Cubic splines. Suppose you are in charge of the design of a roller coaster ride. This

Chapter 1, Problem 32

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Cubic splines. Suppose you are in charge of the design of a roller coaster ride. This simple ride will not make any left or right turns; that is, the track lies in a vertical plane. The accompanying figure shows the ride as viewed from the side. The points (a, , /?,) are given to you, and your job is to connect the dots in a reasonably smooth way. Let aj+ \ > ai.One method often employed in such design problems is the technique of cubic splines. We choose ft (r), a polynomial of degree < 3, to define the shape of the ride between (a, _ i, _ 1) and (a,-,/?/), for / = l,,..,/j.Obviously, it is required that ft (at) = bi and ft (a/_ i) = bj - 1, for i = 1,..., n. To guarantee a smooth ride at the points (ai,bi), we want the first and the second derivatives of ft and //+ 1 to agree at these points:= //+](,) fliat) = f!'+x(a;).and for / = 1,. ,n - 1.Explain the practical significance of these conditions. Explain why, for the convenience of the riders, it is also required thatf\ (00) = /> i.) = 0.Show that satisfying all these conditions amounts to solving a system of linear equations. How many variables are in this system? How many equations? (Note: It can be shown that this system has a unique solution.)