AP Calculus AB · Unit 5 · Analytical Applications of Differentiation

Let $f(t)$ be the number of gallons of water in a cistern $t$ hours after midnight, where $f$ is differentiable for $0 \le t \le 10$. The graph of $f'$ is shown, and selected values of $f$ are given in the table.

(a) Write an equation for the line tangent to the graph of $f$ at the point of inflection of the graph of $f$ on the interval $0 < t < 10$.
(b) Explain why there must be a time $c$, $2 < c < 8$, for which $f'(c) = \dfrac{f(8) - f(2)}{8 - 2}$.
(c) Find the absolute maximum value of $f(t)$ for $0 \le t \le 10$. Justify your answer using the candidates test.
(d) Use the tangent line from part (a) to approximate $f(5.5)$. Is this approximation an overestimate or an underestimate of $f(5.5)$? Give a reason for your answer.