m-3150-001 Midterm exam 1

Andrej Cherkaev

Your name, ........................................................... Solve any three problems:

1. Find the period of the following functions:

a. f_{1}(x) = sin(2x) + cos(4x)  (1)
b. f_{3}(x) = sin(5x) − 2 cos(3x)  (2)
c. f_{2}(x) = cos(3πx)  (3)

2. a. Find the Fourier series of the function f(x) = e^{x}, −π < x < π. b. Differentiate the series and show that f′(x) = f(x).

3. The function 4-periodic is given by formula

f(x) = \{ \begin{matrix}3 & if 0 < x ≤ 1 \\ −1 & if 1 < x ≤ 2 \\ 0 & if 2 < x ≤ 4. \end{matrix}

a. Find the Fourier series.

b. Skatch the graph of the function and of its antiderivative, assuming that integral of the antiderivative over the interval [0, 4] is zero. Write the Fourier series for the antiderivative.

4. Find the even and odd half-range expansions of f(x) = x^{2}, 0 < x < 1. Skatch the graphs of these expansions in the interval −1 < x < 1.