1. Sine and Cosine decomposition.
Suppose you are given a function on an interval, $f(x): [0, 1] \to \R$. Such function $f(x)$ can be expressed as a sum of 'sine waves' and cosine waves and constant
$$ f(x) = a_0 + \sum_{n=1}^\infty a_n \cos(2n \pi x) + b_n \sin(2n \pi x). $$
Can you figure out a way to determine the coefficients $a_n$ and $b_n$?
Test out your method for the following function $$ f(x) = \begin{cases} 1 & 0 < x < 1/2 \cr 0 & 1/2 \leq x \leq 1 \end{cases} $$
find $a_0, a_1, b_1$ and plot the truncated Fourier series $$ a_0 + a_1 \cos(2 \pi x) + b_1 \sin(2 \pi x). $$ How does this resemble your original given function?
2. Consider the following equation, for $t>0$, $$ f'(t) + f(t) = 0 $$ And suppose $f(0) = 1$. Can you solve $f(t)$ for $t > 0$?
3. Consider the following equation, for $t>0$, $$ (d/dt + 1) (d/dt + 2) f(t) = 0 $$ And suppose $f(0) = 1, f'(0)=0$. Can you solve $f(t)$ for $t > 0$?
4. Consider the following equation, for $t>0$, $$ [(d/dt)^2 + 1] f(t) = 0 $$ And suppose $f(0) = 1, f'(0)=0$. Can you solve $f(t)$ for $t > 0$?
5 (bonus, optional). Consider the following equation, for $t>0$, $$ (d/dt + 1) (d/dt + 1) f(t) = 0 $$ And suppose $f(0) = 1, f'(0)=0$. Can you solve $f(t)$ for $t > 0$?