Unit 4: Contextual Applications of Differentiation

AP Calculus AB — 8 practice questions with detailed explanations. Use the links below to open each full question page (ideal for search indexing and deep study).

Unit summary

This unit covers core ideas tested on the AP exam. Each listed item links to a standalone page with the full stem, answer choices where applicable, a step-by-step explanation, and the correct answer. Difficulty labels help you prioritize review.

Questions in this unit

A $13$-foot ladder is leaning against a vertical wall. The bottom of the ladder slides away from the wall at a constant rate of $0.5$ feet per second. At what rate is the top of the ladder sliding down the wall when the bottom of the ladder is $5$ feet from the wall?medium · mcq · 1 pt
The temperature $T$ of a cup of coffee is cooling in a room, modeled by the function $T(t)$, where $T$ is measured in degrees Celsius and $t$ is measured in minutes. Which of the following expressions represents the average rate of change of the coffee's temperature between $t = 2$ and $t = 10$ minutes?hard · mcq · 1 pt
A spherical balloon is being inflated at a rate of $10\pi$ cubic centimeters per second. At what rate is the radius of the balloon increasing when the radius is $5$ cm?easy · stimulus-mcq · 1 pt
A pebble is dropped into a still pond, creating a circular ripple. The radius of the ripple is increasing at a constant rate of $2$ feet per second. Which of the following correctly relates the rate of change of the area $A$ enclosed by the ripple to its radius $r$?hard · mcq · 1 pt
What is the value of $\lim_{x \to 0} \frac{\ln(x + 1)}{x^2 + x}$?easy · mcq · 1 pt
Let $f$ be a differentiable function such that $f(3) = 8$ and $f'(3) = -2$. Which of the following is the best approximation of $f(3.1)$ using local linearity?hard · mcq · 1 pt
A particle moves along the $x$-axis so that its velocity at time $t \ge 0$ is given by $v(t) = t^2 - 5t + 6$. At time $t = 1.5$, is the speed of the particle increasing or decreasing, and why?hard · mcq · 1 pt
The volume of oil in a storage tank is given by the function $V(t)$, where $V$ is measured in gallons and $t$ is measured in hours. Which of the following provides the best interpretation of the equation $V'(5) = -30$?medium · mcq · 1 pt

Sample prompts

Short previews of topics covered (not full question text).

A $13$-foot ladder is leaning against a vertical wall. The bottom of the ladder …
The temperature $T$ of a cup of coffee is cooling in a room, modeled by the func…
A spherical balloon is being inflated at a rate of $10\pi$ cubic centimeters per…
A pebble is dropped into a still pond, creating a circular ripple. The radius of…
What is the value of $\lim_{x \to 0} \frac{\ln(x + 1)}{x^2 + x}$?…
Let $f$ be a differentiable function such that $f(3) = 8$ and $f'(3) = -2$. Whic…
A particle moves along the $x$-axis so that its velocity at time $t \ge 0$ is gi…
The volume of oil in a storage tank is given by the function $V(t)$, where $V$ i…