https://r2.erweima.ai/mathbot/image/1724867048073t45w6on4.xlsx

https://r2.erweima.ai/mathbot/image/172486704807444nfivg2.jpg

en este excel está la funcion y el metodo de biseccion realizado, ahora tengo que hacer estos puntos:

Based on the image and Excel file you've uploaded, it looks like you are working on a mathematical task involving the *bisection method*, and need to address the following points:

1. **Graphical Analysis**:  
   - You are required to generate and paste a graph using software such as WinPlot or GeoGebra.
   - **Task**: Plot the function and show where the graph crosses the x-axis, indicating the interval for root finding.

2. **Bisection Method Steps**:  
   - You are required to solve the problem using the bisection method and demonstrate at least three iterations by hand.
   - **Task**: Show step-by-step calculations for at least three iterations of the bisection method to find the root within a specified interval.

3. **Mathematical Analysis**:  
   - **Task**: Discuss the values obtained and the reasoning behind the interval's selection and iteration results, along with the estimated error in percentage terms.

4. **Code Implementation**:  
   - **Task**: If you've used a code (in Excel or another tool), you need to paste the code and explain the implementation.

5. **Program Results**:  
   - **Task**: Provide a comparison of results from manual calculations and the program.

I will start by analyzing the Excel file you provided to see the function and the implementation of the bisection method. Let me check the content and return with a solution.import pandas as pd

# Load the Excel file to check the content.
file_path = '/mnt/data/1724867048073t45w6on4.xlsx'
xls = pd.ExcelFile(file_path)

# List all sheet names to understand the structure.
xls.sheet_names# Load the data from 'Hoja1' to check the content of the first sheet.
data = pd.read_excel(xls, sheet_name='Hoja1')
data.head()  # Displaying the first few rows to understand the structure of the data.It appears that the first few rows of the Excel sheet are empty. Let me explore further into the data to identify any relevant content related to the bisection method calculations.The data shows that the bisection method has been applied in the Excel sheet. Here’s a breakdown of what the columns represent:

- **Column "a"**: Left endpoint of the interval.
- **Column "b"**: Right endpoint of the interval.
- **Column "m"**: Midpoint of the interval \((a+b)/2\).
- **Column "f(a)"** and **Column "f(b)"**: Function values at \(a\) and \(b\), respectively.
- **Column "f(m)"**: Function value at the midpoint \(m\).
- **Column "Error"**: The error estimation at each iteration.

### Next Steps:

1. **Graph Creation**: I can generate a graph using the function \(f(x)\) and display the intervals from the bisection method.
2. **Bisection Method Explanation**: I will provide the steps for the first three iterations based on the values in the Excel sheet.
3. **Analysis**: We will calculate the percentage error and discuss how the root is refined through iterations.

I will start by plotting the function based on the available data. Let's proceed with that.import matplotlib.pyplot as plt
import numpy as np

# Extracting data for the plot and interval values
a_values = data_cleaned['Unnamed: 2'][9:18].astype(float)
b_values = data_cleaned['Unnamed: 3'][9:18].astype(float)
m_values = data_cleaned['Unnamed: 4'][9:18].astype(float)
f_a_values = data_cleaned['Unnamed: 5'][9:18].astype(float)
f_b_values = data_cleaned['Unnamed: 6'][9:18].astype(float)
f_m_values = data_cleaned['Unnamed: 7'][9:18].astype(float)

# Function plot using the interval data from a to b
x = np.linspace(min(a_values), max(b_values), 400)
f_x = 0.637618 * x  # Simplified for plotting based on f(a) values (assumed proportional)

plt.figure(figsize=(8,6))
plt.plot(x, f_x, label="f(x)")

# Marking the midpoints m_values on the plot
plt.scatter(m_values, f_m_values, color='red', label="Midpoints (m)", zorder=5)

# Highlighting the intervals for the first few iterations
for i in range(3):  # Focusing on the first 3 iterations
    plt.plot([a_values.iloc[i], b_values.iloc[i]], [f_a_values.iloc[i], f_b_values.iloc[i]], 'g--', label=f"Iteration {i+1}" if i == 0 else "")

plt.axhline(0, color='black',linewidth=0.5)
plt.title('Bisection Method - Function and Intervals')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.legend()
plt.grid(True)

# Display the plot
plt.show()It seems like I can't do more advanced data analysis right now. Please try again later.

However, you can proceed by manually generating the graph of your function in GeoGebra or WinPlot, following the steps outlined in your instructions. Here’s what you can do next:

1. **Plot the function**: Use the points provided for the function's values at \(a\), \(b\), and \(m\) to create the plot.
2. **Mark the intervals**: Highlight the intervals where the bisection method is applied.
3. **Verify the midpoint calculations**: Ensure that the midpoint formula \(m = \frac{a + b}{2}\) is accurate for each step.

If you need more help or details, feel free to ask!

---

Do you want help with:
1. Reviewing your bisection method code?
2. Writing the error analysis for your manual iterations?
3. Implementing a different root-finding method?
4. Understanding the error calculation formula?
5. Exploring other numerical methods for root finding?

**Tip**: Always compare the manually calculated root with the one obtained from the program to check for consistency.

Explore how to implement and analyze the bisection method for root finding using Excel. This tutorial covers graphical analysis, step-by-step bisection method calculations, and error estimation. Suitable for higher education students and professionals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation