Answer
a) See graph graph
b) Phase 1: $ \frac{dT}{dt}\gt0$
Phase 2 $ \frac{dT}{dt}=0$
Phase 3: $ \frac{dT}{dt}\leq0$
c) See graph graph
Work Step by Step
a) The initial flow of running water is close to the ambient temperature because the water is sitting in the pipes. When the water for hot water starts flowing from the tap, $T$ increases.
Then $T$ is constant at the same temperature of the heated water.
When there isn't water remaining, $T$ decreases to the ambient temperature.
b) The initial water temperature is close to the room temperature due to the water that was in the pipes. When the water from the hot water tank comes out $ \frac{dT}{dt}$ is positive, but as $t$ increases, $ \frac{dT}{dt}$ starts decreasing and then turns equal to $0$. That will tell us that the temperature of the water is not heating or cooling.
When all the water from the tank has been used, the rate of change is small and negative ($ \frac{dT}{dt}\lt0$).
The last phase is that the tank (that is empty of water) starts cooling to the temperature of the room, so in this phase $ \frac{dT}{dt}$ will increase slowly and turn into 0.
c) Graph
