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5. (30 points) One important application of spline interpolation is the construction of smooth curves that are not necessarily the graph of a function but that have a paramet- ric representation x = x(t) and y = y(t) for t E [a, b]. Hence one needs to determine two splines interpolating (t,, x; ) and (t;, yj), (j = 0, 1, ... , n). The arc length of the curve is a natural choice for the parameter t. However, this is not known a priori and instead the t's are usually chosen as the distances of consecutive points: to = 0, tj = tj-1+\\(xj - xj-1)2 + (yj - yj-1)2, j = 1, 2, . . . , n. Use the values in Table 1 to construct a smooth parametric representation of a curve passing through the points (x;, y;), j = 0, 1, ...,8 by finding the two natural cubic splines interpolating and (t;, yj),j = 0, 1, ...,8, respectively. Tabulate the coefficients of the splines and plot the resulting (parametric) curve. t; C j yi 0 0 1.5 0.75 1 0.618 0.90 0.90 2 0.935 0.60 1.00 1.255 0.35 0.80 4 1.636 0.20 0.45 5 1.905 0.10 0.20 6 2.317 0.50 0.10 7 2.827 1.00 0.20 8 3.330 1.50 0.25