Science:Math Exam Resources/Courses/MATH307/April 2012/Question 01 (a)
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Question 01 (a) 

Suppose you are given a set of N data points (x_{n}, y_{n}), with x_{n} increasing, and you wish to interpolate these points with a spline function ƒ, where ƒ(x) is given by the cubic polynomial p_{n}(x) on each interval (x_{n}, x_{n+1}) for n = 1, ..., N1: (a) Write down the equations required for ƒ(x) to be continuous and to pass through the data points. How many equations does this provide? 
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 

Science:Math Exam Resources/Courses/MATH307/April 2012/Question 01 (a)/Hint 1 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. For ƒ(x) to pass through all the data points, we must have p_{n}(x_{n}) = y_{n} and for ƒ(x) to be continuous, we must have p_{n}(x_{n+1}) = y_{n+1}. Each of these give us N1 equations because there are N1 polynomials. So in total, (N1) + (N1) = 2N  2 equations. 