Euler's method for a cooling cup

You will write Euler's method for Newton's law of cooling and compare it with the exact formula. A cup of coffee cools in a room at $20\ ^\circ\text{C}$ with $k = 0.08\ \text{min}^{-1}$.

The idea. The rate of change is $\dfrac{dT}{dt} = -k(T - 20)$. Euler's method moves forward by a small step $h$ in time and adds $h \times \dfrac{dT}{dt}$ to $T$ each time. The exact answer is $T = 20 + (T_0 - 20)e^{-kt}$. A smaller step should give a smaller error.

Your tasks.

1. Finish rate(T). Return $\dfrac{dT}{dt}$ for the temperature T, using K and ROOM.
2. Finish euler(T0, t_end, h). Start at T0 and repeat the step round(t_end / h) times. Return the final temperature.
3. Finish exact(T0, t). Return the exact temperature at time t.