https://file.aiquickdraw.com/mathbot/file/1726434864262bkdbccwx.jpg

The problem asks to approximate the unique real root \( x^* \) of the function \( f(x) = x^3 + 4x^2 - 10 \) using the **Bisection Method** with an error smaller than \( 10^{-2} \).

The **Bisection Method** works as follows:
1. You choose an initial interval \([a, b]\) such that \( f(a) \) and \( f(b) \) have opposite signs (i.e., \( f(a) \cdot f(b) < 0 \), guaranteeing a root in that interval).
2. You compute the midpoint \( c = \frac{a + b}{2} \) and evaluate \( f(c) \).
3. Based on the sign of \( f(c) \), you choose the subinterval \([a, c]\) or \([c, b]\) where the root lies.
4. This process is repeated until the error \( |b - a| \) is less than the desired tolerance (in this case, \( 10^{-2} \)).

Let's start by identifying an interval where the sign changes for \( f(x) \).

### Step 1: Define the function
\( f(x) = x^3 + 4x^2 - 10 \)

We will now find an initial interval \([a, b]\) and calculate the root step by step using the Bisection Method.Here is a table summarizing the results of applying the Bisection Method to approximate the root of the equation \( f(x) = x^3 + 4x^2 - 10 = 0 \), with an error less than \( 10^{-2} \):

| Iteration | \( a \)   | \( b \)   | \( c = \frac{a+b}{2} \) | \( f(a) \)     | \( f(b) \)     | \( f(c) \)      | \( b - a \)   |
|-----------|-----------|-----------|------------------------|----------------|----------------|-----------------|---------------|
| 1         | 1.0000    | 2.0000    | 1.5000                 | -5.0000        | 14.0000        | 2.3750          | 1.0000        |
| 2         | 1.0000    | 1.5000    | 1.2500                 | -5.0000        | 2.3750         | -1.7969         | 0.5000        |
| 3         | 1.2500    | 1.5000    | 1.3750                 | -1.7969        | 2.3750         | 0.1621          | 0.2500        |
| 4         | 1.2500    | 1.3750    | 1.3125                 | -1.7969        | 0.1621         | -0.8484         | 0.1250        |
| 5         | 1.3125    | 1.3750    | 1.3438                 | -0.8484        | 0.1621         | -0.3510         | 0.0625        |
| 6         | 1.3438    | 1.3750    | 1.3594                 | -0.3510        | 0.1621         | -0.0964         | 0.0313        |
| 7         | 1.3594    | 1.3750    | 1.3672                 | -0.0964        | 0.1621         | 0.0324          | 0.0156        |
| 8         | 1.3594    | 1.3672    | 1.3633                 | -0.0964        | 0.0324         | -0.0321         | 0.0078        |

The root approximation after 8 iterations is \( x^* \approx 1.3633 \) with an error less than \( 10^{-2} \).

Would you like a more detailed explanation of the method or its steps?

Here are 5 follow-up questions you might find useful:
1. How does the Bisection Method compare to other numerical methods like Newton's method?
2. Can the Bisection Method be used to find multiple roots of a function?
3. What is the significance of the sign change in the Bisection Method?
4. How can you ensure the Bisection Method converges to the correct root?
5. What are some limitations of the Bisection Method in practical applications?

**Tip:** Always check the function for continuity and sign changes before applying the Bisection Method!

Hallar una aproximación x* de la única raíz real de la ecuación f(x) = 0, utilizando el método de bisección con un error inferior a 10^-2.

This solution demonstrates how to approximate the unique real root of f(x) = x^3 + 4x^2 - 10 using the Bisection Method with an error smaller than 10^-2. Step-by-step calculations are provided, making it suitable for college-level students studying numerical methods.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation