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

MATH307 December 2010

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q2 (e)  •  Q3 (a)  •  Q3 (b)  •  Q3 (c)  •  Q3 (d)  •  Q4 (a)  •  Q4 (b)  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q6 (a)  •  Q6 (b)  •  Q6 (c)  •  Q6 (d)  •  Q6 (e)  •  Q6 (f)  •  Q7 (a)  •  Q7 (b)  •  Q7 (c)  •  

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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 $x_{1}\leq x\leq x_{4}$, 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. (d) When each pi(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 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?

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Count the number of equations that we gathered in this question so far. How many unknowns do we have in total?

Hint 2
The usual additional equations involve the second derivative at 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. 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: ${\begin{aligned}p''_{1}(x_{1})&=0\\p''_{3}(x_{4})&=0\end{aligned}}$