Group Project 2
A manufacturing company is designing an integrated circuit to control a piece of equipment. Based on experiments, the engineers have concluded that the stress on the equipment can be measured by the function
In this formula, t represents time (the number of hours that the
equipment has been operating), and 
and 
are parameters that
can be varied to represent the various uses of the equipment.
The engineers believe that the machine can be operated safely as long as
this stress measure remains below 3.75. Typically, the machine will be
used for between 3.5 and 10 hours at a time. 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 machine must be able to
estimate the stress function, and shut down the machine if the stress
measure reaches 3.75. 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 machine.
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 machine. 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.
(Use symbolic computing techniques to determine the derivatives of 
.)
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 3.75. 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 3.75. 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: Tuesday, February 24
Final deadline: Friday, March 6, 9:30am
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