Calculus AB
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AP Calculus AB — Rules & Theorems

7 sections · 41 entries · print it and keep it beside your practice sets

AP Calculus does NOT give you a formula sheet — this is the one reference page you have to carry in your head. Treat it as a memorisation checklist rather than an exam-day crutch: cover the right column and recite.

Practise with the sheet, not from memory. The College Board hands out its own version of this page on exam day, so nothing here is worth memorising for its own sake. What earns points is speed: knowing which section a quantity lives in, and reading off the right line without breaking your train of thought. Keep this open (or printed) for every practice set you do.

Limits & continuity

Definition of derivative
f′(a) = lim_(h→0) ( f(a + h) − f(a) ) / h

The limit that defines f′(a)

Alternate form
f′(a) = lim_(x→a) ( f(x) − f(a) ) / (x − a)

Useful when the point is given

Continuity at a
f(a) exists, lim_(x→a) f(x) exists, and lim_(x→a) f(x) = f(a)

All three conditions must hold

L’Hôpital
lim f/g = lim f′/g′

Only for 0/0 or ±∞/±∞ indeterminate forms

Squeeze / special limits
lim_(x→0) (sin x)/x = 1, lim_(x→0) (1 − cos x)/x = 0

Worth knowing by heart

IVT
If f is continuous on [a,b] and k is between f(a) and f(b), then f(c) = k for some c in (a,b).

Intermediate Value Theorem

Differentiation rules

Power
d/dx (xⁿ) = n x^(n−1)

Any real n

Product
(f g)′ = f′ g + f g′

Derivative of a product

Quotient
(f/g)′ = ( f′ g − f g′ ) / g²

Derivative of a quotient

Chain
d/dx f(g(x)) = f′(g(x)) · g′(x)

Composite functions

Inverse function
(f⁻¹)′(a) = 1 / f′( f⁻¹(a) )

Derivative of an inverse at a point

Implicit
d/dx [y²] = 2y (dy/dx)

Differentiate both sides in x, treating y as y(x)

Derivatives to know

Trig
sin x → cos x; cos x → −sin x; tan x → sec²x; cot x → −csc²x; sec x → sec x tan x; csc x → −csc x cot x

The six circular functions

Exponential
d/dx eˣ = eˣ; d/dx aˣ = aˣ ln a

Base e and base a

Logarithmic
d/dx ln x = 1/x; d/dx log_a x = 1/(x ln a)

Natural and general log

Inverse trig
d/dx arcsin x = 1/√(1 − x²); d/dx arccos x = −1/√(1 − x²); d/dx arctan x = 1/(1 + x²)

Three you need

Applications of the derivative

Tangent line
L(x) = f(a) + f′(a)(x − a)

Linearization of f at a

MVT
f′(c) = ( f(b) − f(a) ) / (b − a) for some c in (a,b)

Mean Value Theorem: f continuous on [a,b], differentiable on (a,b)

Extrema
f′ = 0 or undefined ⇒ candidate; f′ changes + to − ⇒ local max

Critical points and the first-derivative test

Concavity
f″ > 0 ⇒ concave up; f″ < 0 ⇒ concave down; f″ changes sign ⇒ inflection point

Second-derivative test

Motion
v(t) = x′(t), a(t) = v′(t); speed = |v(t)|

Position, velocity, acceleration

Speed increases when v and a have the SAME sign. Total distance = ∫|v| dt; displacement = ∫v dt.

Related rates
e.g. V = (4/3)πr³ ⇒ dV/dt = 4πr² (dr/dt)

Differentiate a geometric relationship with respect to t

Integration rules

Power
∫ xⁿ dx = x^(n+1)/(n + 1) + C

For n ≠ −1

Reciprocal
∫ (1/x) dx = ln|x| + C

The n = −1 case

Exponential
∫ eˣ dx = eˣ + C; ∫ aˣ dx = aˣ/ln a + C

Base e and base a

Trig
∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C; ∫ sec²x dx = tan x + C; ∫ sec x tan x dx = sec x + C

Basic antiderivatives

Inverse trig forms
∫ dx/(1 + x²) = arctan x + C; ∫ dx/√(1 − x²) = arcsin x + C

Pattern-match these

u-substitution
∫ f(g(x)) g′(x) dx = ∫ f(u) du, u = g(x)

Reverse chain rule

On a definite integral, convert the limits to u as well.

The Fundamental Theorem & accumulation

FTC part 1
d/dx ∫_a^x f(t) dt = f(x)

Derivative of an accumulation function

With a variable upper limit u(x): d/dx ∫_a^(u(x)) f(t) dt = f(u(x)) · u′(x).

FTC part 2
∫_a^b f′(x) dx = f(b) − f(a)

Evaluating a definite integral

Accumulation
f(b) = f(a) + ∫_a^b f′(x) dx

Net change from a rate

Average value
f_avg = (1/(b − a)) ∫_a^b f(x) dx

Mean value of f on [a,b]

Do not confuse with average rate of change, ( f(b) − f(a) )/(b − a).

Riemann sums

Left, right, and midpoint approximations with n subintervals

Trapezoidal rule
T = (Δx/2)[ y₀ + 2y₁ + 2y₂ + … + 2y_(n−1) + y_n ]

Equal subintervals of width Δx

Overestimates when the graph is concave up, underestimates when concave down.

Areas, volumes & differential equations

Area between curves
A = ∫_a^b [ f(x) − g(x) ] dx

Top minus bottom

Disk method
V = π ∫_a^b [ R(x) ]² dx

Revolving about a horizontal axis

Washer method
V = π ∫_a^b ( [R(x)]² − [r(x)]² ) dx

With a hole

Known cross-sections
V = ∫_a^b A(x) dx

A(x) is the area of the slice

Squares: A = s². Equilateral triangles: A = (√3/4)s². Semicircles: A = (π/8)s² with s the diameter.

Separable ODE
dy/dx = g(x)h(y) ⇒ ∫ dy/h(y) = ∫ g(x) dx

Separate and integrate

Exponential model
y = y₀ e^(k t)

Solution of dy/dt = k y

Slope fields

Plot short segments of slope dy/dx at grid points to sketch solution curves

Practise with the sheet openCheatsheet