Single Variable Calculus, Chapter 3, 3.2, Section 3.2, Problem 54

Suppose that a hot-water faucet is turned on, the temperature $T$ of the water depends on how long the water has been running.
a.) Illustrate a possible graph of $T$ as a function of the time $t$ has elapsed since the faucet was turned on
b.) Describe how the rate of change of $T$ with respect to $t$ varies as $t$ increases.
c.) Sketch the graph of the derivative of $T$.

a.)



b.) Suppose that the faucet and the heater is turned on simultaneously, the
initial temperature of the water is that of the room temperature (let it be $20^\circ\rm{C}$)
suppose that it lasts for 3 seconds until such time that the water starts conducting heat from
the heater and its temperature is increasing rapidly for 1 second from the interval $(3,4)$. After 4 seconds, the water stays
at temperature of the heater, let's say at constant $40^\circ\rm{C}$.
c.)