Formula Sheet

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\begin{center}
  {\Large\bfseries\color{accent} Formula Sheet}\quad
  {\large MATH 2410 Final Exam}\\[2pt]
  \small Provided with the exam. No other notes permitted.
\end{center}

\begin{multicols}{2}
\small

\begin{fgroup}{Differentiation}
\begin{align*}
  (fg)' &= f'g + fg' \\
  \Bigl(\frac{f}{g}\Bigr)' &= \frac{f'g - fg'}{g^2} \\
  \frac{d}{dx} f(g(x)) &= f'(g(x))\,g'(x) \\
  \frac{d}{dx}\arctan x &= \frac{1}{1 + x^2}
\end{align*}
\end{fgroup}

\begin{fgroup}{Integration}
\begin{align*}
  \int u \, dv &= uv - \int v \, du \\
  \int \frac{dx}{a^2 + x^2} &= \frac{1}{a}\arctan\frac{x}{a} + C \\
  \int \frac{dx}{\sqrt{a^2 - x^2}} &= \arcsin\frac{x}{a} + C \\
  \int \sec x \, dx &= \ln\lvert \sec x + \tan x \rvert + C
\end{align*}
\end{fgroup}

\begin{fgroup}{Sequences and series}
\begin{align*}
  \sum_{k=0}^{n-1} ar^k &= a\,\frac{1 - r^n}{1 - r} \quad (r \neq 1) \\
  e^{x} &= \sum_{k=0}^{\infty} \frac{x^k}{k!} \\
  \frac{1}{1 - x} &= \sum_{k=0}^{\infty} x^k \quad (\lvert x \rvert < 1)
\end{align*}
Ratio test: $\lim \bigl|\tfrac{a_{n+1}}{a_n}\bigr| = L$; converges if
$L < 1$, diverges if $L > 1$.
\end{fgroup}

\begin{fgroup}{Trigonometry}
\begin{align*}
  \sin^2\theta + \cos^2\theta &= 1 \\
  \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 \\
  \sin^2\theta = \tfrac{1 - \cos 2\theta}{2}, &\quad
  \cos^2\theta = \tfrac{1 + \cos 2\theta}{2}
\end{align*}
\end{fgroup}

\begin{fgroup}{Linear algebra}
\[
  \det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc,
  \qquad
  \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1}
  = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]
$Av = \lambda v$ with $v \neq 0$ iff $\det(A - \lambda I) = 0$.\\
$\operatorname{tr} A = \sum_i \lambda_i$, \quad
$\det A = \prod_i \lambda_i$.
\end{fgroup}

\begin{fgroup}{Vectors}
\begin{align*}
  u \cdot v &= \lVert u \rVert\, \lVert v \rVert \cos\theta \\
  \operatorname{proj}_v u &= \frac{u \cdot v}{v \cdot v}\, v \\
  \lVert u \times v \rVert &= \lVert u \rVert\, \lVert v \rVert \sin\theta
\end{align*}
\end{fgroup}

\begin{fgroup}{Probability}
\begin{align*}
  P(A \mid B) &= \frac{P(A \cap B)}{P(B)} \\
  P(A \mid B) &= \frac{P(B \mid A)\,P(A)}{P(B)} \\
  \operatorname{Var} X &= \mathbb{E}[X^2] - (\mathbb{E}X)^2
\end{align*}
Normal density:
$f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}\,
e^{-(x - \mu)^2 / (2\sigma^2)}$.
\end{fgroup}

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