Differential Equations     Math 285    Practice Hour Exam 3   Fall, 2011

 

 

 

 

 

 

 

 

 

 

 

 

1.

 

 

 

Let  f(t) = 1  for  0 < t < 1  and let  f(t) = 2 – t  for  1 < t < 2.

 

a.       Set up the Fourier series of f(t) with period 2.  Set up the expressions for the

coefficients but do not evaluate the integrals.  Give a careful graph of the sum of

the series.  Show several periods on your graph.  Pay attention to discontinuities of

the function.

b.      Set up the Fourier sine series of f(t) on the interval  0 < t < 2.  Give a careful graph

of the sum of the series.

c.       Set up the Fourier cosine series of f(t) on the interval  0 < t < 2.  Give a careful graph

of the sum of the series.

                               ∞                                                 1                    2

Answer:    a.    f(t) ~ ao/2 + Σ an cos nπt + bn sin nπt     an = ∫ cos nπt dt + ∫ (2-t) cos nπt dt

                                           n=1                                              0                    1

                                  1                    2      

                          bn = ∫ sin nπt dt + ∫ (2-t) sin nπt dt

                                 0                    1

                                   ∞                              1                      2                                        

Answer:    b.     f(t) = Σ  bn sin nπt     bn = ∫ sin nπt/2 dt + ∫ (2-t) sin nπt/2 dt

                                 n=1                            0                      1

                                             ∞                              1                       2

Answer:    c.     f(t) = ao/2 + Σ  an cos nπt     an = ∫ cos nπt/2 dt + ∫ (2-t) cos nπt/2 dt

                                            n=1                           0                       1

 

 

 

 

 

2.

 

Let  f  be a period 3 function such that  f(t) = 1 – t  on the interval  0 < t < 3.  Find the

Fourier series of  f  and give a graph of its sum.

 

                                     ∞      

Answers:   f(t) ~ - ½ + Σ (3/nπ) sin 2nπt/3

                                    n=1

 

 

 

 

 

 

3.

 

 

 

                          ∞

Given that   t = 2 Σ (-1)n+1 (1/n) sin nt  for  0 < t < π,

                           n=1

 

a.       Does the above equality hold for  - π < t < 0 ?

b.      Graph the right-hand part of the above equality for  - 3π ≤  t ≤ 3π

c.       Differentiate the above equality term wise.  Is the resulting equality correct for

0 < t < π.  Explain.

 

Answers:  a. Yes           c.  No

 

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