Question 1

Use the average rate of change of $C$ over $3 \le t \le 7$ to approximate $C'(5)$. Show the working and give units.

Use a left Riemann sum with the three subintervals in the table to approximate $\displaystyle\int_0^{12} C(t)\,dt$. Then say what $\dfrac{1}{12}\displaystyle\int_0^{12} C(t)\,dt$ means for the coffee.

For $12 \le t \le 20$, the cooling is modelled by $C'(t) = -\dfrac{24.55\,e^{0.01t}}{t}$ degrees Celsius per minute. Find the temperature at $t = 20$. Show the setup.

For that model, $C''(t) = \dfrac{0.2455\,e^{0.01t}\,(100 - t)}{t^2}$. For $12 < t < 20$, is the temperature changing at an increasing rate or a decreasing rate? Give a reason.