Suppose that $f(x)$ is periodic with period $[-\pi,\,\pi)$ and has the following real Fourier coefficients:
$a_0 = -3,\quad a_1 = 1,\quad a_2 = -3,\quad a_3 = 3,\quad \ldots \\ \phantom{b_0 = X,}\quad b_1 = 4,\quad b_2 = 0,\quad b_3 = -3,\quad \ldots$

(A) Write the beginning of the real Fourier series of $f(x)$ (through frequency 3):
$f(x) =$

(B) Give the real Fourier coefficients for the following functions:

      (i) The derivative $f'(x)$
$a_0 =$ $,\ a_1 =$ $,\ a_2 =$ $,\ a_3 =$ $,\ \ldots$
$\phantom{b_0 = X,}\qquad\qquad b_1 =$ $,\ b_2 =$ $,\ b_3 =$ $,\ \ldots$

      (ii) The function $f(x) + 3$
$a_0 =$ $,\ a_1 =$ $,\ a_2 =$ $,\ a_3 =$ $,\ \ldots$
$\phantom{b_0 = X,}\qquad\qquad b_1 =$ $,\ b_2 =$ $,\ b_3 =$ $,\ \ldots$

      (iii) The antiderivative of $(f(x) + 3)$ (with $C=0$)
$a_0 =$ $,\ a_1 =$ $,\ a_2 =$ $,\ a_3 =$ $,\ \ldots$
$\phantom{b_0 = X,}\qquad\qquad b_1 =$ $,\ b_2 =$ $,\ b_3 =$ $,\ \ldots$

      (iv) The function $f(x) + 2\sin\!\left(3x\right)-\cos\!\left(2x\right)$
$a_0 =$ $,\ a_1 =$ $,\ a_2 =$ $,\ a_3 =$ $,\ \ldots$
$\phantom{b_0 = X,}\qquad\qquad b_1 =$ $,\ b_2 =$ $,\ b_3 =$ $,\ \ldots$

      (iv) The function $f(2x)$
$a_0 =$ $,\ a_1 =$ $,\ a_2 =$ $,\ a_3 =$ $,\ \ldots$
$\phantom{b_0 = X,}\qquad\qquad b_1 =$ $,\ b_2 =$ $,\ b_3 =$ $,\ \ldots$