A component in the steering mechanism of an automobile moves during turns. The component will have to withstand the forces that arise during these movements, and these forces are determined by the velocity and acceleration during the motion.

A sensor has been placed on a test version of the steering mechanism, and this sensor records the position of the component at various times. The data values are of the form (t_i,y_i) where t_i records the time, and y_i records the position of the component. Your goal is to estimate the maximum velocity and acceleration of the component.

C Grade:  Two trial data sets, p3dat1.m and p3dat2.m, have been provided. (The first of these is artificial---it is a set of values of Runge's function. It is known that polynomial data fitting works poorly on this data set.) Approximate these data sets using spline functions. For each data set, use three separate approximations: (1) a piecewise-linear spline (obtained using the Matlab function interp1), (2) a piecewise-quadratic spline (obtained using the formulas in Chapter 7 of your textbook), and (3) a piecewise-cubic spline (obtained using the Matlab function spline). Plot your results, and comment on the quality of the approximations.

B Grade:  Plot the first and second derivatives of your approximations. Use the Matlab function fmin to determine the maximum values of these derivatives.

A Grade:  The above data sets are exact, that is, the data values contain no errors other than rounding errors. For these data sets it was appropriate to interpolate the data values with a spline. A more realistic data set can be found in p3dat3.m. In this data set, the y values include measurement errors. Approximate this data set (in the least-squares sense) using a cubic spline. Select the knots of the spline manually. Use the spline to determine the maximum values of the first and second derivative. How sensitive are your answers to the choice of the knots?

Preliminary deadline: Wednesday, October 21

Final deadline: Thursday, November 5, 9:30am