https://r2.erweima.ai/mathbot/image/1724725209518iiy5a17e.jpg

hazme mi tarea de calculo y metodos numericos:
Actividad: Cálculo de raíces
de manera iterativa 
 
Objetivos
 
A través
de esta actividad comprender las
limitaciones inherentes al cálculo analítico y deducir la necesidad del uso de
técnicas de aproximación para la resolución de ecuaciones del ámbito
ingenieril. Para realizarla, puedes emplear la calculadora online WIRIS
(https://calcme.com/a), Matlab
(https://matlab.mathworks.com/)
o Microsoft Excel.
 
Descripción
 
En este
ejercicio debes encontrar los intervalos de convergencia de los cuatro ceros de
una función dada y aplicar los métodos iterativos: bisección, punto fijo y
Newton-Raphson para calcular con precisión 5*10-3 la raíz positiva
más pequeña. La función es: 


The function given in the image is:

\[
f(x) = x^4 + 2x^3 - 4x^2 - 6x + 3
\]

### Objectives
You need to:
1. **Find the intervals of convergence** for the four zeros of the function.
2. **Apply the following iterative methods**:
   - **Bisection Method**
   - **Fixed-Point Iteration**
   - **Newton-Raphson Method**
3. **Calculate the smallest positive root** with a precision of \(5 \times 10^{-3}\).

### Step 1: Finding the Intervals of Convergence
To find the intervals where the roots might lie, you can start by analyzing the function graphically or by evaluating the function at various points.

### Step 2: Apply Iterative Methods

1. **Bisection Method**:
   - Choose an interval \([a, b]\) where the function changes sign, indicating a root.
   - Apply the bisection method iteratively until the desired precision is achieved.

2. **Fixed-Point Iteration**:
   - Rewrite the equation \(f(x) = 0\) in the form \(x = g(x)\).
   - Start with an initial guess \(x_0\) and iterate using \(x_{n+1} = g(x_n)\) until convergence.

3. **Newton-Raphson Method**:
   - Start with an initial guess \(x_0\).
   - Use the iteration formula \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\) where \(f'(x)\) is the derivative of \(f(x)\).
   - Continue iterating until the root is found within the desired precision.

### Step 3: Calculate the Smallest Positive Root
- Using each of the above methods, calculate the smallest positive root to a precision of \(5 \times 10^{-3}\).

Would you like me to start with the specific calculations using one of these methods, or do you have any questions so far?

### Follow-up Questions:
1. What is the importance of finding the interval of convergence for a root?
2. How does the choice of initial guess affect the convergence in Newton-Raphson method?
3. What are the advantages and disadvantages of the bisection method compared to other methods?
4. How do you determine if a fixed-point iteration method will converge?
5. What happens if a function has multiple roots close to each other when using iterative methods?

### Tip:
When using the Newton-Raphson method, ensure that the derivative \(f'(x)\) does not approach zero, as this can lead to divergence or slow convergence.

Explore iterative methods like Bisection, Fixed-Point Iteration, and Newton-Raphson Method for root finding in numerical analysis. Learn how to apply these methods to find the smallest positive root of a function with precision. Suitable for university-level students and engineers.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation