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The AP® Calculus AB formula sheet
The exam hands you nothing: every formula below must live in your head by May. Print this page and drill it until it does.
Limits
Special trig limits
$\lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad \lim_{x \to 0} \frac{1 - \cos x}{x} = 0$Continuity at x = a
$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$Rational limits at infinity
Compare degrees: bottom bigger $\to 0$; equal $\to$ ratio of leading coefficients; top bigger $\to \pm\infty$L'Hospital's Rule (for 0/0 or ∞/∞)
$\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$Derivatives: definitions & rules
Definition
$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$At a point
$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$Power rule
$\frac{d}{dx} x^n = n x^{n-1}$Product rule
$(uv)' = u'v + uv'$Quotient rule
$\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$Chain rule
$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$Inverse function derivative
$(f^{-1})'(b) = \frac{1}{f'(f^{-1}(b))}$Implicit differentiation
Differentiate both sides in $x$; every $y$ term picks up $\frac{dy}{dx}$ by the chain ruleDerivatives to memorize
$\frac{d}{dx} \sin x = \cos x \qquad \frac{d}{dx} \cos x = -\sin x$
$\frac{d}{dx} \tan x = \sec^2 x \qquad \frac{d}{dx} \cot x = -\csc^2 x$
$\frac{d}{dx} \sec x = \sec x \tan x \qquad \frac{d}{dx} \csc x = -\csc x \cot x$
$\frac{d}{dx} e^x = e^x \qquad \frac{d}{dx} a^x = a^x \ln a$
$\frac{d}{dx} \ln x = \frac{1}{x} \qquad \frac{d}{dx} \log_a x = \frac{1}{x \ln a}$
$\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \qquad \frac{d}{dx} \arccos x = \frac{-1}{\sqrt{1 - x^2}}$
$\frac{d}{dx} \arctan x = \frac{1}{1 + x^2}$
Key theorems
Intermediate Value Theorem
If $f$ is continuous on $[a,b]$, it takes every value between $f(a)$ and $f(b)$Mean Value Theorem
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$: $f'(c) = \frac{f(b) - f(a)}{b - a}$ for some $c$ in $(a,b)$Extreme Value Theorem
A continuous function on a closed interval $[a,b]$ attains an absolute max and minCandidates test
Absolute extrema occur only at critical points or endpointsIntegrals & the Fundamental Theorem
FTC part 1
$\frac{d}{dx} \int_a^x f(t)\,dt = f(x)$FTC part 2
$\int_a^b f(x)\,dx = F(b) - F(a)$Power rule for integrals
$\int x^n dx = \frac{x^{n+1}}{n+1} + C \ (n \neq -1)$$\int \frac{1}{x}\,dx = \ln|x| + C \qquad \int e^x dx = e^x + C$
$\int \sin x\,dx = -\cos x + C \qquad \int \cos x\,dx = \sin x + C$
$\int \sec^2 x\,dx = \tan x + C \qquad \int \frac{dx}{1+x^2} = \arctan x + C$
u-substitution
$\int f(g(x)) g'(x)\,dx = \int f(u)\,du$Average value
$f_{avg} = \frac{1}{b-a} \int_a^b f(x)\,dx$Applications
Position, velocity, acceleration
$v(t) = s'(t) \quad a(t) = v'(t)$; speed $= |v(t)|$; speed increases when $v$ and $a$ share a signDisplacement vs distance
Displacement $= \int_a^b v(t)\,dt$; distance $= \int_a^b |v(t)|\,dt$Area between curves
$A = \int_a^b (\text{top} - \text{bottom})\,dx$Disk method
$V = \pi \int_a^b [r(x)]^2 dx$Washer method
$V = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right) dx$Known cross sections
$V = \int_a^b A(x)\,dx$ where $A(x)$ is the cross-sectional areaExponential growth/decay
$\frac{dy}{dt} = ky \implies y = y_0 e^{kt}$Separable differential equations
Separate variables, integrate both sides, solve for the constant with the initial conditionSend this sheet to your inbox
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