AP Calc AB Exam Reference Sheet Cheat Sheet

A printable reference covering limits, continuity, derivative rules, derivative applications, integrals, accumulation, and the Fundamental Theorem of Calculus for grades 11-12.

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This AP Calculus AB exam reference sheet summarizes the core ideas students use most often during review: limits, continuity, derivatives, applications of derivatives, integrals, and accumulation. It is designed to help students quickly connect definitions, rules, and common problem types. A strong reference sheet is useful because AP questions often mix several skills in one problem. Keeping the main formulas in one place supports faster recall and more accurate work.

Key Facts

The limit definition of the derivative is $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$ when the limit exists.
A function is continuous at $x=a$ if $f(a)$ is defined, $\lim_{x\to a}f(x)$ exists, and $\lim_{x\to a}f(x)=f(a)$ .
The product rule is $\frac{d}{dx}[u(x)v(x)]=u'(x)v(x)+u(x)v'(x)$ .
The quotient rule is $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right]=\frac{v(x)u'(x)-u(x)v'(x)}{[v(x)]^2}$ , where $v(x)\ne 0$ .
The chain rule is $\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$ .
Critical numbers occur where $f'(x)=0$ or $f'(x)$ is undefined, as long as $x$ is in the domain of $f$ .
The net change theorem says $\int_a^b f'(x)\,dx=f(b)-f(a)$ .
The Fundamental Theorem of Calculus says if $F(x)=\int_a^x f(t)\,dt$ , then $F'(x)=f(x)$ when $f$ is continuous.

Vocabulary

Limit: A limit describes the value a function approaches as the input approaches a particular number.
Continuity: Continuity at $x=a$ means the graph has no break there and satisfies $\lim_{x\to a}f(x)=f(a)$ .
Derivative: A derivative gives the instantaneous rate of change of a function and the slope of its tangent line.
Critical Number: A critical number is a domain value where $f'(x)=0$ or where $f'(x)$ does not exist.
Definite Integral: A definite integral $\int_a^b f(x)\,dx$ represents signed area and accumulated change over an interval.
Accumulation Function: An accumulation function has the form $F(x)=\int_a^x f(t)\,dt$ and measures total change from $a$ to $x$ .

Common Mistakes to Avoid

Forgetting to check continuity before using the Intermediate Value Theorem is wrong because the theorem only applies when the function is continuous on $[a,b]$ .
Using $f'(x)=0$ as the only test for extrema is wrong because extrema can also occur where $f'(x)$ is undefined or at endpoints of a closed interval.
Dropping the inner derivative in the chain rule is wrong because $\frac{d}{dx}f(g(x))$ must include the factor $g'(x)$ .
Treating $\int_a^b f(x)\,dx$ as total area every time is wrong because the definite integral gives signed area, so regions below the $x$ -axis count as negative.
Confusing position, velocity, and acceleration is wrong because if $s(t)$ is position, then $v(t)=s'(t)$ and $a(t)=v'(t)=s''(t)$ .

Practice Questions

1 Find $\lim_{x\to 2}\frac{x^2-4}{x-2}$ .
2 Differentiate $y=(3x^2-1)^5$ .
3 Evaluate $\int_0^3 (2x+1)\,dx$ .
4 Explain why a function can have a local maximum at a point where $f'(x)$ does not exist.

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