AP Calculus AB · Unit 1 · Limits and Continuity

A substation draws power from the grid for $0 \le t \le 24$ hours. For $0 \le t \le 14$ with $t \neq 4$, the power drawn, in megawatts (MW), is modeled by $$P(t) = \frac{t^2 + 2t - 24}{t - 4};$$ a sensor recalibration recorded $P(4) = 12$. For $14 < t \le 24$, $P(t) = 26 - t$. Selected values of $P(t)$ are shown in the table.

(a) Find the value of $\lim_{t \to 4} P(t)$ algebraically, using the formula $P(t) = \dfrac{t^2 + 2t - 24}{t - 4}$.
(b) Is $P$ continuous at $t = 4$? If not, is the discontinuity removable or nonremovable? Justify your answer.
(c) The engineers claim that, by the Intermediate Value Theorem, there must be a time $t$ in the interval $2 \le t \le 6$ at which $P(t) = 11$ MW. Determine whether the Intermediate Value Theorem can be used to guarantee this claim. Justify your answer, using the value $P(4) = 12$ from the table and the limit found in part (b).
(d) At $t = 14$ hours the substation switches from Feeder A to Feeder B. Determine whether $P'(14)$ exists. Show the work that leads to your answer, including the value of $\lim_{t \to 14^+} P(t)$.