Sample Midterm Questions Math 121, Fall 2004

1. Answer the following questions with a brief explanation to justify your answer.

(a) What is the period of the function sin (7x)?

(b) What is the value of

∫_{0}^{700π} sin^{2} (7x) dx?

(c) What is the value of

∫_{0}^{700π} sin (7x) sin (70x) dx?

2. Suppose that f(x) is the 2π-periodic function defined by

f(x) = \{ \begin{matrix}x & for −π < x ≤ 0, \\ 0 & for 0 < x ≤ π. \end{matrix}

Compute the (real) Fourier series expansion of f(x). What does the Fourier series converge to at x = 0, x = π/2, and x = π? Why?

3. Suppose that

f(x) = 1 − x^{2} 0 < x < 1.

Let f_{p} be the periodic extension of f (with period 1), f_{e} the even periodic extension of f (with period 2), and f_{o} the odd periodic extension of f (with period 2).

(a) Sketch the graphs of f_{p}, f_{e}, and f_{o} on the interval −3 < x < 3.

(b) Write out the corresponding form of the Fourier series for these functions, together with expressions for their Fourier coefficients. (Just write expressions for the coefficients — don’t evaluate any integrals.)

(c) Which Fourier series converges faster — the one for f_{e} or the one for f_{o}? Explain you answer briefly, but don’t do any explicit computations.

4. The function

f(x) = \{ \begin{matrix}0 & for −π < x ≤ 0, \\ sin x & for 0 < x ≤ π. \end{matrix}

Has the Fourier series expansion

f(x) = \frac{1}{π} + \frac{1}{2} sin x − \frac{2}{π} (\frac{cos 2x}{2^{2} − 1} + \frac{cos 4x}{4^{2} − 1} + \frac{cos 6x}{6^{2} − 1} + … + \frac{cos 2nx}{(2n)^{2} − 1} + …)

Apply Parseval’s theorem to this function, and use the result to determine the sum of the infinite series

∑_{n=1}^{∞} \frac{1}{[(2n)^{2} − 1]^{2}} = \frac{1}{3^{2}} + \frac{1}{15^{2}} + \frac{1}{35^{2}} + ….

5. State the orthogonality relation for the functions e^{inx}, where n is an integer.

(a) Use these relations to derive an expression for the Fourier coefficient c_{n} in the complex Fourier expansion of a 2π-periodic function f,

f(x) = ∑_{n=−∞}^{∞} c_{n}e^{inx}.

(b) What can you say about c_{n} if f is an even function?

6. Suppose that f(x) has the complex Fourier series expansion

f(x) = ∑_{n=−∞}^{∞} \frac{1}{1 + in} e^{inx}.

Find the real Fourier series expansion of f(x) (in terms of sines and cosines).