Expert: Paul Klarreich - 1/20/2007

Question
hi there,
I'm a second year physics student.
One weakness i have in mathematics is being able to work with graphs. like i have to plot a graph for

f = 1+2t/T for 0<t<T/2
   1-2t/T for -T/2<t<T/2

and find the fourier coefficients Cn and so forth.
I get stuck on how to draw graphs. Is there a particular way of doing it that i seem to have missed paying attention on while at school?
Can you please suggest a good book for it?

Cheers.

Answer
Questioner:   puneet
Category:  Advanced Math
 
Subject:  graphs
Question:  hi there,
I'm a second year physics student.
One weakness i have in mathematics is being able to work with graphs. like i have to plot a graph for

f = 1+2t/T for 0<t<T/2
  1-2t/T for -T/2<t<T/2

and find the fourier coefficients Cn and so forth.
I get stuck on how to draw graphs. Is there a particular way of doing it that i seem to have missed paying attention on while at school?
Can you please suggest a good book for it?

Cheers.
.....................................
Hi, Puneet,

Probably any precalculus or calculus book will have material on graphing.  The calculus book, obviously, will have more techniques applied to graphs because some of them involve calculus.   

Unfortunately, there is no one particular way to do it -- it's an applied skill, which means that you have to actually think through each example.

Now the one you gave, assuming you are studying Fourier series, has a little bug in it:

f(t) = 1+2t/T for 0    < t < T/2
      1-2t/T for -T/2 < t < T/2

what is f(T/4), for example?  Do you use the first rule, because T/4 is between 0 and T/2, or the second, because T/4 is between -T/2 and T/2?

I think you probably mistyped, and meant

f(t) = 1+2t/T for    0 < t < T/2
      1-2t/T for -T/2 < t <= 0

So here, you will reason:

A. f is defined only on the interval -T/2 to T/2.
B. It has separate rules for the two segments of that interval.
C. One of those segments has the rule:

f(t) = 1+2t/T

If I drew  y = 1 + 2x/T, that would be a straight line graph.  Maybe if I get two points, I can draw it.  How about the end points?

f(0) = 1
f(T/2) = 2.

So I can draw the line segment between (0,1) and (T/2, 2) for this piece.

D. The other segment has the rule:

f(t) = 1-2t/T

If I drew  y = 1 - 2x/T, that would also be a straight line graph.  How about:

f(0) = 1
f(-T/2) = 1 + 1 = 2.

So I can draw the line segment between (0,1) and (T/2, 0) for this piece.

E. Putting those together will produce a kind of 'V'-shaped graph.

---------------------------------------
Now you didn't ask about this, but if you want your Fourier coefficients, you will first note that the graph is symmetric -- it is an even function.  Therefore, its F.S. will have ONLY cosine terms.

AND, because of the symmetry, when you compute:

{T/2  
|    f(t) cos kt dt
}-T/2

You can just do

{T/2  
|    f(t) cos kt dt
}0

and double the answer.