\input{tex/preamble}
\usepackage{lipsum}
\usepackage{showframe}

\begin{document}
    \lipsum[1][1-4]
    
    \begin{equation}
	   a = \lim_{t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt} = \dot{v}
    \end{equation}
    
    \lipsum[1][1-4]

    \begin{equation}
	   J = p_2 - p_1 = m \left( v_2 - v_1 \right) = \int_{t_1}^{t_2} F \, dt
    \end{equation}

    \lipsum[1][1-4]

    \begin{equation}
	   F = \frac{d p}{dt} = \frac{d (mv)}{dt} =  m \frac{dv}{dt} = ma
    \end{equation}

    \lipsum[1][1-4]

    \begin{equation}
	   A = \pi r^2 = \frac{\pi d^2}{4}
    \end{equation}

    \lipsum[1][1-4]

    \begin{equation}
	   f'(x) = \left[ \frac{g(x)}{h(x)} \right]' = \frac{g'(x) \, h(x) - g(x) \, h'(x)}{[h(x)]^2}
    \end{equation}

    \lipsum[2]

    \begin{equation}
	   \int_a^b f(x) \, dx \approx \frac{b-a}{6} \left[ f(a) + 4f \left( \frac{a+b}{2} \right) + f(b) \right]
    \end{equation}

    \lipsum[1][1-4]

    \begin{equation}
	   \varphi = \frac{1+\sqrt{5}}{2} \approx 1.61803398874989...
    \end{equation}

    \lipsum[1][1-4]

    \begin{equation}
	   e \approx 2.718281828459045...
    \end{equation}

    \lipsum[1][1-4]

    \begin{align}
	   \sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \dotso \\
	&= \sum_{n=0}^\infty (-1)^n \frac{x^{2n + 1}}{(2n + 1)!}
    \end{align}

    \lipsum[1]

    \begin{equation}
	   N_\text{A} \approx 6.02214129 \times 10^{23} \frac{1}{\text{mol}} \left( 0.00000027 \times 10^{23} \frac{1}{\text{mol}} \right)
    \end{equation}

    \lipsum[1][1-2]

    \begin{align*}
        \frac{\sin(a+b)}{\cos(a+b)}&=\frac{\sin a \cos b + \cos a \sin b}{\cos a \cos b - \sin a \sin b}
        \\
        \tan(a+b)&=\frac{\sin a \cos b + \cos a \sin b}{\cos a \cos b - \sin a \sin b}
        \\
        &=\frac{\frac{\sin a \cos b}{\cos a \cos b} + \frac{\cos a \sin b}{\cos a \cos b}}{\frac{\cos a \cos b}{\cos a \cos b} - \frac{\sin a \sin b}{\cos a \cos b}}
        \\
        &=\frac{\frac{\sin a}{\cos a} + \frac{\sin b}{\cos b}}{1 - \frac{\sin a \sin b}{\cos a \cos b}}
        \\
        &=\frac{\tan a +\tan b }{1 - \tan a \tan b}
    \end{align*}

    \lipsum[1][1-2]

    \begin{equation*}
        a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}}
    \end{equation*}

    \lipsum[1][1-3]
\end{document}