Page 889 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

lie on the graph of , where x is in radians.

(a) Find the portion of the parabolic runout spline that interpolates these six points for . Maintain an accuracy of
five decimal places in your calculations.

(b) Calculate for the spline you found in part (a). What is the percentage error of with respect to the “exact”
value of ?

The following five points
3.

lie on a single cubic curve.

(a) Which of the three types of cubic splines (natural, parabolic runout, or cubic runout) would agree exactly with the single
cubic curve on which the five points lie?

(b) Determine the cubic spline you chose in part (a), and verify that it is a single cubic curve that interpolates the five
points.

Repeat the calculations in Example 1 using a natural spline to interpolate the five data points.
4.

Repeat the calculations in Example 1 using a cubic runout spline to interpolate the five data points.
5.

Consider the five points (0, 0), (.5, 1), (1, 0), (1.5, −1), and (2, 0) on the graph of .
6.

(a) Use a natural spline to interpolate the data points (0, 0), (.5, 1), and (1, 0).

(b) Use a natural spline to interpolate the data points (.5, 1), (1, 0), and (1.5, −1).

(c) Explain the unusual nature of your result in part (b).

7. (The Periodic Spline) If it is known or if it is desired that the n points , , …, to be interpolated lie

on a single cycle of a periodic curve with period , then an interpolating cubic spline must satisfy

(a) Show that these three periodicity conditions require that