wip midterm

midterm/midterm.tex

@@ -336,4 +336,24 @@ x =
  0 \\ 0 \\ 0 \\ 0 \\ 0 \\
  \end{bmatrix} 
  \]
+ 
+\section*{Question 5}
+\subsection*{Point a)}
+$$f(x) = x \hspace{2cm} K_{abs} = |f'(x)| = 1 \hspace{2cm} K_{rel} = \left|\frac{1 \cdot x}{x}\right| = 1$$
+
+\subsection*{Point b)}
+$$f(x) = \sqrt[3]{x} \hspace{2cm} K_{abs} = |f'(x)| = \frac{1}{3\sqrt[3]{x^2}} \hspace{2cm}
+ K_{rel} = \left|\frac{1}{3\sqrt[3]{x^2}} \cdot \frac{x}{\sqrt[3]{x}}\right| = \frac{1}{3}$$
+
+ \subsection*{Point c)}
+$$f(x) = \frac{1}{x} \hspace{2cm} K_{abs} = |f'(x)| = \frac{1}{x^2} \hspace{2cm}
+ K_{rel} = \left|\frac{-x}{x^2} \cdot \frac{1}{\frac{1}{x}}\right| = 1$$
+ 
+  \subsection*{Point d)}
+$$f(x) = e^x \hspace{2cm} K_{abs} = |f'(x)| = e^x \hspace{2cm}
+ K_{rel} = \left|\frac{xe^x}{e^x}\right| = |x|$$
+ 
+ \subsection*{Point e)}
+ Cases \textit{a)},\textit{b)} and \textit{c)} are well-conditioned for any $x$ since their $K_rel$
+ is not defined by x. Case \textit{d)} is well-conditioned only for $x$s whose absolute value is in the order of magnitude of $1$ or less, since $K_rel$ in this case is exactly $|x|$.
 \end{document}