AP Calculus Formula Sheet

Every formula for the AP Calculus AB and BC exams on one page, grouped by unit. Free to use, print and share with your class.

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AP Calculus AB & BC Formula Sheet

learnappu.com/math/calculus/formula-sheet

You won't get a formula sheet on exam day

The AP Calculus AB and BC exams don't provide one, so everything below needs to be memorized. Sections marked BC only are not tested on AB.

Limits and continuity

Unit 1AB & BC

Special trig limit

x \to 0 lim \frac{sin x}{x} = 1

Special trig limit

x \to 0 lim \frac{1 - cos x}{x} = 0

Definition of e

n \to \infty lim (1 + \frac{1}{n})^{n} = e

Continuity at x = a

f (a) exists, x \to a lim f (x) exists, x \to a lim f (x) = f (a)

All three conditions must hold.

Intermediate Value Theorem

f continuous on [a, b], k between f (a) and f (b) \Rightarrow f (c) = k for some c \in [a, b]

Derivative rules

Units 2–3AB & BC

Definition

Limit definition

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}

Derivative at a point

f^{'} (a) = x \to a lim \frac{f ( x ) - f ( a )}{x - a}

Rules

Power rule

\frac{d}{d x} [x^{n}] = n x^{n - 1}

Product rule

\frac{d}{d x} [f g] = f^{'} g + f g^{'}

Quotient rule

\frac{d}{d x} [\frac{f}{g}] = \frac{f ^{'} g - f g ^{'}}{g ^{2}}

Chain rule

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) g^{'} (x)

Inverse function

(f^{- 1})^{'} (a) = \frac{1}{f ^{'} ( f ^{- 1} ( a ) )}

Implicit differentiation

\frac{d}{d x} [y^{n}] = n y^{n - 1} \frac{d y}{d x}

Differentiate both sides, then solve for dy/dx.

Derivatives to memorize

Sine

\frac{d}{d x} sin x = cos x

Cosine

\frac{d}{d x} cos x = - sin x

Tangent

\frac{d}{d x} tan x = sec^{2} x

Cotangent

\frac{d}{d x} cot x = - csc^{2} x

Secant

\frac{d}{d x} sec x = sec x tan x

Cosecant

\frac{d}{d x} csc x = - csc x cot x

Natural exponential

\frac{d}{d x} e^{x} = e^{x}

Exponential, base a

\frac{d}{d x} a^{x} = a^{x} ln a

Natural log

\frac{d}{d x} ln x = \frac{1}{x}

Log, base a

\frac{d}{d x} lo g_{a} x = \frac{1}{x ln a}

Inverse sine

\frac{d}{d x} arcsin x = \frac{1}{1 - x ^{2}}

Inverse cosine

\frac{d}{d x} arccos x = - \frac{1}{1 - x ^{2}}

Inverse tangent

\frac{d}{d x} arctan x = \frac{1}{1 + x ^{2}}

Applications of derivatives

Units 4–5AB & BC

Tangent line / linearization

y = f (a) + f^{'} (a) (x - a)

L'Hôpital's rule

x \to a lim \frac{f ( x )}{g ( x )} = x \to a lim \frac{f ^{'} ( x )}{g ^{'} ( x )}

Only for the indeterminate forms 0/0 or ∞/∞, and only if the limit on the right exists.

Mean Value Theorem

f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a} for some c \in (a, b)

Requires f continuous on [a, b] and differentiable on (a, b).

Rolle's theorem

f (a) = f (b) \Rightarrow f^{'} (c) = 0 for some c \in (a, b)

Same conditions as the Mean Value Theorem.

Critical point

f^{'} (c) = 0 or f^{'} (c) undefined

c must be in the domain of f.

First derivative test

f^{'} changes + \to - \Rightarrow rel. max; - \to + \Rightarrow rel. min

Second derivative test

f^{'} (c) = 0, f^{''} (c) < 0 \Rightarrow rel. max; f^{''} (c) > 0 \Rightarrow rel. min

If f''(c) = 0, the test is inconclusive.

Concavity and inflection

f^{''} > 0 concave up, f^{''} < 0 concave down

Inflection point: a point on the graph where f'' changes sign.

Extreme Value Theorem

f continuous on [a, b] \Rightarrow f has an absolute max and min on [a, b]

Check critical points and both endpoints (the candidates test).

Related rates

Method

\frac{d V}{d t} = \frac{d V}{d r} \cdot \frac{d r}{d t}

Write an equation linking the quantities, differentiate both sides with respect to t, then substitute the known values.

Motion along a line

Velocity and acceleration

v (t) = x^{'} (t), a (t) = v^{'} (t) = x^{''} (t)

Speed

speed = ∣ v (t) ∣

Speeding up / slowing down

v and a same sign \Rightarrow speeding up

Opposite signs means slowing down.

Integrals

Unit 6AB & BC

Fundamental Theorem of Calculus

FTC (evaluation)

\int_{a}^{b} f (x) d x = F (b) - F (a), F^{'} = f

FTC (derivative of an integral)

\frac{d}{d x} \int_{a}^{x} f (t) d t = f (x)

For f continuous.

FTC with the chain rule

\frac{d}{d x} \int_{a}^{g (x)} f (t) d t = f (g (x)) g^{'} (x)

Net change

f (b) = f (a) + \int_{a}^{b} f^{'} (x) d x

Properties

Reversing limits

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x

Adding intervals

\int_{a}^{b} f d x + \int_{b}^{c} f d x = \int_{a}^{c} f d x

Zero-width interval

\int_{a}^{a} f (x) d x = 0

Antiderivatives to memorize

Power

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C, n \neq = - 1

Reciprocal

\int \frac{1}{x} d x = ln ∣ x ∣ + C

Natural exponential

\int e^{x} d x = e^{x} + C

Exponential, base a

\int a^{x} d x = \frac{a ^{x}}{ln a} + C

Sine

\int sin x d x = - cos x + C

Cosine

\int cos x d x = sin x + C

Secant squared

\int sec^{2} x d x = tan x + C

Cosecant squared

\int csc^{2} x d x = - cot x + C

Secant tangent

\int sec x tan x d x = sec x + C

Cosecant cotangent

\int csc x cot x d x = - csc x + C

Tangent

\int tan x d x = - ln ∣ cos x ∣ + C

Cotangent

\int cot x d x = ln ∣ sin x ∣ + C

Secant

\int sec x d x = ln ∣ sec x + tan x ∣ + C

Cosecant

\int csc x d x = - ln ∣ csc x + cot x ∣ + C

Inverse sine form

\int \frac{d x}{a ^{2} - x ^{2}} = arcsin \frac{x}{a} + C, a > 0

Inverse tangent form

\int \frac{d x}{a ^{2} + x ^{2}} = \frac{1}{a} arctan \frac{x}{a} + C

Techniques and approximations

u-substitution

\int f (g (x)) g^{'} (x) d x = \int f (u) d u, u = g (x)

For definite integrals, change the limits to u-values too.

Left Riemann sum

i = 0 \sum n - 1 f (x_{i}) Δ x, Δ x = \frac{b - a}{n}

Here xᵢ = a + iΔx.

Right Riemann sum

i = 1 \sum n f (x_{i}) Δ x

Midpoint Riemann sum

i = 1 \sum n f (\frac{x _{i - 1} + x _{i}}{2}) Δ x

Trapezoidal rule

\frac{Δ x}{2} [f (x_{0}) + 2 f (x_{1}) + \dots + 2 f (x_{n - 1}) + f (x_{n})]

Definite integral as a limit

\int_{a}^{b} f (x) d x = n \to \infty lim i = 1 \sum n f (x_{i}) Δ x

Differential equations

Unit 7AB & BC

Separation of variables

\frac{d y}{d x} = f (x) g (y) \Rightarrow \int \frac{d y}{g ( y )} = \int f (x) d x

Add + C, then use the initial condition to find it.

Exponential growth and decay

\frac{d y}{d t} = k y \Rightarrow y = y_{0} e^{k t}

Applications of integrals

Unit 8AB & BC

Average value

f_{avg} = \frac{1}{b - a} \int_{a}^{b} f (x) d x

Displacement

\int_{a}^{b} v (t) d t

Total distance traveled

\int_{a}^{b} ∣ v (t) ∣ d t

Position from velocity

x (t) = x (a) + \int_{a}^{t} v (s) d s

Area between curves

A = \int_{a}^{b} [f (x) - g (x)] d x, f \geq g

Volume: disc method

V = π \int_{a}^{b} [R (x)]^{2} d x

Volume: washer method

V = π \int_{a}^{b} ([R (x)]^{2} - [r (x)]^{2}) d x

R = outer radius, r = inner radius, both measured from the axis of rotation.

Rotating around y = k

R (x) = ∣ f (x) - k ∣

For rotation around a vertical line or the y-axis, write everything in terms of y and integrate dy.

Volume: known cross-sections

V = \int_{a}^{b} A (x) d x

Square: A = s². Semicircle: A = (π/8)s², with s = diameter. Equilateral triangle: A = (√3/4)s².

Advanced integration and differential equations

Units 6–8BC only

Integration by parts

\int u d v = uv - \int v d u

Partial fractions (distinct linear factors)

\frac{p x + q}{( x - a ) ( x - b )} = \frac{A}{x - a} + \frac{B}{x - b}

Improper integral (infinite limit)

\int_{a}^{\infty} f (x) d x = b \to \infty lim \int_{a}^{b} f (x) d x

Converges if the limit exists and is finite.

Improper integral (unbounded integrand)

\int_{a}^{b} f (x) d x = c \to a^{+} lim \int_{c}^{b} f (x) d x

When f is unbounded at x = a. Example: the integral of 1/√x from 0 to 1 equals 2.

Euler's method

y_{n + 1} = y_{n} + h \cdot \frac{d y}{d x}_{(x_{n}, y_{n})}, x_{n + 1} = x_{n} + h

Logistic growth

\frac{d P}{d t} = k P (1 - \frac{P}{L})

L is the carrying capacity. P grows fastest when P = L/2.

Arc length of y = f(x)

L = \int_{a}^{b} 1 + [f^{'} (x)]^{2} d x

Parametric, polar and vector functions

Unit 9BC only

Parametric

Slope

\frac{d y}{d x} = \frac{d y / d t}{d x / d t}, \frac{d x}{d t} \neq = 0

Second derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{\frac{d}{d t} ( \frac{d y}{d x} )}{d x / d t}

Arc length

L = \int_{a}^{b} (\frac{d x}{d t})^{2} + (\frac{d y}{d t})^{2} d t

Vector-valued motion

Velocity and acceleration

v (t) = ⟨ x^{'} (t), y^{'} (t)⟩, a (t) = ⟨ x^{''} (t), y^{''} (t)⟩

Speed

∥ v (t) ∥ = [x^{'} (t)]^{2} + [y^{'} (t)]^{2}

Distance traveled

\int_{a}^{b} ∥ v (t) ∥ d t

Polar

Conversion

x = r cos θ, y = r sin θ, r^{2} = x^{2} + y^{2}, tan θ = \frac{y}{x}

Slope

\frac{d y}{d x} = \frac{\frac{d r}{d θ} sin θ + r cos θ}{\frac{d r}{d θ} cos θ - r sin θ}

Area inside r = f(θ)

A = \frac{1}{2} \int_{α}^{β} r^{2} d θ

Area between two polar curves

A = \frac{1}{2} \int_{α}^{β} (r_{outer}^{2} - r_{inner}^{2}) d θ

Infinite sequences and series

Unit 10BC only

Convergence tests

Geometric series

n = 0 \sum \infty a r^{n} = \frac{a}{1 - r}, ∣ r ∣ < 1

Diverges if |r| ≥ 1.

nth-term test for divergence

n \to \infty lim a_{n} \neq = 0 \Rightarrow \sum a_{n} diverges

If the limit is 0, this test tells you nothing.

Harmonic series

n = 1 \sum \infty \frac{1}{n} diverges, n = 1 \sum \infty \frac{( - 1 ) ^{n + 1}}{n} converges

p-series

n = 1 \sum \infty \frac{1}{n ^{p}} converges if p > 1, diverges if p \leq 1

Integral test

\sum a_{n} and \int_{1}^{\infty} f (x) d x both converge or both diverge

f positive, continuous and decreasing, with f(n) = aₙ.

Comparison test

0 \leq a_{n} \leq b_{n} : \sum b_{n} conv. \Rightarrow \sum a_{n} conv.; \sum a_{n} div. \Rightarrow \sum b_{n} div.

Limit comparison test

n \to \infty lim \frac{a _{n}}{b _{n}} = c, 0 < c < \infty \Rightarrow both converge or both diverge

For aₙ, bₙ > 0.

Ratio test

L = n \to \infty lim \frac{a _{n + 1}}{a _{n}} : L < 1 converges absolutely, L > 1 diverges

L = 1 is inconclusive.

Alternating series test

\sum (- 1)^{n} b_{n} converges if b_{n} > 0 decreases and n \to \infty lim b_{n} = 0

Absolute vs conditional convergence

\sum ∣ a_{n} ∣ converges \Rightarrow \sum a_{n} converges absolutely

If Σaₙ converges but Σ|aₙ| diverges, the convergence is conditional.

Error bounds

Alternating series error

∣ S - S_{n} ∣ \leq b_{n + 1}

Sₙ is the sum through the nth term; the error is at most the first omitted term.

Lagrange error bound

∣ R_{n} (x) ∣ \leq \frac{M}{( n + 1 )!} ∣ x - a ∣^{n + 1}

M is the maximum of |f⁽ⁿ⁺¹⁾| between a and x.

Taylor series

Taylor series centered at a

f (x) = n = 0 \sum \infty \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}

A Maclaurin series is a Taylor series centered at a = 0.

nth-degree Taylor polynomial

P_{n} (x) = f (a) + f^{'} (a) (x - a) + \frac{f ^{''} ( a )}{2 !} (x - a)^{2} + \dots + \frac{f ^{(n)} ( a )}{n !} (x - a)^{n}

Radius and interval of convergence

n \to \infty lim \frac{a _{n + 1}}{a _{n}} < 1 \Rightarrow ∣ x - a ∣ < R

Use the ratio test to find R, then test each endpoint separately.

eˣ (all x)

e^{x} = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \dots = n = 0 \sum \infty \frac{x ^{n}}{n !}

sin x (all x)

sin x = x - \frac{x ^{3}}{3 !} + \frac{x ^{5}}{5 !} - \dots = n = 0 \sum \infty \frac{( - 1 ) ^{n} x ^{2 n + 1}}{( 2 n + 1 )!}

cos x (all x)

cos x = 1 - \frac{x ^{2}}{2 !} + \frac{x ^{4}}{4 !} - \dots = n = 0 \sum \infty \frac{( - 1 ) ^{n} x ^{2 n}}{( 2 n )!}

1 / (1 − x)

\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + \dots = n = 0 \sum \infty x^{n}, ∣ x ∣ < 1

ln(1 + x)

ln (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \dots = n = 1 \sum \infty \frac{( - 1 ) ^{n + 1} x ^{n}}{n}, - 1 < x \leq 1

Trig identities you'll still need

PrerequisiteAB & BC

Pythagorean

sin^{2} x + cos^{2} x = 1, 1 + tan^{2} x = sec^{2} x, 1 + cot^{2} x = csc^{2} x

Double angle (sine)

sin 2 x = 2 sin x cos x

Double angle (cosine)

cos 2 x = cos^{2} x - sin^{2} x = 2 cos^{2} x - 1 = 1 - 2 sin^{2} x

Power reducing

sin^{2} x = \frac{1 - cos 2 x}{2}, cos^{2} x = \frac{1 + cos 2 x}{2}

See the formulas in action

Free lessons with worked examples, no account needed.

LimitsHow to evaluate limits, including L'Hôpital's rule DerivativesEvery derivative rule with step-by-step examples IntegralsAntiderivatives, u-substitution and the FTC

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Frequently asked questions

Is there a formula sheet on the AP Calculus exam?

No. Unlike AP Physics, AP Chemistry and AP Statistics, the AP Calculus AB and BC exams do not give you a formula sheet. Every formula on this page has to be memorized, which is why it's worth practicing with them until using them is automatic.

Which formulas do I need for AP Calculus AB?

Everything on this page not marked BC only: limits and continuity, the derivative rules and basic derivatives, applications of derivatives, the Fundamental Theorem of Calculus, basic antiderivatives, u-substitution, Riemann sums, separable differential equations, and area and volume.

What extra formulas are on AP Calculus BC?

BC adds integration by parts, partial fractions, improper integrals, Euler's method, logistic growth, arc length, parametric, polar and vector-valued functions, and the whole of Unit 10: series convergence tests, error bounds and Taylor and Maclaurin series.

How do I get this as a PDF?

Use the Print or save as PDF button, then choose Save as PDF as the destination in the print window. The page is formatted to print cleanly without the site navigation.

What's the best way to memorize calculus formulas?

Use them rather than reread them. Cover the right-hand side of each formula and write it from memory, then do a few practice problems that need it. Formulas you derive or apply a few times stick far better than ones you only read, especially the derivative and antiderivative pairs, which you can learn together.

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