https://file.aiquickdraw.com/mathbot/file/1726412734637app9aytj.jpg

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.

Find roots of equations using the Bisection Method for the following problems:
1. Solve x - (1/2) cos^2(x) = 0
2. Find the real root of x^3 + 3x^2 - 8 = 0
3. Solve f(x) = x^3 + 4x^2 - 10
4. Approximate the root of x - 2^(-x) = 0
5. Find the root of f(x) = x^3 + 4x^2 - 10 in [1,2] to an error of 10^-4
6. Find the number of iterations needed to solve x^3 + 4x^2 - 10 = 0 to a precision of 10^-3
7. Find the real root of f(x) = x^4 + 2x^2 + 10x - 20
8. Solve f(x) = -0.5x^5 + 2.5x^4 + 4.5 using Bisection, graphical and quadratic formula methods.

Learn how to solve polynomial and trigonometric equations using the Bisection Method. Includes step-by-step guides for approximating roots with high precision and error estimates.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation