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\begin{document}
	
	\mbox{}
	\title{Problem Set 10: Curve Sketching}
	
	\maketitle
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\section{Introduction} \label{sec:intro}
	\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
	
%	\begin{problem}
%		Find the equation of a tangent line to the graph of the function $f(x) = x^2 + 4x + 3$ passing by the point $P = (0,-2)$.
%	\end{problem}

	\begin{problem}
		Sketch the graph of the following functions showing vertical and horizontal asymptotes, intervals of increase/decrease, local maximum and minimum values, intervals of concavity, and inflection points.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $f(x) = 1 + 1/x -1/x^2$;
			\item $y = e^{-x^2}$;
			\item $g(x) = x - \frac{1}{6} x^2 - \frac 23 \ln x$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Show that the curve $(1+x)/(1+x^2)$ has three points of inflection and they all lie on one straight line.
	\end{problem}

\begin{problem}
	Sketch the graph of the following functions showing domain, intercepts, symmetry, asymptotes, intervals of increase/decrease, local maximum and minimum values, intervals of concavity, and inflection points.
	\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
		\item $f(x) = \frac{\sin x}{1 + \cos x}$;
		\item $y = \ln(1 + x^3)$;
		\item $g(x) = e^{\arctan x}$.
	\end{enumerate}
\end{problem}
	
	%\vspace{20pt}
	
	\begin{thebibliography}{20}
		
		\bibitem{jS15} J.~Stewart: \emph{Single Variable Calculus} 8th Edition, Cengage Learning, Boston 2015.
		
	\end{thebibliography}
	
\end{document}