Unit 5: Analytical 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

Consider the curve given by the equation $x^2 + y^2 = 2x + 4y - 4$. At what points on the curve is the tangent line horizontal?easy · mcq · 1 pt
If $f'(c) = 0$ and $f''(c) < 0$, which of the following must be true about the function $f$ at $x = c$?easy · mcq · 1 pt
Let $g(x) = x^4 - 4x^3$. On what open interval(s) is the graph of $g$ concave down?medium · mcq · 1 pt
The derivative of a function $f$ is given by $f'(x) = x^2(x-3)$. For what value(s) of $x$ does the graph of $f$ have a relative minimum?easy · mcq · 1 pt
Let $f(x) = x^3 - 3x^2 - 9x + 5$ on the closed interval $[-2, 4]$. What is the absolute maximum value of $f$ on this interval?easy · mcq · 1 pt
Given the curve defined by the equation $x^2 + xy + y^2 = 7$, find the slope of the tangent line to the curve at the point $(2, 1)$.hard · mcq · 1 pt
A farmer wants to build a rectangular pen adjacent to a straight river. The pen must have an area of 800 square feet. The side of the pen adjacent to the river requires no fencing, while the other three sides do. What is the minimum amount of fencing required to enclose the other three sides of the pen?medium · mcq · 1 pt
Let $f$ be a function continuous on the closed interval $[-2, 5]$ and differentiable on the open interval $(-2, 5)$. If $f(-2) = 1$ and $f(5) = 22$, which of the following statements must be true?hard · mcq · 1 pt

Sample prompts

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

Consider the curve given by the equation $x^2 + y^2 = 2x + 4y - 4$. At what poin…
If $f'(c) = 0$ and $f''(c) < 0$, which of the following must be true about the f…
Let $g(x) = x^4 - 4x^3$. On what open interval(s) is the graph of $g$ concave do…
The derivative of a function $f$ is given by $f'(x) = x^2(x-3)$. For what value(…
Let $f(x) = x^3 - 3x^2 - 9x + 5$ on the closed interval $[-2, 4]$. What is the a…
Given the curve defined by the equation $x^2 + xy + y^2 = 7$, find the slope of …
A farmer wants to build a rectangular pen adjacent to a straight river. The pen …
Let $f$ be a function continuous on the closed interval $[-2, 5]$ and differenti…