FORMULAS

BASIC ARITHMETIC

$\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a + b}{c}$

$\dfrac{a}{b} \times \dfrac{x}{y} = \dfrac{ax}{by}$

$\dfrac{a}{b} \div \dfrac{x}{y} = \dfrac{ay}{bx}$

$n(x - y) = nx - ny$

$|-x| = x$

$\dfrac{ax + ay}{a} = \dfrac{x + y}{1}$

$a\left(\dfrac{x}{y}\right) = \dfrac{ax}{y}$

$\dfrac{a}{\left(\dfrac{x}{y}\right)} = \dfrac{ay}{x}$

Square Root Rule

$\begin{align} x^2 & = a \\ x & = \pm \sqrt{a} \\ \\ \end{align}$

LINEAR & GEOMETRY

$Slope = \dfrac{\Delta\ rise}{\Delta\ run} = \dfrac{y_2 - y_1}{x_2 - x_1}$

Slope, y-intercept

$ƒ(x) = y = mx + b$

m = slope
b = y-intercept Point, slope

$y_2 - y_1 = m(x_2 - x_1)$

Midpoint

$\left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right)$

Distance

$d = \sqrt{\left(x_2 - x_1\right)^2 + \left(y_2 - y_1\right)^2}$

Circle

$(x - h)^2 + (y - k)^2 = r^2$

r = radius
(h, k) center

QUADRATIC FUNCTIONS AND RELATIONS

Standard Form

$y = ax^2 + bx + c$

vertex at $\color{gray}{\dfrac{-b}{2a}}$ Factored Form

$y = a(x - r)(x - s)$

x = r, s Vertex Form

$y = a(x - h)^2 + k$

a > 0 = opens up
a < 0 = opens down
vertex = (h, k) Factoring

$x^2 + (a + b)x + ab \\ \\ = (x + a)(x + b)$

$cnx^2 + (a + b)x + ab \\ \\ = (cx + \tfrac{a}{n})(nx + \tfrac{b}{c})$

Quadratic Formula

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Axis of Symmetry

$x = \dfrac{-b}{2a}$

Discriminant

$b^2 - 4ac$

positive = 2 x-intercepts
zero = 1 x-intercept
negative = 0 x-intercepts Perfect Squares

$(x + b)^2 = x^2 + 2bx + b^2$

$(x - b)^2 = x^2 - 2bx + b^2$

Difference of Squares

$x^2 - b^2 = (x + b)(x - b)$

$x^{2n} - b^{2n} = (x^n + b^n)(x^n - b^n)$

Complete the Square for Vertex Form

$y = a\left(x^2 + \dfrac{b}{a}x + \left(\dfrac{b}{2a}\right)^2 - \left(\dfrac{b}{2a}\right)^2\right) + c$

Complete the Square for Zeros

$\begin{align} ax^2 + bx + c & = 0 \\ \\ \dfrac{a}{a}x^2 + \dfrac{b}{a}x + \dfrac{c}{a} & = 0 \\ \\ \dfrac{a}{a}x^2 + \dfrac{b}{a}x & = -\dfrac{c}{a} \\ \\ \dfrac{a}{a}x^2 + \dfrac{b}{a}x + \left(\dfrac{b}{2a}\right)^2 & = -\dfrac{c}{a} + \left(\dfrac{b}{2a}\right)^2 \\ \\ \left(x + \dfrac{b}{2a}\right)^2 & = \dfrac{-c}{a} + \left(\dfrac{b}{2a}\right)^2 \\ \\ x + \dfrac{b}{2a} & = \pm \sqrt{\dfrac{-c}{a} + \left(\dfrac{b}{2a}\right)^2} \\ \\ x & = \pm \sqrt{ \dfrac{-c}{a} + \dfrac{b^2}{4a^2} } - \dfrac{b}{2a} \\ \\ x & = -\dfrac{b}{2a} \pm \sqrt{ \dfrac{b^2}{4a^2} - \dfrac{c}{a} } \\ \\ x & = -\dfrac{b}{2a} \pm \sqrt{ \dfrac{b^2}{4a^2} - \dfrac{c}{a}\left(\dfrac{4a}{4a}\right) } \\ \\ x & = -\dfrac{b}{2a} \pm \sqrt{ \dfrac{b^2}{4a^2} - \dfrac{4ac}{4a^2} } \\ \\ x & = -\dfrac{b}{2a} \pm \sqrt{ \dfrac{1}{4a^2}\left(b^2 - 4ac\right) } \\ \\ x & = -\dfrac{b}{2a} \pm \dfrac{1}{2a}\sqrt{b^2 - 4ac} \\ \\ x & = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \end{align}$

EXPONENT LAWS

$(ax)^0 = 0$

$(x^a)(x^b) = x^{a + b}$

$\dfrac{x^a}{x^b} = x^{a - b}$

$(x^a)^b = x^{ab}$

$\left(\dfrac{ab}{cd}\right)^x = \dfrac{a^xb^x}{c^xd^x}$

$\left(x^{\tfrac{1}{b}}\right)^a = x^{\tfrac{a}{b}}$

$x^{-a} = \dfrac{1}{x^{+a}}$

$\dfrac{1}{x^{-a}} = x^{+a}$

$\left(\dfrac{x}{y}\right)^{-a} = \left(\dfrac{y}{x}\right)^{+a}$

ROOT RADICAL LAWS

$\sqrt{a}\sqrt{b} = \sqrt{ab}$

$\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$

$\sqrt[b]{x^a} = x^{\tfrac{a}{b}}$

$\sqrt[b]{x^b} = x$

if b is odd

$\sqrt[b]{x^b} = \pm\ x$

if b is even

BASIC TRIGONOMETRY

$A˚ + B˚ + C˚ = 180˚$

$a^2 + b^2 = c^2$

SohCahToa

$\sinθ = \dfrac{opposite}{hypotenuse}$

$\cosθ = \dfrac{adjacent}{hypotenuse}$

$\tanθ = \dfrac{opposite}{adjacent}$

$\begin{align} \sinθ & = a \\ θ & = \sin^{-1}(a) \end{align}$

Sine Law

$\dfrac{\sin A˚}{a} = \dfrac{\sin B˚}{b} = \dfrac{\sin C˚}{c}$

$\dfrac{a}{\sin A˚} = \dfrac{b}{\sin B˚} = \dfrac{c}{\sin C˚}$

Cosine Law

$a^2 = b^2 + c^2 - 2bc·\cos A˚$

$b^2 = a^2 + c^2 - 2ac·\cos B˚$

$c^2 = a^2 + b^2 - 2ab·\cos C˚$

Angles Corresponding = 'F'
Interior Alternate = 'Z'
Exterior Alternate = 'X'
Consecutive Interior = 'C'

TRIGONOMETRY II

$degrees \xrightarrow{ \ × \dfrac{π}{180˚} \ } radians$

$radians \xrightarrow{ \ × \dfrac{180˚}{π} \ } degrees$

Reciprocal Identities

$\cscθ = \dfrac{1}{\sinθ} = \dfrac{hypotenuse}{opposite}$

$\secθ = \dfrac{1}{\cosθ} = \dfrac{hypotenuse}{adjacent}$

$\cotθ = \dfrac{1}{\tanθ} = \dfrac{adjacent}{opposite}$

Trig Functions

$y = a · \sin \Big(k (x - c)\Big) + d$

a = amplitude
-a = reflection across x-axis
|a| > 1 = vertical stretch
0 < |a| < 1 = vertical compression

k = period T = 2π/k, tan is π/k
-k = reflection across y-axis
|k| > 1 = horizontal compression
0 < |k| < 1 = horizontal stretch

c = phase shift = c/k
-c = horizontal phase shift left
+c = horizontal phase shift right

d = vertical shift
-d = vertical shift down
+d = vertical shift up

SPECIAL TRIANGLES

THE UNIT CIRCLE

Coordinate Form: (x, y)

$\sin(θ) = y$

$\cos(θ) = x$

$\tan(θ) = \dfrac{y}{x}$

TRIGONOMETRY III

Quotient Identities

$\tanθ = \dfrac{\sinθ}{\cosθ} \quad\quad\quad\quad \cotθ = \dfrac{\cosθ}{\sinθ}$

Pythagorean Identities

$\sin^2θ + \cos^2θ = 1$

$\sec^2θ - \tan^2θ = 1$

$\csc^2θ - \cot^2θ = 1$

Periodic

$\sin(θ + 2π ·n) = \sinθ$

$\cos(θ + 2π ·n) = \cosθ$

$\tan(θ + π ·n) = \tanθ$

Compound Angle (Addition, Subtraction)
(Sum, Difference)

$\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$

$\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$

$\tan(a \pm b) = \dfrac{\tan a \pm \tan b}{1 \mp \tan a \tan b}$

Cofunction Identities Q1

$\sin \left( \dfrac{π}{2} - θ \right) = \cosθ$

$\cos \left( \dfrac{π}{2} - θ \right) = \sinθ$

$\tan \left( \dfrac{π}{2} - θ \right) = \cotθ$

Reciprocal Cofunction

$\csc \left( \dfrac{π}{2} - θ \right) = \secθ$

$\sec \left( \dfrac{π}{2} - θ \right) = \cscθ$

$\cot \left( \dfrac{π}{2} - θ \right) = \tanθ$

$\sin \left( \dfrac{π}{2} + θ \right) = \cos θ$

$\cos \left( \dfrac{π}{2} + θ \right) = -\sin θ$

$\tan \left( \dfrac{π}{2} + θ \right) = -\cot θ$

$\sin \left( \dfrac{3π}{2} - θ \right) = -\cos θ$

$\cos \left( \dfrac{3π}{2} - θ \right) = -\sin θ$

$\tan \left( \dfrac{3π}{2} - θ \right) = \cot θ$

$\sin \left( \dfrac{3π}{2} + θ \right) = -\cos θ$

$\cos \left( \dfrac{3π}{2} + θ \right) = \sin θ$

$\tan \left( \dfrac{3π}{2} + θ \right) = -\cot θ$

Even/Odd

$\sin(-θ) = -\sinθ$

$\cos(-θ) = \cosθ$

$\tan(-θ) = -\tanθ$

Double Angle Formulas

$\sin(2x) = 2\sin x \cos x$

$\begin{align} \cos(2x) & = \cos^2x - \sin^2x \\ \\ & = 2\cos^2x - 1 \\ \\ & = 1 - 2\sin^2x \end{align}$

$\tan(2x) = \dfrac{2\tan x}{1 - \tan^2x}$

Half Angle Formulas

$\sin\left( \dfrac{x}{2} \right) = \pm \sqrt{\dfrac{1 - \cos\,x}{2}}$

$\cos\left( \dfrac{x}{2} \right) = \pm \sqrt{\dfrac{1 + \cos\,x}{2}}$

$\tan\left( \dfrac{x}{2} \right) = \pm \sqrt{\dfrac{1 - \cos\,x}{1 + \cos\,x}}$

Co-Related Angle Q1 → Q2

$\sin \left( π - θ \right) = \sinθ$

$\cos \left( π - θ \right) = -\cos θ$

$\tan \left( π - θ \right) = -\tan θ$

Co-Related Angle Q1 → Q3

$\sin \left( π + θ \right) = -\sin θ$

$\cos \left( π + θ \right) = -\cos θ$

$\tan \left( π + θ \right) = \tan θ$

Co-Related Angle Q1 → Q4

$\sin \left( 2π - θ \right) = -\sin θ$

$\cos \left( 2π - θ \right) = \cos θ$

$\tan \left( 2π - θ \right) = -\tan θ$

SEQUENCES & SERIES

Arithmetic

$t_n = a + d(n - 1)$

n = term number
t_n = term value
d = common difference
a = first term
S_n = sum

$S_n = \dfrac{n}{2}[2a + d(n - 1)]$

Geometric

$t_n = a\cdot r^{n-1}$

r = common ratio

$S_n = \dfrac{a(r^n - 1)}{r - 1}$

Geometric Sum to Infinity

$S_∞ = \dfrac{a}{1 - r} \quad\quad |r| < 1$

FINANCIAL MATH

FV = future value
PV = present value
R = initial amount (or PV)
t = time in years
i = rate / 100%
n = compounding periods per year
     Daily: n = 365
     Weekly: n = 52
     Bi-weekly: n = 26
     Monthly: n = 12
     Quarterly: n = 4
     Semi-annually: n = 2
     Annually: n = 1
Simple Interest

$I = R \left(i\right)(t)$

Compound Interest

$FV = R \left(1 + \dfrac{i}{n}\right)^{n × t}$

Annuity (Deposit)

$FV = \dfrac{R\left[\left(1 + \dfrac{i}{n}\right)^{n × t} - 1\right]}{\dfrac{i}{n}}$

Annuity (Withdrawal)

$PV = \dfrac{R\left[1 - \left(1 + \dfrac{i}{n}\right)^{-n × t}\right]}{\dfrac{i}{n}}$

COMPLEX NUMBERS

$i^2 = -1$

$i = \sqrt{-1}$

$\sqrt{-a} = i\sqrt{a}$

$ai + bi = (a + b)i$

$(i)(-i) = 1$

LOG LAWS

$\log_b (1) = 0$

$\log_m (m)^n = n$

$\ln (e^{an}) = an$

$(b)^{\log_b (a)} = a$

$(e)^{\ln (a)} = a$

$y = \log_b x \quad ↔ \quad x = b^{\ y}$

$\log(ab) = \log(a) + \log(b)$

$\log\left(\dfrac{a}{b}\right) = \log(a) - \log(b)$

$\log _m (n) = \dfrac{\log _b (n)}{\log _b (m)}$

$\log _a (b) = \dfrac{1}{\log _b (a)}$

$(m)^{\log _a (n)} = (n)^{\log _a (m)}$

POLYNOMIALS

Symmetry

Even: ƒ(x) = ƒ(-x)

Odd: ƒ(-x) = -ƒ(x)

Neither:
ƒ(x) ≠ ƒ(-x),
ƒ(-x) ≠ -ƒ(x)

Sum of Cubes

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Difference of Cubes

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Perfect Cubes (Think: Pascal's Triangle)

$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

COMBINATION AND COMPOSITION OF FUNCTIONS

Combinations

(ƒ + g)(x) = ƒ(x) + g(x)

(ƒ − g)(x) = ƒ(x) − g(x)

(ƒ × g)(x) = ƒ(x) × g(x)

$\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}$

Compositions

(ƒ ◦ g)(x) = ƒ(g(x))

(ƒ ◦ g ◦ h)(x) = ƒ(g(h(x)))

(ƒ ◦ ƒ^-1)(x) = (ƒ^-1 ◦ ƒ)(x)

(g^-1 ◦ ƒ^-1)(x) = (ƒ ◦ g)^-1(x)

CALCULUS

VECTORS, LINES, PLANES

Vector Addition, Resultant

$\text{If:} \ \ \vec{A} = \big< \ i, \ j \ \big>, \ \ \vec{B} = \big< \ x, \ y \ \big> \\ \\ \text{Then:} \ \ \vec{A}+\vec{B} \ = \ \big< \ i+x, \ j+y \ \big>$

Direction Vectors

$\text{For:} \ \ \vec{A} = \big< \ i, \ j, \ k \ \big>, \ \ \vec{B} = \big< \ x, \ y, \ z \ \big> \\ \\ \vec{AB} \ = \ \big\langle x-i, \ y-j, \ z-k \big\rangle \\ \\ \vec{BA} \ = \ \big\langle i-x, \ j-y, \ k-z \big\rangle \\ \\$

Cartesian 2D Line Equation

$Ax + By + C = 0$

'Normal vector' from Cartesian coefficients:

$\vec{d} = \big< \ A, \ B \ \big>$

(There is no Cartesian 3D Line Equation) Cartesian (Scalar) Plane Equation The Cartesian equation is a scalar.

$Ax + By + Cz + D = 0$

Planes are parallel when their normals are collinear
Planes are coincident if D₁ = D₂. Derive 'normal vector' from Cartesian coefficients:

$\vec{d} = \big< \ A, \ B, \ C \ \big>$

Vector Plane Equation

$(x, y, z) = (\text{PV}) + (\text{SM 1})\Big\langle\text{Vec 1}\Big\rangle + (\text{SM 2})\Big\langle\text{Vec 2}\Big\rangle$

(x, y, z) = a generic point on the plane
PV = position vector [any point on plane (in relation to origin)]
SM = scalar multiple
Vec 1 & Vec 2 = coplanar vectors Parametric Equations

$\text{For:} \ \ \vec{V} \ = \ (i, \ j, \ k) \ + \ t(f, \ g, \ h) \\ \\ x \ = \ i \ + \ t(f) \\ \\ y \ = \ j \ + \ t(g) \\ \\ z \ = \ k \ + \ t(h)$

Symmetric Equation From the parametric equations...

$\begin{align} t \ & = \ t \ = \ t \\ \\ \dfrac{x - i}{f} & = \dfrac{y - j}{g} = \dfrac{z - k}{h} \end{align}$

Magnitude

$\text{For:} \ \ \vec{A} = \big< \ i, \ j, \ k \ \big>, \ \ \vec{B} = \big< \ x, \ y, \ z \ \big> \\ \\ \text{Magnitude,} \ \ \Big|\,\vec{A}\,\Big| \ = \ \sqrt{i^2 \ + \ j^2 \ + \ k^2}$

Dot Product

$\vec{A} \ \bullet \ \vec{B} \ = \ \Big|\,\vec{A}\,\Big| \ \Big|\,\vec{B}\,\Big| \cos\,\theta$

Perpendicular: Dot Product = 0 Dot Product Example

$\text{For:} \ \ \vec{A} = \big< \ i, \ j , \ k \ \big>, \ \ \vec{B} = \big< \ x, \ y, \ z \ \big> \\ \\ \vec{A} \ \bullet \ \vec{B} \ = \ \Big|\,\vec{A}\,\Big| \ \Big|\,\vec{B}\,\Big| \cos\theta \\ \\ (i)(x) + (j)(y) + (k)(z) = \sqrt{i + j + k} \,\sqrt{x + y+ z}\,\cos\,\theta$

Dot Product Rule Conventions

$\vec{A} \ \bullet \ \vec{B} \ = \ \vec{B} \ \bullet \ \vec{A}$

$\vec{A} \ \bullet \ \vec{A} \ = \ \Big|\,\vec{A}\,\Big|^2$

$\vec{A} \ \bullet \ \left(\vec{B} \ + \ \vec{C}\right) \ = \ \vec{A} \ \bullet \ \vec{B} \ + \ \vec{A} \ \bullet \ \vec{C}$

Scalar Projection

$\vec{A} \ \text{on} \ \vec{B}=\dfrac{\vec{A}\bullet\vec{B}}{\Big|\,\vec{B}\,\Big|}$

Vector Projection

$\begin{align} \vec{A} \ \text{on} \ \vec{B}&=(\text{scalar projection})(\text{unit vector}) \\ \\ &=\dfrac{\vec{A}\bullet\vec{B}}{\Big|\,\vec{B}\,\Big|}\left(\dfrac{\vec{B}}{\Big|\,\vec{B}\,\Big|}\right) \\ \\ &=\dfrac{\left(\vec{A}\bullet\vec{B}\right)\left(\vec{B}\right)}{\Big|\,\vec{B}\,\Big|^2} \end{align}$

Direction Cosines

$\text{For:} \ \ \vec{A} = \big< \ x, \ y \ , \ z \ \big>$

Perpendicular: Direction Cosines = 0

$\cos\,\alpha = \dfrac{x}{\sqrt{x^2+y^2+z^2}}$

angle with the x-axis

$\cos\,\beta = \dfrac{y}{\sqrt{x^2+y^2+z^2}}$

angle with the y-axis

$\cos\,\gamma = \dfrac{z}{\sqrt{x^2+y^2+z^2}}$

angle with the z-axis

$\cos^2\,\alpha + \cos^2\,\beta + \cos^2\,\gamma = 1$

Matrix Determinant

$\begin{vmatrix} a & b \\ c & d \end{vmatrix} \ = \ (a)(d) - (b)(c)$

Cross Product Procedure [Matrix format not shown]

$\text{For:} \ \ \vec{A} \, = \, \big\langle a_1,\, a_2,\, a_3 \big\rangle, \ \ \vec{B} = \big\langle b_1,\, b_2,\, b_3 \big\rangle \\ \\ \vec{A} \, × \, \vec{B} \, = \, \big\langle a_2b_3 \, - \, a_3b_2,\ a_3b_1 \, - \, a_1b_3,\ a_1b_2 \, - \, a_2b_1 \big\rangle$

Cross Product Magnitude

$\begin{align} \text{Magnitude,} \ \ \Big|\,\vec{A} \times \vec{B}\,\Big| \ &= \Big|\,\vec{A}\,\Big| \ \Big|\,\vec{B}\,\Big| \sin\,\theta \\ \\ &=\Big|\,\vec{A}\,\Big| \ \Big|\,\vec{B}\,\Big|\sqrt{1-\cos^2\,\theta} \\ \\ &=\Big|\,\vec{A}\,\Big| \ \Big|\,\vec{B}\,\Big|\sqrt{1-\left(\dfrac{\vec{A} \ \bullet \ \vec{B}}{\Big|\,\vec{A}\,\Big| \ \Big|\,\vec{B}\,\Big|}\right)^2} \\ \\ \end{align}$

Cross Product Conventions

$\vec{A} \ \times \ \vec{A} \ = 0$

$\vec{A} \ \times \ \vec{B} \ = \ -\vec{B} \ \times \ \vec{A}$

$\vec{A} \times \left(\vec{B} + \vec{C}\right) \ = \vec{A} \times \vec{B} \ + \ \vec{A} \times \vec{C}$

Distance: 2 Points in 2-Dimensions

$d = \sqrt{(x_2 - x_1)^2 - (y_2 - y_1)^2}$

Distance: 2 Points in 3-Dimensions

$d = \sqrt{(x_2 - x_1)^2 - (y_2 - y_1)^2 - (z_2 - z_1)^2}$

Distance: Point and Line in 2-Dimensions

$d = \dfrac{\big|(A)(x) \ + \ (B)(y) \ + \ (C)\big|}{\sqrt{A^2 \ + \ B^2}}$

Distance: Point and Line in 3-Dimensions

$d = \dfrac{\big| \vec{m} \ × \ \vec{NP} \big|}{\vec{m}}$

Where 'N' is the point, and 'P' is a point on the line. Distance: Point and Plane in 3-Dimensions

$d = \dfrac{\big|(A)(x) \ + \ (B)(y) \ + \ (C)(z) + (D) \big|}{\sqrt{A^2 \ + \ B^2 \ + \ C^2}}$

Cartesian equation of plane: Ax + By + Cz + D = 0
Point: (x, y, z)
Magnitude of vector of plane: $\sqrt{A^2 \ + \ B^2 \ + \ C^2}$

LIMITS

Sum & Difference Rule

$\lim\limits_{ x\rightarrow a }{ \Big( f(x) \pm g(x) \Big) } = \lim\limits_{ x\rightarrow a }{ f(x) } \pm \lim\limits_{ x\rightarrow a }{ g(x) }$

Product Rule

$\lim\limits_{ x\rightarrow a }{ \Big( f(x) · g(x) \Big) } = \lim\limits_{ x\rightarrow a }{ f(x) } · \lim\limits_{ x\rightarrow a }{ g(x) }$

Constant (C) Multiple Rule

$\lim\limits_{ x\rightarrow a }{ \Big( C · f(x) \Big) } = C · \lim\limits_{ x\rightarrow a }{ f(x) }$

Quotient Rule

$\lim\limits_{ x\rightarrow a }{ \dfrac{f(x)}{g(x)} } = \dfrac{\lim\limits_{ x\rightarrow a }{ f(x) }}{\lim\limits_{ x\rightarrow a }{ g(x) }}$

Power Rule

$\lim\limits_{ x\rightarrow a }{ \Big( f(x) \Big)^{\frac{m}{n}} } = \left( \lim\limits_{ x\rightarrow a }{ f(x) } \right)^{\frac{m}{n}}$

Where m & n are integers, and n ≠ 0

RATE OF CHANGE

Average Rate of Change

$m = \dfrac{f(x_2) - f(x_1)}{x_2 - x_1}$

(Approximate) Instantaneous Rate of Change

$m = \dfrac{f(x + h) - f(x - h)}{(x + h) - (x - h)}$

For small values of 'h' Instantaneous Rate of Change (Difference Quotient)

$m = \lim\limits_{ h \rightarrow 0 } {\dfrac{f(x + h) - f(x)}{h}}$

Instantaneous Velocity

$v(t) = \lim\limits_{ h \rightarrow 0 } {\dfrac{s(t + h) - s(t)}{h}}$

[where s(t) is position] Average Velocity

$= \dfrac{s(t_2) - s(t_1)}{t_2 - t_1}$

[where s(t) is position] Instantaneous Acceleration

$a(t) = \lim\limits_{ h \rightarrow 0 } {\dfrac{v(t + h) - v(t)}{h}}$

CURVE SKETCHING

Intervals

Positive interval: f(x) > 0

Negative interval: f(x) < 0

Slopes

Increasing: f'(x) > 0

Decreasing: f'(x) < 0

Local Max/Min (Critical/Stationary)

Point at 'x' where: f'(x) = 0

'Jump' Discontinuity

$\lim\limits_{ x \rightarrow c^{-} }{ } \ne \lim\limits_{ x \rightarrow c^{+} }{ }$

$f’(c^{-}) \ne f’(c^{+})$

Oblique Asymptote
Degree Numerator = Degree Denominator + 1 Point of Inflection (POI) at 'c'

When opposite concavity (sign) on:
f''(c^-) & f''(c⁺)

- Vertical POI at 'x' when f'(x) = undefined
- same slopes (limit, or derivative) on either side of 'x' Concavity

Concave up (slope increasing) at c: f''(c) > 0

Local minimum at c: f''(c) > 0, and f'(c) = 0

Neither concave up nor concave down: f''(x) = 0

Concave down (slope decreasing) at c: f''(c) < 0

Local maximum at c: f''(c) < 0, and f'(c) = 0

Horizontal Point of Inflection
- f'(x) = 0 at the point
- opposite slopes either side
- same concavity on either side Vertical Point of Inflection
- f'(x) Does Not Exist (DNE) at the point
- same slopes either side
- opposite concavity on either side Oblique Point of Inflection
- f'(x) defined at the point
- same slopes either side
- opposite concavity on either side Cusp Points

- f'(x) Does Not Exist (DNE) at the point
- opposite slopes either side
- same concavity on either side Corner Points

- f'(x) Does Not Exist (DNE) at the point
- opposite slopes either side

DERIVATIVES & APPLICATIONS

Power Rule

$\begin{align} y &= x^a \\ \dfrac{dy}{dx} &= a(x)^{a - 1} \end{align}$

Product Rule

$\begin{align} y &= u·v \\ \dfrac{dy}{dx} &= u\,’v + u\,v\,’ \end{align}$

Chain Rule (Substitution)

$\begin{align} F(x) &= f[g(x)] \\ F\,’(x) &= f\,’[g(x)]·g\,’(x) \end{align}$

Chain Rule (Leibniz)

$\dfrac{dy}{dx} = \dfrac{du}{dx} · \dfrac{dy}{du}$

Chain Rule (Composition)

$\begin{align} F(x) &= f ∘ g \\ F\,’(x) &= (f\,’ ∘ g) · g\,’ \end{align}$

Quotient Rule

$\begin{align} f(x) & = \dfrac{a}{b} \\ f\,’(x) & = \dfrac{a\,’·b - a·b\,’}{b\,^2} \end{align}$

Displacement (d) & Position (s)

$\begin{align} \vec{d} & = \Delta \vec{s} \\ \\ & = \vec{s_2} - \vec{s_1} \end{align}$

Velocity (v) & Position (s)

$\vec{v} = \vec{s}\,’$

Acceleration (a) & Velocity (v)

$\vec{a} = \vec{v}\,’ = \vec{s}\,’\,’$

Jerk (j) & Acceleration (a)

$\vec{j} = \vec{a}\,’$

$\vec{j} = \vec{v}\,’\,’$

$\vec{j} = \vec{s}\,’\,’\,’$

Marginal Revenue Function

$R(x) = (price)(\#\ sold) \\ \\ R\,’(x) = Marginal\ Revenue$

Marginal Profit Function

$P(x) = Revenue - Cost \\ \\ P\,’(x) = Marginal\ Profit$

Exponential

$\begin{align} y &= b^{\,ax} \\ y\,’ &= b^{\,ax}\Big[\ln(b)\Big] \cdot \dfrac{d}{dx} \, (ax) \end{align}$

Natural Exponential

$\begin{align} \dfrac{d}{dx} & \, e^{\, g(x)} \\ & = e^{\, g(x)} \cdot \dfrac{d}{dx} \, g(x) \end{align}$

Logarithmic

$\begin{align} y &= \ln(ax) \\ y\,’ &= \dfrac{1}{ax} \cdot \dfrac{d}{dx} \, (ax) \end{align}$

$\begin{align} y &= \log_a\,(kx) \\ y\,’ &= \dfrac{1}{(kx)·\ln(a)} \cdot \dfrac{d}{dx} \, (kx) \end{align}$

Common Trig

$\begin{align} y &= \sin(x) \\ y\,’ &= \cos(x) \end{align}$

$\begin{align} y &= \cos(x) \\ y\,’ &= -\sin(x) \end{align}$

$\begin{align} y &= \tan(x) \\ y\,’ &= \sec^2(x) \end{align}$

$\begin{align} y &= \csc(x) \\ y\,’ &= -\csc(x)·\cot(x) \end{align}$

$\begin{align} y &= \sec(x) \\ y\,’ &= \sec(x)·\tan(x) \end{align}$

$\begin{align} y &= \cot(x) \\ y\,’ &= -\csc^2(x) \end{align}$

$\begin{align} y &= \sin^{-1}(x) \\ y\,’ &= \dfrac{\sqrt{1 - x^2}}{1 - x^2} \end{align}$

$\begin{align} y &= \cos^{-1}(x) \\ y\,’ &= -\dfrac{\sqrt{1 - x^2}}{1 - x^2} \end{align}$

$\begin{align} y &= \tan^{-1}(x) \\ y\,’ &= \dfrac{1}{1 + x^2} \end{align}$

INTEGRALS

Integration as Antiderivative

$\displaystyle\int{\dfrac{d^2y}{dx^2}} = \dfrac{dy}{dx}$

$\displaystyle\int{f’’(x)} = f’(x)$

$\displaystyle\int{\dfrac{dy}{dx}} = y$

$\displaystyle\int{f’(x)} = f(x)$

Rules a, b, n, c = constants

$\displaystyle\int{\left( a \right)}\, dx = ax + c$

$\displaystyle\int{a \cdot f(x)}\, dx = a \cdot \displaystyle\int{f(x)}\, dx$

$\displaystyle\int{\left( kx^n \right)}\, dx = \dfrac{kx^{n+1}}{n + 1} + c$

$\displaystyle\int{\left( ax + b \right)^n}\, dx = \dfrac{(ax + b)^{n+1}}{a(n+1)} + c$

$\displaystyle\int{\left( \dfrac{n}{ax + b} \right)}\, dx = \dfrac{n}{a} \ln\Big|ax+b\Big| + c$

$\displaystyle\int{\left( e^{\ ax + b} \right)}\, dx = \dfrac{1}{a} e^{ax + b} + c$

$\displaystyle\int{\left( a^x \right)}\, dx = \dfrac{a^x}{ln\ a} + c$

$\displaystyle\int{\left( \ln x \right)}\, dx = x \ln x - x + c$

$\displaystyle\int{\Big( \sin(ax + b) \Big)}\, dx = \dfrac{-\cos(ax + b)}{a} + c$

$\displaystyle\int{\Big( \cos(ax + b) \Big)}\, dx = \dfrac{\sin(ax + b)}{a} + c$

$\displaystyle\int{\Big[ f(x) \pm g(x) \Big]}\, dx = \displaystyle\int{f(x)}\, dx \ \pm \ \displaystyle\int{g(x)}\, dx$

Integration By Substitution [This formula is a general guide, but not a precise procedure. Your steps will differ based on the question.]

$\displaystyle\int{f(u) \, u’ }\,dx = \displaystyle\int{f(u)}·\,du$

Integration By Parts (Shown in several different forms)

$\displaystyle\int{u \cdot v’} = u \cdot v - \displaystyle\int{u’ \cdot v}$

$\displaystyle\int{u \cdot d(v)} = u \cdot v - \displaystyle\int{d(u) \cdot v}$

$\displaystyle\int{f(x) g’(x) \, dx} = f(x) \, g(x) - \displaystyle\int{f’(x) g(x) \, dx}$

Definite Integrals

$\displaystyle\int_{a}^{b}{f(x)}\,dx = -\displaystyle\int_{b}^{a}{f(x)}\,dx$

$\displaystyle\int_{a}^{b}{f(x)}\,dx \ + \ \displaystyle\int_{b}^{c}{f(x)}\,dx \ = \ \displaystyle\int_{a}^{c}{f(x)}\,dx$

STATISTICS & PROBABILITY

STATISTICS

Mean

$\displaystyle{ µ = \frac{ \sum x_{ i } }{ n } }$

Variance

$\displaystyle{ σ^2 = \frac{ \sum (x_{ i }-µ)^{ 2 } }{ n } }$

Standard Deviation

$\displaystyle{ σ = \sqrt{ \frac{ \sum (x_{ i }-µ)^{ 2 } }{ n } } }$

Combination
(Binomial Coefficient) "n choose k"

$\left( \begin{matrix} n \\ k \end{matrix} \right) \ = \ C(n, k) \ = \ _n C _k \ = \ ^n C _k \ = \ C _{n, k}$

$\left( \begin{matrix} n \\ k \end{matrix} \right) \ = \ \dfrac{n\,!}{k\,! (n - k)\,!}$

Equivalent notations Binomial Formula

$\displaystyle{ (a+b)^{ n }=\sum _{ k \, = \, 0 }^{ n }{ \left( \begin{matrix} n \\ k \end{matrix} \right) \, a\, ^{ n-k }\, b\, ^{ k } } }$

Binomial Expansion

$(a + b)^n = a^n + \left( \begin{matrix} n \\ 1 \end{matrix} \right) \, a\,^{n - 1} \, b\,^1 + \left( \begin{matrix} n \\ 2 \end{matrix} \right) \, a\,^{n - 2} \, b\,^2 + \, ... \, + \left( \begin{matrix} n \\ n \end{matrix} \right) \, a\,^{n - n} \, b\,^k$

Bernoulli Probability Events that are binary
(one, or the other)

$P(x) \ = \ p^x(1-p^{1-x})$

Binomial Probability 'n' trials of Bernoulli events

$P(x) \ = \ \dbinom{n}{x} p^x(1-p^{1-x})$

Geometric perform trials until first 'success'

$P(x) \ = \ p(1 - p^{x-1})$

Permutation

$P(n, k) \ = \ _n P _k \ = \ ^n P _k \ = \ P _{n, k}$

$P _{n, k} \ = \ \dfrac{n\,!}{(n - k)\,!} \ = \ (n - 0) \times (n - 1) \times \dots \times (n - r + 1)$

PROBABILITY

Venn: Non-disjoint

Sum of Event and its Complement

$P(A) + P(A\ ’) = 1$

Complement can be shown as: A' or A^c Union

$(A \cup B)$

'A' union 'B' equals either 'A', or 'B', or both. Intersection

$(A \cap B)$

'A' intersect 'B' equals both 'A' and 'B'. Addition Rule of Combined Events Non-disjoint only

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$P(A \cup B \cup C) = P(A) + P(B) + P(C) - \\ \quad P(A \cap B) - P(A \cap C) - P(B \cap C) + \\ \quad P(A \cap B \cap C)$

Conditional Probability

P('A given B'); when outcome 'A' is dependent on outcome 'B'
E.g.) Selecting cards *without* replacement.

$P(A\, |\, B) = \dfrac{P(A \cap B)}{P(B)}$

$P(A\ ’\, |\, B) = \dfrac{P(A\ ’ \cap B)}{P(B)}$

De Morgan's Laws

$P(A\ ’ \cap B\ ’) = P(A \cup B)\ ’ = 1 \ - \ P(A \cup B)$

$P(A\ ’ \cup B\ ’) = P(A \cap B)\ ’ = 1 \ - \ P(A \cap B)$

Other Stuff

You don't have to memorize these.
You can work them out by looking at the Venn diagram.

$P(A) = P(A \cap B\ ’) + P(A \cap B)$

$P(A) = P(A \cup B) - P(A\ ’ \cap B)$

Independent Events

The outcome of one event doesn't affect the chance of another event.
E.g.) Selecting cards *with* replacement.

$P(A\, |\, B) = P(A) \\ \\ P(A\, |\, B\ ’) = P(A) \\ \\ P(B\, |\, A) = P(B) \\ \\ P(B\, |\, A\ ’) = P(B)$

Product Rule for Independent Events

$P(A \cap B) = P(A) \times P(B)$

Product Rule for Dependent Events

$P(A \cap B) = P(A) \times P(B\, |\, A)$

General Rules About Reversibility

$P(A \cap B) = P(B \cap A)$

Intersections are (equivalent when) reversible

$P(A \ | \ B) \ne P(B \ | \ A)$

Conditionals are not (equivalent when) reversible Mutually Exclusive (Disjoint)
Two event outcomes cannot occur at the same time.
E.g.) In cards, the events of selecting an 'Ace' and a 'King'

$P(A \cap B) = 0 \\ \\ P(A \cup B) = P(A) + P(B)$

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FORMULAS

BASIC ARITHMETIC

LINEAR & GEOMETRY

QUADRATIC FUNCTIONS AND RELATIONS

EXPONENT LAWS

ROOT RADICAL LAWS

BASIC TRIGONOMETRY

TRIGONOMETRY II

SPECIAL TRIANGLES

THE UNIT CIRCLE

TRIGONOMETRY III

SEQUENCES & SERIES

FINANCIAL MATH

COMPLEX NUMBERS

LOG LAWS

POLYNOMIALS

COMBINATION AND COMPOSITION OF FUNCTIONS

CALCULUS

VECTORS, LINES, PLANES

LIMITS

RATE OF CHANGE

CURVE SKETCHING

DERIVATIVES & APPLICATIONS

INTEGRALS

STATISTICS & PROBABILITY

STATISTICS

PROBABILITY

CLASS PAGE SETTINGS

Streamlined Question View

Level of Difficulty

KEYBOARD SHORTCUTS

LINK

FAVORITES

OPTIONS

CLASS OPTIONS

FORMULAS

BASIC ARITHMETIC

LINEAR & GEOMETRY

QUADRATIC FUNCTIONS AND RELATIONS

EXPONENT LAWS

ROOT RADICAL LAWS

BASIC TRIGONOMETRY

TRIGONOMETRY II

SPECIAL TRIANGLES

THE UNIT CIRCLE

TRIGONOMETRY III

SEQUENCES & SERIES

FINANCIAL MATH

COMPLEX NUMBERS

LOG LAWS

POLYNOMIALS

COMBINATION AND COMPOSITION OF FUNCTIONS

CALCULUS

VECTORS, LINES, PLANES

LIMITS

RATE OF CHANGE

CURVE SKETCHING

DERIVATIVES & APPLICATIONS

INTEGRALS

STATISTICS & PROBABILITY

STATISTICS

PROBABILITY

CLASS PAGE SETTINGS

Streamlined Question View

Level of Difficulty

KEYBOARD SHORTCUTS

LINK