The temperature T(t) (in degrees Celsius) of river water is monitored over a 24-hour period, where t=0 corresponds to midnight and t=12 corresponds to noon. Because heating and cooling occur gradually, T varies smoothly. Measurements show T(0)=12.4 and T(12)=18.7.
a. Explain why the model must supply a temperature value for every time t with 0≤t≤12.
b. Give a convincing argument that during the morning there is a time when the water temperature is exactly 15∘C.
c. Suppose a malfunction occasionally causes the temperature sensor to skip readings, producing abrupt jumps in the reported data. Discuss how such jumps would affect the certainty of your conclusion in part (b).