^3 + 4x^2 - 10 = 0$ with precision of $10^{-3}$ given $a_1 = 1$ and $b_1 = 2$.

7. Exercise 7:
   Use the Bisection Method to find a real root of the polynomial $f(x) = x^4 + 2x^2 + 10x - 20$.

8. Exercise 8:
   Determine the real roots of $f(x) = -0.5x^5 + 2.5x^4 + 4.5$:
   a. Graphically,
   b. Using the quadratic formula,
   c. Using the Bisection Method with three iterations to determine the largest root. Compute the estimated error $e_2$ and the true error $e_1$ for each iteration.
   d. Determine the roots of $f(x) = 5x^3 - 5x^2 + 6x = 2$:
   i. Graphically,
   ii. Using the Bisection Method to locate the root, with initial values $x_0 = 0$ and $x_1 = 1$.

Would you like me to solve any specific exercise or provide a step-by-step explanation for the Bisection Method?

5 Additional Questions for Deeper Understanding:

1. What are the key conditions for applying the Bisection Method?
2. How does the Bisection Method ensure convergence to a root?
3. Can the Bisection Method be applied to any continuous function? Why or why not?
4. How is the error bound for each iteration in the Bisection Method determined?
5. In what scenarios might other root-finding methods (like Newton's method) be preferred over the Bisection Method?

Tip:

Always check that the function changes sign over the interval $[a, b]$ to ensure that the Bisection Method can be applied.