Write a program that uses Lagrangian polynomials for interpolation. You must use only three nodes such that the interpolating polynomial is at most quadratic. Using this program find approximate value of f (x) = 5x at x =1.5. The nodes are at points x 0 = 0, x 1 = 1, x 2 = 2.
Lagrangian polynomial interpolation is a technique used to estimate the value of a function at a given point based on known data points. In this case, we are asked to use Lagrangian polynomials to interpolate the function \( f(x) = 5x \) at \( x = 1.5 \), using three nodes: \( x_0 = 0 \), \( x_1 = 1 \), and \( x_2 = 2 \).
The Lagrangian interpolation polynomial for three nodes can be expressed as:
\[
L(x) = L_0(x) f(x_0) + L_1(x) f(x_1) + L_2(x) f(x_2)
\]
Where \( L_0(x) \), \( L_1(x) \), and \( L_2(x) \) are the Lagrange basis polynomials for each node, and \( f(x_0) ______ ____ ____ _______ _________ _____.
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Write a program to interpolate using Newton's Backward difference formula using only three points. Solve Problem asked in Question 4 using Newton’s Interpolating polynomial using Backward difference (instead of Lagrangian Polynomial).
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Write a program that uses Lagrangian polynomials for interpolation. You must use only three nodes such that the interpolating polynomial is at most quadratic. Using this program find approximate value of f (x) = 5x at x =1.5. The nodes are at points x 0 = 0, x 1 = 1, x 2 = 2.
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