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

MATH307 December 2010

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Question 03 (a)

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 pi(x) is a polynomial. (a) What equations, written in terms of pi(x) and possibly their derivatives, express the condition that ƒ(x) goes through the given points? Do the equations imply that ƒ(x) is continuous?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The condition that $f(x)$ goes through all the points means that $f(x_{i})=y_{i}$. Rewrite this in terms of $p_{i}(x)$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Since the piece-wise function above has inclusive restrictions, we can see that both polynomials, $p_{1}$ and $p_{2}$, have the same value $f(x_{2})$. The same goes for polynomials $p_{2}$ and $p_{3}$. Thus, we need to satisfy $\displaystyle p_{i}(x)=p_{i+1}(x)$ Therefore, ${\begin{aligned}p_{1}(x_{2})=p_{2}(x_{2})\\p_{2}(x_{3})=p_{3}(x_{3})\end{aligned}}$ Hence these equations imply that $f(x)$ is continuous as they show that both polynomials surrounding the given point equal each other at $x_{i}$. As well, to show that the function goes through the given points, we need to write the following six equations: ${\begin{aligned}p_{1}(x_{1})=y_{1}\\p_{1}(x_{2})=y_{2}\\p_{2}(x_{2})=y_{2}\\p_{2}(x_{3})=y_{3}\\p_{3}(x_{3})=y_{3}\\p_{3}(x_{4})=y_{4}\end{aligned}}$

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