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AP Calculus AB Core Formulas & Theorems

Essential limits, derivative rules, integral formulas, and fundamental theorems for AP Calculus AB.

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Definition of the Derivative

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Definition of the Derivative

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

Product Rule

\frac{d}{d x} [u \cdot v] = u^{'} v + u v^{'}

Quotient Rule

\frac{d}{d x} [\frac{u}{v}] = \frac{u ^{'} v - u v ^{'}}{v ^{2}}

Chain Rule

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

Derivative of

sin (x)

and

cos (x)

\frac{d}{d x} sin (x) = cos (x), \frac{d}{d x} cos (x) = - sin (x)

Derivative of

tan (x)

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

Derivative of

sec (x)

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

Derivative of

ln (x)

\frac{d}{d x} ln (x) = \frac{1}{x} (x > 0)

Derivative of

e^{x}

and

a^{x}

\frac{d}{d x} e^{x} = e^{x}, \frac{d}{d x} a^{x} = a^{x} ln (a)

Derivative of

arctan (x)

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

Derivative of

arcsin (x)

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

Intermediate Value Theorem (IVT)

f

is continuous on

[a, b]

and

u

is between

f (a)

and

f (b)

, there exists at least one

c \in (a, b)

such that

f (c) = u

Mean Value Theorem (MVT)

f

is continuous on

[a, b]

and differentiable on

(a, b)

, then there exists

c \in (a, b)

such that $

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

Rolle's Theorem

f

is continuous on

[a, b]

, differentiable on

(a, b)

, and

f (a) = f (b)

, there exists at least one

c \in (a, b)

such that

f^{'} (c) = 0

Fundamental Theorem of Calculus, Part 1

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

Fundamental Theorem of Calculus, Part 2

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

Average Value of a Function on

[a, b]

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

L'Hôpital's Rule

lim \frac{f ( x )}{g ( x )}

yields indeterminate

\frac{0}{0}

\frac{\pm \infty}{\pm \infty}

, then $

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

Trapezoidal Rule for

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

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

Power Rule for Integration

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

Integral of

\frac{1}{x}

\int \frac{1}{x} d x = ln ∣ x ∣ + C

Volume of Revolution (Disk Method)

V = π \int_{a}^{b} [R (x)]^{2} d x

Volume of Revolution (Washer Method)

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

Inflection Point Condition

Occurs where

f^{''} (x)

changes sign (from positive to negative or vice versa) and

f

is continuous.

Position, Velocity, and Acceleration

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

S p ee d i s

|v(t)|$.

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