Science:Math Exam Resources/Courses/MATH307/December 2010/Question 06 (f)
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Question 06 (f) 

Suppose you expanded the same function ƒ(x) = x as in part (a) except on the interval . What would be the form (i.e., do not compute the coefficients) of the Fourier series valid for this interval? What points on the plane would you plot to produce a frequencyamplitude plot from this new Fourier series? (Give the answer in terms of the coefficients in the new expansion.) 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

The Fourier coefficients to the function over an interval are given by How can we transform these integrals to be over instead? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. If we extend the domain from to , we can write the new Fourier coefficients using the techniques from part (a) as We could go and directly compute these integrals but we want to relate them to the coefficients from part (a) computed on a domain . To do this, we can use a substitution and let . Then . With this transformation we have and and so putting everything together, where we have recognized that the new integral is precisely what we computed to get the coefficients on a period . Therefore on the Fourier coefficients double and Therefore we have that and the points to plot for the frequency amplitude are . 