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\title{Homework 12}
\author{Menyoung Lee}
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\begin{document}
\maketitle
\subsection*{CP6}
\subsubsection*{a}
\subsubsection*{b}
\subsubsection*{c}

\subsection*{CP1, 7, 2, 8}
\begin{verbatim}
trap(f,a,b,n) = (b-a)/n * sum(f(a:(b-a)/n:b)) - (b-a)/(2*n) * (f(a) + f(b)).
rect(f,a,b,n) = (b-a)/n * sum(f(a+(b-a)/(2*n):(b-a)/n:b-(b-a)/(2*n))).
simp(f,a,b,n) = (2*rect(f,a,b,n)+trap(f,a,b,n))/3.
\end{verbatim}
\subsubsection*{a}
\[f(x) = \frac{x}{\sqrt{x^2+9}\]
\[\int_0^4 f(x) = 2.\]
\paragraph*{1}
\[Trap_16 = \frac{4}{16}\sum_{0}^{16}{f(x_i) - \frac{4}{32}(f(0)+f(4)) = 1.99863818147028, \epsilon = 1.36\times10^{-3}\]
\[Trap_32 = \frac{4}{32}\sum_{0}^{16}{f(x_i) - \frac{4}{64}(f(0)+f(4)) = 1.99965967807791, \epsilon = 3.40\times10^{-4}\]
\paragraph*{7}
\paragraph*{2}
\paragraph*{8}
\subsubsection*{b}
\[f(x) = \frac{x^3}{\sqrt{x^2+1}\]
\paragraph*{1}
\[Trap_16 = \frac{4}{16}\sum_{0}^{16}{f(x_i) - \frac{4}{32}(f(0)+f(4)) = 1.99863818147028, \epsilon = 1.36\times10^{-3}\]
\[Trap_32 = \frac{4}{32}\sum_{0}^{16}{f(x_i) - \frac{4}{64}(f(0)+f(4)) = 1.99965967807791, \epsilon = 3.40\times10^{-4}\]
\paragraph*{7}
\paragraph*{2}
\paragraph*{8}
\subsubsection*{c}
\[f(x) = x\exp{x}\]
\paragraph*{1}
\[Trap_16 = \frac{4}{16}\sum_{0}^{16}{f(x_i) - \frac{4}{32}(f(0)+f(4)) = 1.99863818147028, \epsilon = 1.36\times10^{-3}\]
\[Trap_32 = \frac{4}{32}\sum_{0}^{16}{f(x_i) - \frac{4}{64}(f(0)+f(4)) = 1.99965967807791, \epsilon = 3.40\times10^{-4}\]
\paragraph*{7}
\paragraph*{2}
\paragraph*{8}
\subsubsection*{d}
\[f(x) = x^2\ln{x}\]
\paragraph*{1}
\[Trap_16 = \frac{4}{16}\sum_{0}^{16}{f(x_i) - \frac{4}{32}(f(0)+f(4)) = 1.99863818147028, \epsilon = 1.36\times10^{-3}\]
\[Trap_32 = \frac{4}{32}\sum_{0}^{16}{f(x_i) - \frac{4}{64}(f(0)+f(4)) = 1.99965967807791, \epsilon = 3.40\times10^{-4}\]
\paragraph*{7}
\paragraph*{2}
\paragraph*{8}
\subsubsection*{e}
\[f(x) = x^2\sin{x}\]
\paragraph*{1}
\[Trap_16 = \frac{4}{16}\sum_{0}^{16}{f(x_i) - \frac{4}{32}(f(0)+f(4)) = 1.99863818147028, \epsilon = 1.36\times10^{-3}\]
\[Trap_32 = \frac{4}{32}\sum_{0}^{16}{f(x_i) - \frac{4}{64}(f(0)+f(4)) = 1.99965967807791, \epsilon = 3.40\times10^{-4}\]
\paragraph*{7}
\paragraph*{2}
\paragraph*{8}
\subsubsection*{f}
\[f(x) = \frac{x^3}{\sqrt{x^4-1}\]
\paragraph*{1}
\[Trap_16 = \frac{4}{16}\sum_{0}^{16}{f(x_i) - \frac{4}{32}(f(0)+f(4)) = 1.99863818147028, \epsilon = 1.36\times10^{-3}\]
\[Trap_32 = \frac{4}{32}\sum_{0}^{16}{f(x_i) - \frac{4}{64}(f(0)+f(4)) = 1.99965967807791, \epsilon = 3.40\times10^{-4}\]
\paragraph*{7}
\paragraph*{2}
\paragraph*{8}
\subsubsection*{g}
\[f(x) = \frac{1}{\sqrt{x^2+4}\]
\paragraph*{1}
\[Trap_16 = \frac{4}{16}\sum_{0}^{16}{f(x_i) - \frac{4}{32}(f(0)+f(4)) = 1.99863818147028, \epsilon = 1.36\times10^{-3}\]
\[Trap_32 = \frac{4}{32}\sum_{0}^{16}{f(x_i) - \frac{4}{64}(f(0)+f(4)) = 1.99965967807791, \epsilon = 3.40\times10^{-4}\]
\paragraph*{7}
\paragraph*{2}
\paragraph*{8}
\subsubsection*{h}
\[f(x) = \frac{x}{\sqrt{x^4+1}\]
\paragraph*{1}
\[Trap_16 = \frac{4}{16}\sum_{0}^{16}{f(x_i) - \frac{4}{32}(f(0)+f(4)) = 1.99863818147028, \epsilon = 1.36\times10^{-3}\]
\[Trap_32 = \frac{4}{32}\sum_{0}^{16}{f(x_i) - \frac{4}{64}(f(0)+f(4)) = 1.99965967807791, \epsilon = 3.40\times10^{-4}\]
\paragraph*{7}
\paragraph*{2}
\paragraph*{8}
\end{document}