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

MATH307 December 2010
Other MATH307 Exams

### Question 03 (d)

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 ${\displaystyle x_{1}\leq x\leq x_{4}}$, whose graph interpolates these points. Assume that

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

where each pi(x) is a polynomial.

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

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

the equations written in parts (a), (b) and (c) above are equivalent to a system of linear equations in the unknowns ai, bi, ci and di, 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?

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