Q1: Which of the following functions $f: \mathbb{R} \rightarrow \mathbb{R}$ is $1$-periodic?

A1: $f(x) = 5$.

A2: $f(x) = \sin(x)$

A3: $f(x) = \sin(\frac{x}{2 \pi})$

A4: $f(x) = \cos(\pi x)$

Q2: Which of the following functions $f: \mathbb{R} \rightarrow \mathbb{R}$ is not $2\pi$-periodic?

A1: $f(x) = \cos(\frac{1}{2} x)$

A2: $f(x) = \sin( 2 x)$

A3: $f(x) = \sin(3 x)$

A4: $f(x) = \cos(x)$

Q3: How can the trigonometric polynomial $f(x) = 2 \cos(x) + 4 \sin(2x)$ be written as a complex linear combination with exponential functions?

A1: $f(x) = e^{ix} + e^{-ix} + \frac{2}{i}( e^{i 2 x} - e^{-i 2 x}) $

A2: $f(x) = 3 e^{ix} + e^{-ix} + e^{i 2 x} - 3 e^{-i 2 x} $

A3: $f(x) = e^{ix} + \frac{2}{i} e^{i 2 x} $

A4: $f(x) = 2 e^{ix} + 2 e^{-ix} + \frac{4}{i}( e^{i 2 x} - e^{-i 2 x}) $