


For this module, there are two computer algebra system files for each system. If you have your CAS open, save your file and close the program. When you click on the button below corresponding to your CAS, this will launch the CAS and will load a file corresponding to Parts 57 of this module.

In Part 4 we discovered integral formulas for the coefficients of trigonometric polynomials. In this part we will use those formulas to approximate general functions of period 2*pi.
Now suppose we have a 2*piperiodic function that is not a trigonometric polynomial. For example, consider the function obtained by extending the absolute value function
to be periodic of period 2*pi. Because of the shape of the graph, this function is often referred to as a "triangular wave (function)."
Our approach here will be to use the same formulas as above to calculate the coefficients a_{k} and b_{k}. Then we will see how closely the trigonometric polynomials
approximate f. Our hope is that, as n increases, the approximations will be increasingly like the function f itself.
Let f be the "triangular wave" function.
(b) Find a general formula for a_{k} where k is an odd positive integer.
(c) Find a general formula for a_{k} where k is an even positive integer.
(d) Notice that f is an even function. Use that fact to predict the values of the b_{k}.
Compare the "triangular wave function" to your approximation with n = 5 by evaluating them at various values of t between 0 and pi.
to be 2*piperiodic.
Using Steps 1  3 as a model, find and compare the Fourier approximations to the sawtooth wave function. Your worksheet should include a discussion of odd and even functions and their relationship to the general formulas for a_{k} or b_{k}. Your worksheet should also include graphs of the sawtooth wave function together with your approximations for various values of n. In addition, compare the sawtooth wave function with your approximations by evaluating both g and the approximation at appropriate points. Describe what happens to the graph of your approximations near the points of discontinuity of g. What is the value of your approximation at the points of discontinuity of g?
to be 2pi periodic.
to be 2pi periodic.
Follow the same steps as for the previous examples. Again, the general formulas for a_{k} and b_{k} are difficult to analyze. Look for values of k for which a_{k} or b_{k} are equal to 0.



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