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

$f(x) = a_0 + \sum_{n=1}^\infty a_n \cos(n \pi x) + b_n \sin(n \pi x).$

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(\pi x) + b_1 \sin(\pi x). $$ How does the 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 $t > 0$?

For the following problems, you need to apply Laplace transformation to the equation, and get the $F(p)$ for the desired $f(t)$, and do the inverse Laplace transformation to get back the desired answer.

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 $t > 0$?