AP Calculus AB — Limits and Derivatives flashcards

10 of 68 cards free to study here · 5 topics · the whole deck is built into Willys

Units 1–5 of AP Calculus AB: limits and continuity, the derivative and its rules, chain rule, implicit and inverse differentiation, motion and related rates, linearization, L'Hospital's rule, MVT and EVT, curve sketching and optimization.

Get the app All AP Calculus AB decks

Learn deck · part of AP Calculus AB · written against: College Board AP Calculus AB Course and Exam Description (Units 1–8). Updated 2026-10-10.

Study the free sample

10 cards from the deck as they are. Flip, rate yourself, filter by topic; your progress stays in this browser.

The 10 sample cards as a list

What does the statement $\lim_{x \to a} f(x) = L$ claim, and what does it say about the value $f(a)$ ⁠?

As $x$ gets close to $a$ from both sides (without equaling $a$ ⁠), the outputs $f(x)$ get as close as you like to L. It says nothing about $f(a)$ ⁠: the function may be undefined at $a$ or have a different value there.
Limits and Continuity › Evaluating Limits
Both one-sided limits of $f$ at $x = 2$ exist. When does the two-sided limit at 2 still fail to exist?

When the left-hand and right-hand limits are different numbers, as at a jump. The two-sided limit exists only when $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x)$ ⁠.
Limits and Continuity › Evaluating Limits
What are the two limit forms of the definition of $f'(a)$ ⁠?

$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$ ⁠, or equivalently $\lim_{x \to a} \frac{f(x) - f(a)}{x - a}$ ⁠. Each is the limit of secant slopes, giving the slope of the tangent line.
Differentiation Definition and Basic Rules › Definition of the Derivative
What is the value of $\lim_{h \to 0} \frac{\sin\left(\frac{\pi}{3} + h\right) - \sin\left(\frac{\pi}{3}\right)}{h}$ ⁠, and how do you see it quickly?

Recognize it as the definition of the derivative of $\sin x$ at $x = \frac{\pi}{3}$ ⁠, so it equals $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$ ⁠.
Differentiation Definition and Basic Rules › Definition of the Derivative
What does the chain rule say, and what is the derivative of $\sin\left(x^2\right)$ ⁠?

$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$ ⁠: derivative of the outside evaluated at the inside, times the derivative of the inside. So the answer is $2x \cos\left(x^2\right)$ ⁠.
Composite Implicit and Inverse Functions
What is the derivative of $\left(3x^2 + 1\right)^5$ ⁠, and what is the usual mistake?

$5\left(3x^2 + 1\right)^4 \cdot 6x = 30x\left(3x^2 + 1\right)^4$ ⁠.

Watch out: forgetting the inner derivative $6x$ is a common chain-rule error.
Composite Implicit and Inverse Functions
$W(t)$ is the number of gallons in a tank at time $t$ hours, and $W'(3) = -4$ ⁠. What does that mean in context?

At $t = 3$ hours, the amount of water is decreasing at 4 gallons per hour. A derivative's units are output units per input unit.
Contextual Applications of Differentiation › Rates of Change and Motion
A particle's position is $x(t)$ ⁠. How are velocity and acceleration defined, and when is the particle at rest?

Velocity is $v(t) = x'(t)$ and acceleration is $a(t) = v'(t) = x''(t)$ ⁠; the particle is at rest when $v(t) = 0$ ⁠.
Contextual Applications of Differentiation › Rates of Change and Motion
What does the Mean Value Theorem state, including its hypotheses?

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ ⁠, then $f'(c) = \frac{f(b) - f(a)}{b - a}$ for at least one $c$ in $(a, b)$ ⁠: some tangent is parallel to the secant.
Analytical Applications of Differentiation › MVT and EVT
A car's position is differentiable, and it covers 120 miles in 2 hours. What does the MVT guarantee?

At some instant the car's velocity is exactly 60 mph, the average velocity over the trip.
Analytical Applications of Differentiation › MVT and EVT

Unlock the whole deck

You are studying a free 10-card sample. All 68 cards of AP Calculus AB — Limits and Derivatives, the other 2 AP Calculus AB decks, and spaced-repetition scheduling are built into Willys. 14-day free trial, then $14.99 a year.

Get the app

In Willys, tap Exam Library at the top of your Decks (or Browse Exam Library… under the + button), find the exam on its shelf or by search, tick it and tap Add.

More AP Calculus AB decks

Integrals and Applications Practice Questions

← All AP Calculus AB decks, with the study guide and topic outline.

About these cards

Review status: Second-reader reviewed, with corrections applied.

Checked against:

Topic outline follows the College Board AP Calculus AB and BC Course and Exam Description (effective Fall 2020), Units 1–5 (the AP name is used nominatively; no course or exam text reproduced).
OpenStax Calculus Volumes 1 and 2, consulted as references; no text copied.

These are independent study notes written for Willys. They are not official exam material, may contain mistakes, and are no substitute for your course, your teacher or the exam owner's own publications. Found an error? Tell us — corrections ship as a new version of the deck.

AP® is a trademark registered by the College Board, which is not affiliated with, and does not endorse, this product.

Use of these pages is covered by our Terms of Use.