Science:Math Exam Resources/Courses/MATH307/December 2010/Question 03 (d)/Solution 1

In part (a) we were asked to find the different equations that can be drawn from the condition that $f(x)$ goes through all the given points. This yielded the following 6 equations:

${\begin{aligned}p_{1}(x_{1})&=y_{1}\\p_{1}(x_{2})&=y_{2}\\p_{2}(x_{2})&=y_{2}\\p_{2}(x_{3})&=y_{3}\\p_{3}(x_{3})&=y_{3}\\p_{3}(x_{4})&=y_{4}\end{aligned}}$

In part (b) we were then asked to find the equations that can be found from the condition that the function, $f(x)$, was also first order differentiable. This yielded 2 more equations:

${\begin{aligned}p'_{1}(x_{2})&=p'_{2}(x_{2})\\p'_{2}(x_{3})&=p'_{3}(x_{3})\end{aligned}}$

This same method can be applied in part (c) to find the conditions that the function is second order differentiable yielding 2 more equations:

${\begin{aligned}p''_{1}(x_{2})&=p''_{2}(x_{2})\\p''_{2}(x_{3})&=p''_{3}(x_{3})\end{aligned}}$

Therefore, we were able to find 10 equations in total using the conditions set forth in (a), (b), and (c). However, we have three polynomials and each of these polynomials have 4 unknowns ($a_{i}$, $b_{i}$, $c_{i}$, and $d_{i}$ ; $i=1,2,3$), so we have a total of 12 unknowns that we need to solve for. It follows that we would need 2 more equations in order to have the same number of equations as unknowns.

These equations are usually prescribed depending on the context of the problem. The usual condition is that the second derivative values at the endpoints should be 0. Thus, the last 2 equations that we need are:

${\begin{aligned}p''_{1}(x_{1})&=0\\p''_{3}(x_{4})&=0\end{aligned}}$