Science:Math Exam Resources/Courses/MATH307/December 2010/Question 03 (d)/Statement

Suppose we are given 4 points (x₁, y₁), (x₂, y₂), (x₃, y₃) and (x₄, y₄) in the plane and we want to find a function ƒ(x), defined for $x_{1}\leq x\leq x_{4}$, whose graph interpolates these points. Assume that

$f(x)={\begin{cases}p_{1}(x)&{\text{for }}x_{1}\leq x\leq x_{2}\\p_{2}(x)&{\text{for }}x_{2}\leq x\leq x_{3}\\p_{3}(x)&{\text{for }}x_{3}\leq x\leq x_{4}\end{cases}}$

where each p_i(x) is a polynomial.

(d) When each p_i(x) is a cubic polynomial of the form

$\displaystyle a_{i}(x-x_{1})^{3}+b_{i}(x-x_{i})^{2}+c_{i}(x-x_{i})+d_{i}$

the equations written in parts (a), (b) and (c) above are equivalent to a system of linear equations in the unknowns a_i, b_i, c_i and d_i, i = 1, 2, 3. How many more equations are needed if there are to be the same number of equations as unknowns? What equations are usually added and why?