Bookkeeping with the first law

You will write three functions that keep the energy books of a closed system.

The idea. The first law says $\Delta U = q + w$. Energy into the system is positive. For a whole cycle the system returns to its start, so $\Delta U = 0$.

Your tasks.

1. In delta_u, return $\Delta U$ from $q$ and $w$.
2. In heat_needed, return the heat $q$ for a given $\Delta U$ and $w$.
3. In missing_work, a cycle has the heats in heats and the works in known_works, with one work missing. Return the missing work.