**Objective**: This activity is designed to help you understand the concept of Lagrange interpolation, a polynomial interpolation method used in numerical analysis and applied mathematics.

**Material**:

- Internet access
- Pen and paper
- A computer for running Python code

**Instructions**:

**Part 1: Understanding Lagrange Interpolation**

**Research**: Start by researching the definition and basics of Lagrange interpolation. Some key points to understand are:- What is Lagrange interpolation?
- How does it work?
- What is its mathematical formula?
- When is it used?

**Conceptual Understanding**: Once you’ve read about Lagrange interpolation, try to explain the concept to someone else or write down your understanding. This helps consolidate your learning.**Examples**: Look up examples of Lagrange interpolation. This could be mathematical examples where you’re given a set of points and you need to find the interpolating polynomial using the Lagrange formula.

**Part 2**

**Setup your development environment**. Open your web browser and go to https://replit.com/. Sign up for a new account if you don’t already have one, or log in.**Start a new Python project**. Click on the “+ New Repl” button on the top right, select “Python” as the language, give your project a name, and click on “Create Repl”.**Write the Lagrange interpolation function**. In the code editor, write a function named`lagrange_interpolation`

. This function should take in three parameters: a list of x values (`x_values`

), a list of y values (`y_values`

), and the x value to interpolate (`x`

). Use the provided code snippet for this step:

`def lagrange_interpolation(x_values, y_values, x):`

n = len(x_values)

y = 0

for i in range(n):

p = 1

for j in range(n):

if i != j:

p *= (x - x_values[j]) / (x_values[i] - x_values[j])

y += y_values[i] * p

return y

4. **Test the Lagrange interpolation function**. Still in the same Python file, create a list of x values (`x_values`

) and a list of y values (`y_values`

). Call the `lagrange_interpolation`

function with these lists and the x value to interpolate. Print the result. Use the provided code snippet for this step:

`x_values = [0, 1, 2, 3]`

y_values = [1, 2, 1, 10]

print(lagrange_interpolation(x_values, y_values, 2.5)) # Outputs: 3.75

**Run the code**. Click the ‘Run’ button at the top of the page. The output of the code will be displayed on the right side of the page.**Analyze the results**.- Try changing the
`x`

value in the`lagrange_interpolation`

function call. What happens when you interpolate at a value of x that is within your list of x values? What happens when you interpolate at a value of x that is outside your list of x values? - Try changing the
`y_values`

. How do the results change when you alter these values?

- Try changing the
**Write a short report**. After performing the above steps, write a short report explaining your findings. Include the answers to the questions asked in the analysis step. Also, discuss any challenges you faced and how you overcame them.