AP Calculus AB · Unit 2 · Differentiation: Definition and Fundamental Properties

The number of gallons of water in a cistern $t$ hours after a storm begins is modeled by a differentiable function $W$, with selected values shown in the table for $0 \le t \le 8$. The depth of water in a channel that feeds the cistern is $H(t) = t\sqrt{t+9}$ inches for $t \ge 0$.

(a) Use the average rate of change of $W$ over the interval $2 \le t \le 5$ to approximate $W'(3)$. Show the computations that lead to your answer, and indicate units of measure.
(b) Find an equation for the line tangent to the graph of $H$ at $t = 7$.
(c) Using appropriate units, explain the meaning of $H'(7) = 4.875$ in the context of this problem.
(d) Use the line tangent to the graph of $H$ at $t = 7$ to approximate $H(7.5)$. Is this approximation an overestimate or an underestimate of $H(7.5)$? Give a reason for your answer.