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

MATH307 December 2010

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Question 03 (b)

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. (b) What equations, written in terms of pi(x) and possibly their derivatives, express the condition that $\displaystyle f'(x)$ 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 function ƒ'(x) is continuous if the slope to the left of the points xi approaches the slope to the right of the points. This only applies to the inner points of the interval, not the endpoints.

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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. The function $\displaystyle f'(x)$ being continuous implies that the slopes of the adjacent polynomials must be the same at $x_{i}$, for i = 2, 3. In our case this implies ${\begin{aligned}p'_{1}(x_{2})=p'_{2}(x_{2})\\p'_{2}(x_{3})=p'_{3}(x_{3})\end{aligned}}$

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