https://file.aiquickdraw.com/mathbot/file/1730115964599thikbw2s.jpg

This exercise involves a series of questions about modeling sales data with polynomial interpolation. Let's break down each part.

1. **Problem Statement Overview**:
   - The table lists sales (in millions of euros) for a store, "Doces da Lu," for the years 2000 to 2005.
   - It requests a quadratic polynomial interpolation based on specific data points and also includes a cubic model for comparison.

2. **Question Analysis**:

   ### Part (a)
   - **Objective**: Create a quadratic model that fits the data for the years 2000, 2002, and 2004.
   - **Transformation**: Set \( z = x - 2000 \) to simplify calculations, where \( p(z) = a_0 + a_1 z + a_2 z^2 \).
   - **Steps**:
     1. Substitute the values for years 2000 (\( z = 0 \), sales = 62), 2002 (\( z = 2 \), sales = 70), and 2004 (\( z = 4 \), sales = 94).
     2. Set up a system of linear equations for the coefficients \( a_0 \), \( a_1 \), and \( a_2 \).

   ### Part (b)
   - **Objective**: Solve the system of equations obtained in part (a) using Gaussian elimination.
   - **Solution Steps**:
     1. Write out the system based on substitutions in part (a).
     2. Apply Gaussian elimination to find \( a_0 \), \( a_1 \), and \( a_2 \).
   - **Result**: Determine the interpolating polynomial \( p(z) \) and convert it back to \( p(x) \) if needed.

   ### Part (c)
   - **Objective**: Evaluate a cubic model that uses years 2000, 2001, 2002, and 2004, where \( p(z) = 62 - 32z + 26z^2 - 4z^3 \).
   - **Sub-question**: Estimate whether this model reasonably predicts sales for 2003 and justify the answer.

Would you like to see a detailed step-by-step solution for each part? 

### Related Questions:
1. How do you derive the system of equations for polynomial interpolation?
2. What is the process of Gaussian elimination for solving linear equations?
3. How does changing variables simplify polynomial interpolation?
4. What are the criteria to determine if a polynomial model is a good fit for a given data set?
5. How do you interpret the coefficients in a polynomial model?

**Tip**: When interpreting a polynomial model, always check if the degree of the polynomial aligns well with the number of data points to avoid overfitting.

The store 'Doces da Lu' has sales data from 2000 to 2005. You are asked to perform polynomial interpolation based on specific years. Follow these steps:

(a) Find a quadratic polynomial model using data from 2000, 2002, and 2004.

(b) Solve the system of equations obtained in part (a) using Gaussian elimination to find the polynomial coefficients.

(c) Analyze a cubic model based on 2000, 2001, 2002, and 2004 data, given by the polynomial p(z) = 62 - 32z + 26z^2 - 4z^3. Determine if this model accurately predicts sales for 2003 and justify.

Learn how to use polynomial interpolation to model sales data for the store 'Doces da Lu' from 2000 to 2005. This exercise guides you through creating quadratic and cubic polynomial models, solving linear systems with Gaussian elimination, and assessing model accuracy. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation