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

(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
$x^2 + (a + b)x + ab \\ \\ = (x + a)(x + b)$
$cnx^2 + (a + b)x + ab \\ \\ = (cx + \tfrac{a}{n})(nx + \tfrac{b}{c})$
$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
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

## 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
$\csc \left( \dfrac{π}{2} - θ \right) = \secθ$
$\sec \left( \dfrac{π}{2} - θ \right) = \cscθ$
$\cot \left( \dfrac{π}{2} - θ \right) = \tanθ$
Q2
$\sin \left( \dfrac{π}{2} + θ \right) = \cos θ$
$\cos \left( \dfrac{π}{2} + θ \right) = -\sin θ$
$\tan \left( \dfrac{π}{2} + θ \right) = -\cot θ$
Q3
$\sin \left( \dfrac{3π}{2} - θ \right) = -\cos θ$
$\cos \left( \dfrac{3π}{2} - θ \right) = -\sin θ$
$\tan \left( \dfrac{3π}{2} - θ \right) = \cot θ$
Q4
$\sin \left( \dfrac{3π}{2} + θ \right) = -\cos θ$
$\cos \left( \dfrac{3π}{2} + θ \right) = \sin θ$
$\tan \left( \dfrac{3π}{2} + θ \right) = -\cot θ$
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
tn = term value
d = common difference
a = first term
Sn = 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

$\text{If:} \ \ \vec{A} = \big< \ i, \ j \ \big>, \ \ \vec{B} = \big< \ x, \ y \ \big> \\ \\ \text{Then:} \ \ \vec{A}+\vec{B} \ = \ \big< \ i+x, \ j+y \ \big>$
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 Ac 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'
$P(A \cap B) = 0 \\ \\ P(A \cup B) = P(A) + P(B)$
Percent complete:
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