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<!DOCTYPE html>
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<title>Mathy</title>
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<div id='editor'>\text{add latex here}</div>
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Examples
</li>
<li>
<a
href='#'
class='example'
data-text='
\lim_{x \to \infty}{\frac{1}{x^2}}
'
>1. Some Limit</a>
</li>
<!-- li>
<a
href='#'
class='example'
data-text='
\begin{align*}
&= \int_0^2{x^2 + 5x + 2}\,\text{d}x \\
&= \int{x^2 + 5x + 2}\,\text{d}x \\
&= \frac{1}{3}x^3 + \frac{5}{2}x^2 + 2x + c \\
&= \frac{1}{3}2^3 + \frac{5}{2}2^2 + 2\times2 -
(\frac{1}{3}0^3 + \frac{5}{2}0^2 + 2\times0) \\
&= \frac{8}{3} + \frac{20}{2} + 4 \\
&= 14 + \frac{8}{3} \\
&= \frac{50}{3}
\end{align*}
'
>Test 2</a>
</li -->
<li>
<a
href='#'
class='example'
data-text='
\text{Quadratic Formula} \\
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
'
>2. Quadratic Formula</a>
</li>
<li>
<a
href='#'
class='example'
data-text='
\text{Cross Product via Matrix Determinant} \\
\begin{align*}
\vec{a} \times \vec{b} & = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
\vec{a}_x & \vec{a}_y & \vec{a}_z \\
\vec{b}_x & \vec{b}_y & \vec{b}_z
\end{vmatrix} \\
& =
\begin{vmatrix}
\vec{a}_y & \vec{a}_z \\
\vec{b}_y & \vec{b}_z
\end{vmatrix} \hat{i} -
\begin{vmatrix}
\vec{a}_x & \vec{a}_z \\
\vec{b}_x & \vec{b}_z
\end{vmatrix} \hat{j} +
\begin{vmatrix}
\vec{a}_x & \vec{a}_y \\
\vec{b}_x & \vec{b}_y
\end{vmatrix} \hat{k} \\
& =
(\vec{a}_y\vec{b}_z - \vec{a}_z\vec{b}_y)\hat{i} -
(\vec{a}_x\vec{b}_z - \vec{a}_z\vec{b}_x)\hat{j} +
(\vec{a}_x\vec{b}_y - \vec{a}_x\vec{b}_y)\hat{k} \\
& =
\langle
\vec{a}_y\vec{b}_z - \vec{a}_z\vec{b}_y\text{, }
\vec{a}_x\vec{b}_z - \vec{a}_z\vec{b}_x\text{, }
\vec{a}_x\vec{b}_y - \vec{a}_y\vec{b}_z
\rangle \\
\vec{a} & = \langle2, 1, -1\rangle \\
\vec{b} & = \langle-3, 4, 1\rangle \\
\vec{a} \times \vec{b} & = \langle
(1)(1) - (-1)(4),
(2)(1) - (-1)(-3),
(2)(4) - (1)(-3)
\rangle \\
& = \langle
5, 5, 11
\rangle \\
\vec{b} \times \vec{a} & = \langle
(4)(-1) - (1)(1),
(-3)(-1) - (1)(2),
(-3)(1) - (4)(2)
\rangle \\
& = \langle
-5, -5, -11
\rangle
\end{align*}
'
>3. Cross Product</a>
</li>
<li>
<a
href='#'
class='example'
data-text="
\text{Derivative Rules} \\
\begin{align*}
% sum/difference rule
\frac{\text{d}}{\text{d}x} \,
f(x) \pm g(x) &=
\frac{\text{d}}{\text{d}x} \, f(x) \pm
\frac{\text{d}}{\text{d}x} \, g(x) &
\text{Sum/Difference Rule} \\
% constant factor rule
\frac{\text{d}}{\text{d}x} \,
k f(x) &=
k \frac{\text{d}}{\text{d}x} \, f(x) &
\text{Constant Factor Rule} \\
% constant rule
\frac{\text{d}}{\text{d}x} \,
k &=
0 &
\text{Constant Rule} \\
% power rule
\frac{\text{d}}{\text{d}x} \,
x^n &=
nx^{n-1} &
\text{Power Rule} \\
% exponent rule
\frac{\text{d}}{\text{d}x} \,
b^x &=
b^xln(b) &
\text{Exponent Rule} \\
% chain rule
\frac{\text{d}}{\text{d}x} \,
f(g(x)) &=
% (f \cdot g)(x) &=
f'(g(x))g'(x) &
\text{Chain Rule} \\
% product rule
\frac{\text{d}}{\text{d}x} \,
f(x)g(x) &=
f'(x)g(x) + f(x)g'(x) &
\text{Product Rule} \\
% quotient rule
\frac{\text{d}}{\text{d}x} \,
\frac{f(x)}{g(x)} &=
\frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} &
\text{Quotient Rule} \\
% logarithm rule
\frac{\text{d}}{\text{d}x} \,
log_b{x} &=
\frac{1}{x ln(b)} &
\text{Logarithm Rule} \\
\end{align*}
"
>4. Derivative Rules</a>
</li>
<li>
<a
href='#'
class='example'
data-text="
% https://math.vanderbilt.edu/schectex/courses/cubic/
\text{The Cubic Formula} \\
\begin{align*}
% first term
x &= \sqrt[3]{
% first term, first subterm
\left (
\frac{-b^3}{27a^3} +
\frac{bc}{6a^2} -
\frac{d}{2a}
\right )
+
\sqrt{
% first term, second subterm
\left (
\frac{-b^3}{27a^3} +
\frac{bc}{6a^2} -
\frac{d}{2a}
\right )^2
+
% first term, third subterm
\left (
\frac{c}{3a} -
\frac{b^2}{9a^2}
\right )^3
}
} \\
&+
% second term
\sqrt[3]{
% first term, second subterm
\left (
\frac{-b^3}{27a^3} +
\frac{bc}{6a^2} -
\frac{d}{2a}
\right )
-
\sqrt{
% second term, second subterm
\left (
\frac{-b^3}{27a^3} +
\frac{bc}{6a^2} -
\frac{d}{2a}
\right )^2
+
% second term, third subterm
\left (
\frac{c}{3a} -
\frac{b^2}{9a^2}
\right )^3
}
} \\
&-
% third part
\frac{b}{3a}
\end{align*}
"
>5. Cubic Formula</a>
</li>
<li>
<a
href='#'
class='example'
data-text="
\text{Linear Regression} \\
\begin{align*}
m &= \frac{
\sum(x_i - \bar{x})(y_i - \bar{y} )
}{
\sum(x_i - \bar{x})^2
} \\
b &= \bar{y} - m\bar{x} \\
y &= mx + b
\end{align*}
"
>6. Linear Regression</a>
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