A manufacturing company is designing an motor to control a piece of equipment. Based on experiments in which the engineers increase the demands on the motor at a controlled rate, the engineers have concluded that the stress on the motor can be measured by the function

In this formula, t represents the time (measured in hours) during which the demands on the motor increase, and and are parameters that can be varied to represent the various uses of the motor.

The engineers believe that the motor can be operated safely as long as this stress measure remains below 8.00. In normal use, the parameters will take on the values and . (All of the data in this project can be measured accurately to three significant digits.)

The integrated circuit that controls the motor must be able to  estimate the stress function, and shut down the motor if the stress measure reaches 8.00. Unfortunately, computing the stress function is time consuming, and cannot be done directly since most of the computing power of the integrated circuit must be used to operate the motor. Instead, the stress function will be approximated by a polynomial obtained from only a small number of values of .

 C Grade:  Plot for . Use quad8 to estimate the integral. Use polyfit to approximate by a polynomial of "adequate" degree; the Chebyshev points should be used for interpolation. (You may judge the "adequacy" of the polynomial by looking at plots.) Use fzero to determine when , and hence when to shut down the motor. Perform this calculation for three equally-spaced values of each of the parameters and (9 cases in all).

 B Grade:  For the B-grade section, assume that and . Plot the estimated error of the polynomial for various choices of degree (you can determine this error numerically). Use symbolic computing techniques to determine the theoretical bound on the error for a cubic polynomial. What degree of polynomial must you use if you wish to have three-digit accuracy in the approximation? Let be the time when the stress measure equals 8.00. What are the scaling factors for with respect to the values of used to obtain the polynomial? What are the scaling factors for with respect to the values of used to obtain the polynomial?

 A Grade (Version 1):  Modify your software so that it determines the earliest possible time that the stress measure could equal 8.00. You must take into account all sources of error in the project.

 A Grade (Version 2):  Modify your software so that it determines to three significant digits as efficiently as possible. You can choose whether you wish to measure efficiency using time ( tic and toc) or using numbers of calculations ( flops).

Preliminary deadline: Wednesday, September 30

Final deadline:  Thursday, October 8, 9:30am