AP Calculus AB Formula Sheet Cheat Sheet

A printable reference covering limits, continuity, derivatives, integrals, the Fundamental Theorem of Calculus, and AP Calculus AB applications for grades 11-12.

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This AP Calculus AB formula sheet covers the core tools students use to analyze change, motion, area, and accumulation. It brings together limits, continuity, derivative rules, graph interpretation, integrals, and major theorems in one organized reference. Students need this cheat sheet to review formulas quickly and connect procedures to AP-style reasoning. It is especially useful when preparing for free-response questions that require both computation and explanation. The main ideas are built around limits, rates of change, and accumulated change. Limits support continuity, derivatives describe instantaneous rates and slopes, and integrals measure net accumulation or signed area. The Fundamental Theorem of Calculus connects derivatives and integrals through functions such as $F(x)=\int_a^x f(t)\,dt$ . Many AP Calculus AB problems combine formulas with interpretation, units, and justification.

Key Facts

The limit definition of the derivative is $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ .
A function $f$ is continuous at $x=a$ when $\lim_{x\to a}f(x)=f(a)$ and both sides of the equality exist.
The power rule is $\frac{d}{dx}\left(x^n\right)=nx^{n-1}$ for real values of $n$ where the derivative is defined.
The product rule is $\frac{d}{dx}\left(uv\right)=u'v+uv'$ and the quotient rule is $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^2}$ .
The chain rule is $\frac{d}{dx}\left[f(g(x))\right]=f'(g(x))g'(x)$ .
A definite integral gives net accumulation, so $\int_a^b f(x)\,dx$ represents signed area or total change when $f$ is a rate.
The Fundamental Theorem of Calculus states that if $F'(x)=f(x)$ , then $\int_a^b f(x)\,dx=F(b)-F(a)$ .
If $A(x)=\int_a^x f(t)\,dt$ , then $A'(x)=f(x)$ wherever $f$ is continuous.

Vocabulary

Limit: A limit describes the value a function approaches as the input approaches a specified number.
Continuity: Continuity at $x=a$ means the function value exists, the limit exists, and $\lim_{x\to a}f(x)=f(a)$ .
Derivative: A derivative measures the instantaneous rate of change of a function and the slope of its tangent line.
Critical Point: A critical point occurs where $f'(x)=0$ or where $f'(x)$ is undefined, provided $f(x)$ is defined.
Definite Integral: A definite integral $\int_a^b f(x)\,dx$ measures signed area or net accumulated change over $[a,b]$ .
Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus connects differentiation and integration by stating that accumulation functions have derivatives related to their integrands.

Common Mistakes to Avoid

Forgetting the chain rule, such as differentiating $\sin(3x)$ as $\cos(3x)$ , is wrong because the inner derivative must be included, giving $3\cos(3x)$ .
Treating $\int_a^b f(x)\,dx$ as total area is wrong when $f(x)$ is below the $x$ -axis because the definite integral gives signed area.
Using a derivative test without checking intervals is wrong because the sign of $f'(x)$ must be analyzed on both sides of a critical point to justify increasing, decreasing, or extrema.
Assuming a function is continuous because it is defined at $x=a$ is wrong because continuity also requires $\lim_{x\to a}f(x)=f(a)$ .
Dropping the constant in an indefinite integral, such as writing $\int 2x\,dx=x^2$ , is incomplete because the full family of antiderivatives is $x^2+C$ .

Practice Questions

1 Find $\frac{d}{dx}\left(x^3\sin x\right)$ .
2 Evaluate $\int_1^4 2x\,dx$ .
3 If $A(x)=\int_2^x \sqrt{t^2+1}\,dt$ , find $A'(3)$ .
4 Explain how the sign of $f'(x)$ and the sign of $f''(x)$ describe the shape and behavior of the graph of $f$ .

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