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\begin{document}
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\stepcounter{section}
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{\newpage
\clearpage
\samepage \begin{equation}f\left(x\right) = f\left(x_0\right) + \left(x - x_0\right) f^{'}{\mid}_{x_0} + {1 \over 2} \left(x - x_0\right)^2 f^{''}{\mid}_{x_0}
 + {1 \over {3 !}}\left(x - x_0\right)^3 f^{'''}{\mid}_{x_0} + \ \ldots
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}f\left(x_0 \pm dx \right) = f\left(x_0\right) \pm  dx f^{'}{\mid}_{x_0} + {1 \over 2} dx^2 f^{''}{\mid}_{x_0}
 \pm  {1 \over {3 !}} dx^3 f^{'''}{\mid}_{x_0} + \ \ldots
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}f_{i+1} - f_{i}  = dx f^{'}{\mid}_{x_i}  +  {1 \over 2} dx^2 f^{''}{\mid}_{x_i} + {1 \over {3 !}}dx^3 f^{'''}{\mid}_{x_i}
+ \ldots
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}f^{'}{\mid}_{x_i} =  {{f_{i+1} - f_{i}} \over {dx}}  - {1 \over 2} dx f^{''}{\mid}_{x_i} + \ldots
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}f^{'}{\mid}_{x_i} =  {{f_{i} - f_{i-1}} \over {dx}}  + {1 \over 2} dx f^{''}{\mid}_{x_i} + \ldots
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}f_{i+1} - f_{i-1}  =  2 dx f^{'}{\mid}_{x_i}  +  { 2 \over {3!} } dx^3 f^{'''}{\mid}_{x_i} + \ldots
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}f^{'}{\mid}_{x_i} = {{f_{i+1} - f_{i-1}} \over {2 dx}} - { 2 \over {3!} } dx^2 f^{'''}{\mid}_{x_i} + \ldots
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}f_1 = f_0 + dx f^{'}{\mid}_{x_0} + {1 \over 2} dx^2 f^{''}{\mid}_{x_0}
 +  {1 \over 6} dx^3 f^{'''}{\mid}_{x_0} + \ \ldots
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}f_2 = f_0 + 2 dx f^{'}{\mid}_{x_0} + 2 dx^2 f^{''}{\mid}_{x_0}
 +  {8 \over 6} dx^3 f^{'''}{\mid}_{x_0} + \ \ldots
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}4 f_1 - f_2 = 3 f_0 + 2 dx f^{'}{\mid}_{x_0} - {2 \over 3} dx^3 f^{'''}{\mid}_{x_0} + \ldots
 \end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}f^{'}{\mid}_{x_0} =  {{4 f_1 -3 f_0 - f_2} \over { 2 dx }}  -  {1 \over 3} dx^2 f^{'''}{\mid}_{x_0} + \ldots
 \end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}f_{i+1} + f_{i-1}  =  2 f_i +  dx^2 f^{''}{\mid}_{x_i} +  {1 \over 12} dx^4 f^{''''}{\mid}_{x_i}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}f^{''}{\mid}_{x_i}  =  {{f_{i+1} -2 f_i + f_{i-1} } \over { dx^2 }} - {1 \over 12} dx^2 f^{''''}{\mid}_{x_i} + \ldots
\end{equation}
}


\end{document}
