Math Cheatsheet Template

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\begin{document}
\begin{multicols}{4}

\section{Calculus}

\subsection{Derivatives}
$(x^n)' = nx^{n-1}$\\
$(\sin x)' = \cos x$\\
$(\cos x)' = -\sin x$\\
$(\tan x)' = \sec^2 x$\\
$(e^x)' = e^x$\\
$(\ln x)' = \frac{1}{x}$

\textbf{Chain rule:} $(f(g(x)))' = f'(g(x)) \cdot g'(x)$

\textbf{Product rule:} $(fg)' = f'g + fg'$

\textbf{Quotient rule:} $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

\subsection{Integrals}
$\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)\\
$\int \frac{1}{x} dx = \ln|x| + C$\\
$\int e^x dx = e^x + C$\\
$\int \sin x dx = -\cos x + C$\\
$\int \cos x dx = \sin x + C$

\textbf{Integration by parts:}
$$\int u dv = uv - \int v du$$

\textbf{Substitution:}
$$\int f(g(x))g'(x)dx = \int f(u)du$$

\subsection{Taylor Series}
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

$$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$$

$$\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$$

\section{Linear Algebra}

\subsection{Matrix Operations}
\textbf{Determinant (2×2):}
$$\det\begin{pmatrix}a & b\\c & d\end{pmatrix} = ad - bc$$

\textbf{Inverse (2×2):}
$$A^{-1} = \frac{1}{\det A}\begin{pmatrix}d & -b\\-c & a\end{pmatrix}$$

\textbf{Eigenvalues:} $\det(A - \lambda I) = 0$

\textbf{Trace:} $\text{tr}(A) = \sum_{i=1}^n a_{ii} = \sum \lambda_i$

\subsection{Vector Spaces}
\textbf{Dot product:} $\vect{a} \cdot \vect{b} = |\vect{a}||\vect{b}|\cos\theta$

\textbf{Cross product:}
$$\vect{a} \times \vect{b} = \begin{vmatrix}\vect{i} & \vect{j} & \vect{k}\\a_1 & a_2 & a_3\\b_1 & b_2 & b_3\end{vmatrix}$$

$|\vect{a} \times \vect{b}| = |\vect{a}||\vect{b}|\sin\theta$

\textbf{Projection:} $\text{proj}_{\vect{b}}\vect{a} = \frac{\vect{a} \cdot \vect{b}}{|\vect{b}|^2}\vect{b}$

\section{Probability}

\subsection{Basics}
$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

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

\textbf{Bayes' theorem:}
$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$

\subsection{Distributions}
\textbf{Binomial:} $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$\\
$E[X] = np$, $\text{Var}(X) = np(1-p)$

\textbf{Poisson:} $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$\\
$E[X] = \lambda$, $\text{Var}(X) = \lambda$

\textbf{Normal:} $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$\\
$E[X] = \mu$, $\text{Var}(X) = \sigma^2$

\textbf{Exponential:} $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$\\
$E[X] = \frac{1}{\lambda}$, $\text{Var}(X) = \frac{1}{\lambda^2}$

\section{Trigonometry}

\subsection{Identities}
$\sin^2 x + \cos^2 x = 1$\\
$1 + \tan^2 x = \sec^2 x$\\
$1 + \cot^2 x = \csc^2 x$

\textbf{Double angle:}\\
$\sin(2x) = 2\sin x \cos x$\\
$\cos(2x) = \cos^2 x - \sin^2 x$\\
$\cos(2x) = 2\cos^2 x - 1 = 1 - 2\sin^2 x$

\textbf{Sum formulas:}\\
$\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$\\
$\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$

\subsection{Special Values}
\begin{tabular}{c|c|c|c}
$x$ & $\sin x$ & $\cos x$ & $\tan x$ \\\hline
$0$ & $0$ & $1$ & $0$ \\
$\pi/6$ & $1/2$ & $\sqrt{3}/2$ & $1/\sqrt{3}$ \\
$\pi/4$ & $\sqrt{2}/2$ & $\sqrt{2}/2$ & $1$ \\
$\pi/3$ & $\sqrt{3}/2$ & $1/2$ & $\sqrt{3}$ \\
$\pi/2$ & $1$ & $0$ & $\infty$ \\
\end{tabular}

\section{Complex Numbers}

\textbf{Euler's formula:} $e^{i\theta} = \cos\theta + i\sin\theta$

$e^{i\pi} + 1 = 0$

\textbf{Polar form:} $z = re^{i\theta}$ where $r = |z|$, $\theta = \arg(z)$

\textbf{De Moivre:} $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$

\section{Series \& Sequences}

\textbf{Geometric series:}
$$\sum_{k=0}^{\infty} ar^k = \frac{a}{1-r} \text{ for } |r| < 1$$

\textbf{Arithmetic series:}
$$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$$

$$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$

\textbf{Convergence tests:}\\
Ratio: $L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$ (conv. if $L<1$)

Root: $L = \lim_{n\to\infty}\sqrt[n]{|a_n|}$ (conv. if $L<1$)

\section{Differential Equations}

\textbf{Separable:} $\frac{dy}{dx} = f(x)g(y)$\\
$\Rightarrow \int \frac{dy}{g(y)} = \int f(x)dx$

\textbf{Linear 1st order:} $y' + P(x)y = Q(x)$\\
$\mu(x) = e^{\int P(x)dx}$\\
$y = \frac{1}{\mu(x)}\left(\int \mu(x)Q(x)dx + C\right)$

\textbf{2nd order homogeneous:} $ay'' + by' + cy = 0$\\
Char. eq: $ar^2 + br + c = 0$\\
If $r_1 \neq r_2$ real: $y = c_1e^{r_1x} + c_2e^{r_2x}$\\
If $r_1 = r_2 = r$: $y = (c_1 + c_2x)e^{rx}$\\
If $r = \alpha \pm \beta i$: $y = e^{\alpha x}(c_1\cos\beta x + c_2\sin\beta x)$

\end{multicols}
\end{document}