Use appropriate Lagrange interpolating polynomials of degrees one, two, and three to approximate each of the following: a. f(8.4) if f(8.1) = 16.94410, f(8.3) = 17.56492, f(8.6) = 18.50515, f(8.7) = 18.8209

help pls

can anyone just tell which rule i can use

To approximate the value of f(8.4) using Lagrange interpolating polynomials, we can utilize the following steps:

Step 1: Determine the degree of polynomial needed
In this case, we are asked to approximate using polynomials of degrees one, two, and three. We will calculate the interpolating polynomial for each degree.

Step 2: Calculate the Lagrange interpolating polynomial
To calculate the Lagrange interpolating polynomial for a specific degree, we need to identify the appropriate formulas.

For a polynomial of degree one, the formula is:
P₁(x) = [(x - x₂) / (x₁ - x₂)] * f(x₁) + [(x - x₁) / (x₂ - x₁)] * f(x₂)

For a polynomial of degree two, the formula is:
P₂(x) = [(x - x₂)(x - x₃) / (x₁ - x₂)(x₁ - x₃)] * f(x₁) + [(x - x₁)(x - x₃) / (x₂ - x₁)(x₂ - x₃)] * f(x₂) + [(x - x₁)(x - x₂) / (x₃ - x₁)(x₃ - x₂)] * f(x₃)

For a polynomial of degree three, the formula is:
P₃(x) = [(x - x₂)(x - x₃)(x - x₄) / (x₁ - x₂)(x₁ - x₃)(x₁ - x₄)] * f(x₁) + [(x - x₁)(x - x₃)(x - x₄) / (x₂ - x₁)(x₂ - x₃)(x₂ - x₄)] * f(x₂) + [(x - x₁)(x - x₂)(x - x₄) / (x₃ - x₁)(x₃ - x₂)(x₃ - x₄)] * f(x₃) + [(x - x₁)(x - x₂)(x - x₃) / (x₄ - x₁)(x₄ - x₂)(x₄ - x₃)] * f(x₄)

Step 3: Substitute the given values and evaluate
Substitute the given x-values for x₁, x₂, x₃, and x₄ into their respective interpolating polynomial formulas. Multiply each term by f(x) and evaluate the result to approximate f(8.4).

For example, to find the degree one polynomial:
P₁(x) = [(x - 8.3) / (8.1 - 8.3)] * 16.94410 + [(x - 8.1) / (8.3 - 8.1)] * 17.56492

Repeat this process for the degree two and three polynomials to approximate f(8.4).