Let $f$ be the function satisfying $f'(x) = -3x f(x)$, for all real numbers $x$, with $f(1) = 4$ and $\lim_{x \rightarrow \infty} f(x) = 0$.
(a) Evaluate $\displaystyle\int_{1}^{\infty} -3x f(x)\, dx$. Show the work that leads to your answer.
(b) Use Euler's method, starting at $x = 1$ with a step size of $0.5$, to approximate $f(2)$.
(c) Write an expression for $y = f(x)$ by solving the differential equation $\dfrac{dy}{dx} = -3xy$ with the initial condition $f(1) = 4$.