Precalculus Cheatsheet

This cheatsheet generally covers anything that is typically taught before calculus. I also include linear algebra and a bit of complex numbers.

The Discrete Mathematics and Probability and Statistics cheatsheets also contain some precalculus missing from this cheatsheet.

Partial Fraction Decomposition

A proper rational function can be rewritten as a sum of partial fractions.

For each irreducible factor in the denominator, the partial fractions are as follows:

\begin{array}{ccccc} \substack{\text{\underline{irreducible factor in denominator}}} &&&& \substack{\text{\underline{partial fractions}}} \\[1ex] \displaystyle {(\memphR{ax + b})}^k & & \Longrightarrow & \quad & %\displaystyle \sum_{n=1}^k{\frac{A_n}{\memphR{(ax + b)}^n}} \displaystyle \frac{A_1}{\memphR{ax + b}} + \frac{A_2}{\parens{\memphR{ax + b}}^2} + \dots + \frac{A_k}{\parens{\memphR{ax + b}}^k} \\[3ex] \displaystyle {(\memphR{ax^2 + bx + c})}^k & & \Longrightarrow & \quad & %\displaystyle \sum_{n=1}^k{\frac{A_n x + B_n}{\memphR{(ax^2 + bx + c)}^n}} \displaystyle \frac{A_1 x + B_1}{\memphR{ax^2 + bx + c}} + \frac{A_2 x + B_2}{\parens{\memphR{ax^2 + bx + c}}^2} + \dots + \frac{A_k x + B_k}{\parens{\memphR{ax^2 + bx + c}}^k} \end{array}

For improper rational functions, you must first convert it to a proper rational function.

Hint: You can use complex numbers to further reduce some factors. Example: $\parens{1 + x^2} = \parens{1 + i x} \parens{1 - i x}$

Exponentiation and Logarithm Identities

$\displaystyle x = a^y$	$\iff$	$\displaystyle \log_a{x} = y$
$\displaystyle a^{\log_a{x}} = x$	$\iff$	$\displaystyle \log_a{a^x} = x$
		$\overset{\text{\textbf{Change of Base Law}}}{\boxed{\log_a{x} = \frac{\log_u{x}}{\log_u{a}}}}$
$\displaystyle a^0 = 1$	$\longrightarrow$	$\displaystyle \log_a{1} = 0$
$\displaystyle a^1 = a$	$\longrightarrow$	$\displaystyle \log_a{a} = 1$
$\displaystyle a^{-1} = \frac{1}{a}$	$\longrightarrow$	$\displaystyle \log_a{\frac{1}{a}} = -1$
$\displaystyle a^x a^y = a^{x+y}$	$\longrightarrow$	$\displaystyle \log_a{xy} = \log_a{x} + \log_a{y}$
$\displaystyle a^x b^x = \parens{ab}^x$
$\parens{a^x}^y = a^{xy}$	$\longrightarrow$	$\displaystyle \log_a{x^n} = n \log_a{x}$
$\displaystyle \frac{1}{a^y} = a^{-y}$		$\displaystyle \log_a{\frac{1}{y}} = -\log_a{y}$
$\displaystyle \frac{a^x}{a^y} = a^{x-y}$	$\longrightarrow$	$\displaystyle \log_a{\frac{x}{y}} = \log_a{x} - \log_a{y}$
$\displaystyle \frac{a^x}{b^x} = \parens{\frac{a}{b}}^x$
$\displaystyle a^{\frac{1}{2}} = \sqrt{a}$	$\longrightarrow$	$\displaystyle \log_a{\sqrt{a}} = \frac{1}{2}$
$\displaystyle a^{\frac{1}{y}} = \sqrt[y]{a}$	$\longrightarrow$	$\displaystyle \log_a{\sqrt[y]{a}} = \frac{1}{y}$
$\displaystyle a^{\frac{x}{y}} = \parens{\sqrt[y]{a}}^x$	$\longrightarrow$	$\displaystyle \log_a{\parens{\sqrt[y]{a}}^x} = \frac{x}{y}$

Pascal’s Triangle

\begin{matrix} n = 0 && 1 && \binom{0}{0} \\ n = 1 && 1 \quad 1 && \binom{1}{0} \:\: \binom{1}{1} \\ n = 2 && 1 \quad 2 \quad 1 && \binom{2}{0} \:\: \binom{2}{1} \:\: \binom{2}{2} \\ n = 3 && 1 \quad 3 \quad 3 \quad 1 && \binom{3}{0} \:\: \binom{3}{1} \:\: \binom{3}{2} \:\: \binom{3}{3} \\ n = 4 && 1 \quad 4 \quad 6 \quad 4 \quad 1 && \binom{4}{0} \:\: \binom{4}{1} \:\: \binom{4}{2} \:\: \binom{4}{3} \:\: \binom{4}{4} \end{matrix}

Binomial Theorem

\begin{align*} \parens{x + y}^n &= \binom{n}{0} x^n y^0 + \binom{n}{1} x^{n-1} y^1 + \dots + \binom{n}{n} x^0 y^n \\ &= \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k} \end{align*}

See Discrete Mathematics for the binomial coefficient.

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \qquad\quad \MathOverLabel{\text{\underline{discriminant}}}{\Delta = b^2 - 4ac}

Quadratic Factorization

x^2 + bx + c = \parens{x+v} \parens{x+u} \qquad\quad %\MathOverLabel{\text{\ul{sums and products}}}{ % Too much space. Not worth it. \begin{array}{c} b = u+v \\ c = uv \end{array} %}

Completing The Square

x^2 + b x + c = \parens{x + \xmemphR{\brackets{\memphR{\frac{b}{2}}}}}^2 + c - \xmemphR{\brackets{\memphR{\frac{b}{2}}}^2}

Quadratic/Cubic Identities

\begin{align*} a^2 - b^2 &= \parens{a + b} \parens{a - b} \\ a^3 - b^3 &= \parens{a - b} \parens{a^2 + ab + b^2} \\ a^3 + b^3 &= \parens{a + b} \parens{a^2 - ab + b^2} \end{align*}

sin/cos In Terms Of Exponentials

Derived from Euler’s formula.

\sin{x} = \frac{e^{ix} - e^{-ix}}{2i} \qquad\quad \cos{x} = \frac{e^{ix} + e^{-ix}}{2}

Hyperbolic Function Exponential Definitions

\begin{align*} \sinh{x} &= \frac{e^x - e^{-x}}{2} = \frac{e^{2x} - 1}{2 e^x} = \frac{1 - e^{-2x}}{2 e^{-x}} \\ %\quad \forall x \in \mathbb{R} \\ \cosh{x} &= \frac{e^x + e^{-x}}{2} = \frac{e^{2x} + 1}{2 e^x} = \frac{1 + e^{-2x}}{2 e^{-x}} \\ %\quad \forall x \in \mathbb{R} \\ \tanh{x} &= \frac{\sinh{x}}{\cosh{x}} = \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{e^{2x} - 1}{e^{2x} + 1} \end{align*}

Pythagorean Theorem

a^2 = b^2 + c^2

Pythagorean Identities

\begin{gather*} \sin^2{\theta} + \cos^2{\theta} = 1 \\ \tan^2{\theta} + 1 = \sec^2{\theta} \quad \parens{\cos{\theta} \ne 0} \\ \cot^2{\theta} + 1 = \csc^2{\theta} \quad \parens{\sin{\theta} \ne 0} \end{gather*}

Complementary Angles

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

Compound Angles

\begin{align*} \sin{\parens{\alpha \pm \beta}} &= \sin{\alpha} \cos{\beta } \pm \cos{\alpha} \sin{\beta } \\ \cos{\parens{\alpha \pm \beta}} &= \cos{\alpha} \cos{\beta } \mp \sin{\alpha} \sin{\beta } \\ \tan{\parens{\alpha \pm \beta}} &= \frac{\tan{\alpha} \pm \tan{\beta}}{1 \mp \tan{\alpha} \tan{\beta}} \end{align*}

Double-Angle Formulae

Derived from compound angle formulae.

\begin{align*} \sin{2 \theta} &= 2 \sin{\theta} \cos{\theta} \\ \cos{2 \theta} &= \cos^2{\theta} - \sin^2{\theta} \, \overbrace{= 2 \cos^2{\theta} - 1} \\ &\phantom{{}= \cos^2{\theta} - \sin^2{\theta}} \, \underbrace{= 1 - 2 \sin^2{\theta}}_{ %\substack{ % \text{simplified using} \\ \sin^2{\theta} + \cos^2{\theta} = 1 %} }\\ \tan{2 \theta} &= \frac{2 \tan{\theta}}{1 - \tan^2{\theta}} \end{align*}

Half-Angle Formulae

Derived from the $\cos{2 \theta}$ compound angle formula.

\sin^2{\theta} = \frac{1}{2} - \frac{1}{2} \cos{2 \theta} \qquad\enspace \cos^2{\theta} = \frac{1}{2} + \frac{1}{2} \cos{2 \theta}

Products to Sums

Derived by adding compound angle formulae.

\begin{alignat*}{8} &2 \sin&&{A} && \cos&&{B} &&= \sin&&{\parens{A+B}} &&+ \sin&&{\parens{A-B}} \\ &2 \cos&&{A} && \sin&&{B} &&= \sin&&{\parens{A+B}} &&- \sin&&{\parens{A-B}} \\ &2 \cos&&{A} && \cos&&{B} &&= \cos&&{\parens{A+B}} &&+ \cos&&{\parens{A-B}} \\ -&2 \sin&&{A} && \sin&&{B} &&= \cos&&{\parens{A+B}} &&- \cos&&{\parens{A-B}} \end{alignat*}

Sums to Products

Derived by reversing Products to Sums.

\begin{alignat*}{4} \sin{S} + \sin{T} &= \phantom{+} 2 \sin&&{\parens{\frac{S+T}{2}}} \cos&&{\parens{\frac{S-T}{2}}} \\ \sin{S} - \sin{T} &= \phantom{+} 2 \cos&&{\parens{\frac{S+T}{2}}} \sin&&{\parens{\frac{S-T}{2}}} \\ \cos{S} + \cos{T} &= \phantom{+} 2 \cos&&{\parens{\frac{S+T}{2}}} \cos&&{\parens{\frac{S-T}{2}}} \\ \cos{S} - \cos{T} &= - 2 \sin&&{\parens{\frac{S+T}{2}}} \sin&&{\parens{\frac{S-T}{2}}} \end{alignat*}

Hyperbolic: Difference of Squares

\begin{gather*} \cosh^2{x} - \sinh^2{x} = 1 \\ 1 - \tanh^2{x} = \sech^2{x} \\ \coth^2{x} - 1 = \csch^2{x} \end{gather*}

Hyperbolic: Sum and Difference Formulae

\begin{align*} \sinh{\parens{\alpha \pm \beta}} &= \sinh{\alpha} \cosh{\beta } \pm \cosh{\alpha} \sinh{\beta } \\ \cosh{\parens{\alpha \pm \beta}} &= \cosh{\alpha} \cosh{\beta } \pm \sinh{\alpha} \sinh{\beta } \\ \tanh{\parens{\alpha \pm \beta}} &= \frac{\tanh{\alpha} \pm \tanh{\beta}}{1 \pm \tanh{\alpha} \tanh{\beta}} \end{align*}

Hyperbolic: Double-Angle Formulae

\begin{align*} \sinh{2x} &= 2 \sinh{x} \cosh{x} \\ \cosh{2x} &= \cosh^2{x} + \sinh^2{x} \\ \tanh{2x} &= \frac{2 \tanh{x}}{1 + \tanh^2{x}} \end{align*}

Arithmetic Progression and Series

\begin{gather*} a_n = a_1 + \parens{n - 1} d \\ S_n = \sum_{k = 1}^{n} a_k = \frac{n}{2} \brackets{2 a_1 + \parens{n - 1} d} = \frac{n}{2} \brackets{a_1 - a_n} \end{gather*}

Imaginary Unit

i^2 = -1

Euler’s Formula

e^{i \theta} = \cos{\theta} + i \sin{\theta}, \qquad \forall \theta \in \Reals

Complex Number Representation

\begin{gather*} a, b, r, \theta \in \mathbb{R} \\ \begin{alignedat}{2} z &= a + ib & \qquad & \text{(Rectangular Form)} \\ &= r(\cos{\theta} + i \sin{\theta}) & \qquad & \text{(Polar Form)} \\ &= r e^{i \theta} & \qquad & \text{(Exponential Form)} \\ &= r \phase{\theta} & \qquad & \text{(Steinmetz Notation)} \\ &\mathbin{\Exn{=}} \Exn{r \cis{\theta}} & \qquad & \Exn{\text{(cis Form) \footnotesize [rarely used]}} \end{alignedat}% \end{gather*}

\begin{alignat*}{3} \MyRe{(z)} &= a &&= r \cos{\theta} \phantom{\frac{}{a}} & \qquad & \text{(Real Part)} \\ \MyIm{(z)} &= b &&= r \sin{\theta} \phantom{\frac{b}{a}} & \qquad & \text{(Imaginary Part)} \\ \abs{z} &= r &&= \sqrt{a^2 + b^2} \phantom{\frac{b}{a}} & \qquad & \text{(Modulus)} \\ \arg{(z)} &= \theta &&= \atantwo{\parens{b, a}} & \qquad & \text{(Argument)} \end{alignat*}

Complex Conjugate

\complexconjugate{a + ib} = \parens{a + ib}^* = a - ib

Useful identities:

\begin{gather*} z \complexconjugate{z} = \abs{z}^2 \\ \parens{a + ib} \complexconjugate{\parens{a + ib}} = \parens{a + ib} \parens{a - ib} = a^2 + b^2 \end{gather*}

Dot Product / Scalar Product

In $\Reals^n$ :

\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + \cdots + a_n b_n = \sum_{k=0}^n{a_k b_k}

In $\Reals^2$ or $\Reals^3$ :

\mathbf{a} \cdot \mathbf{b} = \abs{\mathbf{a}} \abs{\mathbf{b}} \cos{\theta} ,\qquad \theta \in [0, \pi]

Useful geometric properties:

$\mathbf{a} \cdot \mathbf{a} = \abs{a}^2$ , hence $\abs{\mathbf{a}} = \sqrt{\mathbf{a} \cdot \mathbf{a}}$ .
Vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R}^n$ are orthogonal if $\mathbf{a} \cdot \mathbf{b} = 0$ .

Cross Product / Vector Product

The cross product is only defined in $\mathbb{R}^3$ .

\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \times \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} = \begin{pmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1 \end{pmatrix}

Calculation using determinants:

\begin{gather*} \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \times \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} = \begin{vmatrix} \mathbf{e}_1 & \mathbf{e}_2 & \mathbf{e}_3 \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \\ = \mathbf{e}_1 \begin{vmatrix} \memphR{a_2} & \memphB{a_3} \\ \memphB{b_2} & \memphR{b_3} \end{vmatrix} - \mathbf{e}_2 \begin{vmatrix} \memphR{a_1} & \memphB{a_3} \\ \memphB{b_1} & \memphR{b_3} \end{vmatrix} + \mathbf{e}_3 \begin{vmatrix} \memphR{a_1} & \memphB{a_2} \\ \memphB{b_1} & \memphR{b_2} \end{vmatrix} \\ \begin{aligned} = \mathbf{e}_1 &\parens{\memphR{a_2 b_3} - \memphB{a_3 b_2}} \\ &- \mathbf{e}_2 \parens{\memphR{a_1 b_3} - \memphB{a_3 b_1}} \\ &+ \mathbf{e}_3 \parens{\memphR{a_1 b_2} - \memphB{a_2 b_1}} \end{aligned} \end{gather*}

Useful geometric properties:

Vector $\mathbf{a} \times \mathbf{b}$ is orthogonal to $\mathbf{a}$ and $\mathbf{b}$ .
$\abs{\mathbf{a} \times \mathbf{b}} = \abs{\mathbf{a}} \abs{\mathbf{b}} \sin{\theta} = \text{area of a parallelogram}$

Cramer’s Rule

Consider the following linear system with $n \times n$ invertible matrix $A$ :

A \mathbf{x} = \mathbf{b} ,\qquad A \in M_{nn},\ \mathbf{x} \in \mathbb{R}^n,\ \mathbf{b} \in \mathbb{R}^n

The system has a unique solution:

x_k = \frac{\det{\parens{B_k}}}{\det{\parens{A}}} ,\qquad \forall k = 1, \dots, n,

where $B_k$ is the matrix obtained from $A$ by replacing the $\Nth{k}{th}$ column with the vector $\mathbf{b}$ .

Cramer’s Rule, $2 \times 2$ Matrix

\begin{dcases} \xmemphB{a_{11}} x + \xmemphB{a_{12}} y = \xmemphR{b_1} \\ \xmemphB{a_{21}} x + \xmemphB{a_{22}} y = \xmemphR{b_2} \end{dcases} \quad \Rightarrow \quad \xmemphB{ \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} } \begin{bmatrix} x \\ y \end{bmatrix} = \xmemphR{ \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} }

x = \frac{ \xmemphB{ \begin{vmatrix} \xmemphR{b_1} & a_{12} \\ \xmemphR{b_2} & a_{22} \end{vmatrix} } }{ \xmemphB{ \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} } } ,\qquad y = \frac{ \xmemphB{ \begin{vmatrix} a_{11} & \xmemphR{b_1} \\ a_{21} & \xmemphR{b_2} \end{vmatrix} } }{ \xmemphB{ \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} } }

Cramer’s Rule, $3 \times 3$ Matrix

\begin{dcases} \xmemphB{a_{11}} x + \xmemphB{a_{12}} y + \xmemphB{a_{13}} z = \xmemphR{b_1} \\ \xmemphB{a_{21}} x + \xmemphB{a_{22}} y + \xmemphB{a_{23}} z = \xmemphR{b_2} \\ \xmemphB{a_{31}} x + \xmemphB{a_{32}} y + \xmemphB{a_{33}} z = \xmemphR{b_3} \end{dcases}

\xmemphB{ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} } \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \xmemphR{ \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} }

x = \frac{ \xmemphB{ \begin{vmatrix} \xmemphR{b_1} & a_{12} & a_{13} \\ \xmemphR{b_2} & a_{22} & a_{23} \\ \xmemphR{b_3} & a_{32} & a_{33} \end{vmatrix} } }{ \xmemphB{ \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} } } ,\qquad y = \frac{ \xmemphB{ \begin{vmatrix} a_{11} & \xmemphR{b_1} & a_{13} \\ a_{21} & \xmemphR{b_2} & a_{23} \\ a_{31} & \xmemphR{b_3} & a_{33} \end{vmatrix} } }{ \xmemphB{ \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} } } ,

z = \frac{ \xmemphB{ \begin{vmatrix} a_{11} & a_{12} & \xmemphR{b_1} \\ a_{21} & a_{22} & \xmemphR{b_2} \\ a_{31} & a_{32} & \xmemphR{b_3} \end{vmatrix} } }{ \xmemphB{ \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} } }

Symbols and Units

Name	Symbol			Decimal Multiplier
yotta	$\si{\yotta\sinounit}$	${10}^{24}$	${1000}^{8}$	$1\;000\;000\;000\;000\;000\;000\;000\;000$
zetta	$\si{\zetta\sinounit}$	${10}^{21}$	${1000}^{7}$	$\phantom{0\;00}1\;000\;000\;000\;000\;000\;000\;000$
exa	$\si{\exa\sinounit}$	${10}^{18}$	${1000}^{6}$	$\phantom{0\;000\;00}1\;000\;000\;000\;000\;000\;000$
peta	$\si{\peta\sinounit}$	${10}^{15}$	${1000}^{5}$	$\phantom{0\;000\;000\;00}1\;000\;000\;000\;000\;000$
tera	$\si{\tera\sinounit}$	${10}^{12}$	${1000}^{4}$	$\phantom{0\;000\;000\;000\;00}1\;000\;000\;000\;000$
giga	$\si{\giga\sinounit}$	${10}^{9}$	${1000}^{3}$	$\phantom{0\;000\;000\;000\;000\;00}1\;000\;000\;000$
mega	$\si{\mega\sinounit}$	${10}^{6}$	${1000}^{2}$	$\phantom{0\;000\;000\;000\;000\;000\;00}1\;000\;000$
kilo	$\si{\kilo\sinounit}$	${10}^{3}$	${1000}^{1}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;00}1\;000$
hecto	$\si{\hecto\sinounit}$	${10}^{2}$	${1000}^{2/3}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000}\;100$
deca	$\si{\deca\sinounit}$	${10}^{1}$	${1000}^{1/3}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;0}10$
		${10}^{0}$	${1000}^{0}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;00}1$
deci	$\si{\deci\sinounit}$	${10}^{-1}$	${1000}^{-1/3}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;00} 0.1$
centi	$\si{\centi\sinounit}$	${10}^{-2}$	${1000}^{-2/3}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;00} 0.01$
milli	$\si{\milli\sinounit}$	${10}^{-3}$	${1000}^{-1}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;00} 0.001$
micro	$\si{\micro\sinounit}$	${10}^{-6}$	${1000}^{-2}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;00} 0.000\;001$
nano	$\si{\nano\sinounit}$	${10}^{-9}$	${1000}^{-3}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;00} 0.000\;000\;001$
pico	$\si{\pico\sinounit}$	${10}^{-12}$	${1000}^{-4}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;00} 0.000\;000\;000\;001$
femto	$\si{\femto\sinounit}$	${10}^{-15}$	${1000}^{-5}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;00} 0.000\;000\;000\;000\;001$
atto	$\si{\atto\sinounit}$	${10}^{-18}$	${1000}^{-6}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;00} 0.000\;000\;000\;000\;000\;001$
zepto	$\si{\zepto\sinounit}$	${10}^{-21}$	${1000}^{-7}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;00} 0.000\;000\;000\;000\;000\;000\;001$
yocto	$\si{\yocto\sinounit}$	${10}^{-24}$	${1000}^{-8}$	$\phantom{0\;000\;000\;000\;000\;000\;000\;000\;00} 0.000\;000\;000\;000\;000\;000\;000\;001$

$\myAlpha$	$\alpha$		Alpha
$\myBeta$	$\beta$		Beta
$\Gamma$	$\gamma$		Gamma
$\Delta$	$\delta$		Delta
$\myEpsilon$	$\epsilon$	$\varepsilon$	Epsilon
$\myZeta$	$\zeta$		Zeta
$\myEta$	$\eta$		Eta
$\Theta$	$\theta$	$\vartheta$	Theta
$\myIota$	$\iota$		Iota
$\myKappa$	$\kappa$	$\varkappa$	Kappa
$\Lambda$	$\lambda$		Lambda
$\myMu$	$\mu$		Mu
$\myNu$	$\nu$		Nu
$\Xi$	$\xi$		Xi
$\myOmicron$	$\myomicron$		Omicron
$\Pi$	$\pi$	$\varpi$	Pi
$\myRho$	$\rho$	$\varrho$	Rho
$\Sigma$	$\sigma$	$\varsigma$	Sigma
$\myTau$	$\tau$		Tau
$\Upsilon$	$\upsilon$		Upsilon
$\Phi$	$\phi$	$\varphi$	Phi
$\myChi$	$\chi$		Chi
$\Psi$	$\psi$		Psi
$\Omega$	$\omega$		Omega
$\myDigamma$	$\digamma$		Digamma

Appendix: Extended Cheatsheet

Basic Number Sets

\MathOverLabel{\text{\footnotesize{Natural Numbers}}}{ \myNaturalsSet = \braces{1, 2, 3, \dots} } \qquad\,\, \MathOverLabel{\text{\footnotesize{Integers}}}{ \myIntegerSet = \braces{\dots, -2, -1, 0, 1, 2, \dots} }

\MathOverLabel{\text{\footnotesize{Rational Numbers}}}{\myRationalsSet} \qquad \MathOverLabel{\text{\footnotesize{Real Numbers}}}{\myReals} \qquad \MathOverLabel{\text{\footnotesize{Complex Numbers}}}{\myComplex}

Polynomial

A function $f : \myReals \to \myReals$ is a polynomial of degree $n$ if:

\begin{gather*} f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0 \\ \forall x \in \myReals \end{gather*}

where:

degree:	$n = 0, 1, 2, 3, \dots$
coefficients:	$a_0, a_1, \dots, a_{n-1}, a_n \in \myReals$
leading coefficient:	$a_n \ne 0$

The most common polynomials are named:

	Name	Form
Degree 0	constant	$a$
Degree 1	linear	$a x + b$
Degree 2	quadratic	$a x^2 + b x + c$
Degree 3	cubic	$a x^3 + b x^2 + c x + d$
Degree 4	quartic	$a x^4 + b x^3 + c x^2 + d x + e$

A monic polynomial is one where the leading coefficient is $1$ .

Rational Function

A function $f$ is a rational function if it can be written in the form:

f(x) = \frac{P(x)}{Q(x)}

where $P$ and $Q$ are polynomial functions, and $Q$ is not the zero function.

The domain of $f$ excludes zeroes of the denominator:

\dom{(f)} = \braces{x \in \myReals : Q(x) \ne 0}

The fundamental theorem of calculus is so useful that you should just know it intuitively.

First Fundamental Theorem of Calculus

Let $f$ be a continuous real-valued function defined on $[a, b]$ , and let $F$ be defined by:

F : [a, b] \to \myReals \qquad F(x) = \int_a^x{f(t) \,\diff{t}} .

$F$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and has a derivative $F'$ given by:

F'(x) = f(x) \qquad \forall x \in (a, b) .

Second Fundamental Theorem of Calculus

Let $F$ be a real-valued function on $[a, b]$ , and $F$ be an antiderivative of $f$ on $[a, b]$ . Then:

\int_a^b{f(t) \,\diff{t}} = F(b) - F(a)

TODO: Look into this more. I’m not 100% certain on details such as whether the second fundamental theorem requires

f

to be continuous, and whether or not

f

being Riemann integrable is significant here.

Appendix: Discussion

Completing the Square

Suppose we have a quadratic function:

f(x) = a x^2 + b x + c

We want to express the same quadratic in the form:

f(x) = A(x + B)^2 + C

We can expand this form, then equate coefficients:

f(x) = A x^2 + 2ABx + A B^2 + C

A = a \qquad 2AB = b \qquad A B^2 + C = c

Solving for $A$ , $B$ , and $C$ , we get our function:

f(x) = a \parens{x + \brackets{\frac{b}{2a}}}^2 + \brackets{c - \frac{b^2}{4a}}

Thus, we have our general form for completing the square.

However, it can be a bit unwieldy. If we instead assume we have a monic (i.e. $a=1$ ), then:

f(x) = \parens{x + \xmemphR{\brackets{\frac{b}{2}}}}^2 + c - \xmemphP{\brackets{\frac{b}{2}}^2}

This form is easier for me to memorize since you just remember where $\memphR{\frac{b}{2}}$ and its square appears.

Though, if one forgets the square, you can guess the form of the outside by expanding $\parens{x + \memphR{\frac{b}{2}}}^2$ :

\begin{align*} f(x) &= \parens{x + \xmemphR{\brackets{\frac{b}{2}}}}^2 + c + \text{something} \\ &= \parens{x^2 + 2 \xmemphR{\brackets{\frac{b}{2}}} + \xmemphP{\brackets{\frac{b}{2}}^2}} + c + \text{something} \end{align*}

This something must be such that the constant doesn’t change:

f(x) = \parens{x^2 + 2 \xmemphR{\brackets{\frac{b}{2}}} + \xmemphP{\brackets{\frac{b}{2}}^2}} + c - \xmemphP{\brackets{\frac{b}{2}}^2}