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\begin{document}
	
	\mbox{}
	\title{Problem Set 5: Notion and Formal Definition of Derivatives}
	
	\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}
		Find and equation of the tangent line to the curve at the given point.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $y = 4x - 3x^2, \quad (2,-4)$;
			\item $y = \sqrt{x}, \quad (1,1)$;
			\item $y = \frac{2x + 1}{x+2}, \quad (1,1)$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Sketch the graph of a function $f$ where the domain is $(-2,2)$, $f'(0) = -2$, $\lim_{x \to 2^-} f(x) = \infty$, $f$ is continuous at all points in its domain except $\pm 1$, and $f$ is odd.
	\end{problem}

	\begin{problem}
		If $f(x) = 3x^2 - x^3$, find $f'(1)$ and use to find an equation of the tangent line to the curve $y = 3x^2 - x^3$ at the point $(1,2)$.
	\end{problem}

	\begin{problem}
		Find $f'(a)$ by using the definition of derivative.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $f(x) = 2x^3 + x$;
			\item $f(x) = x^{-2}$;
			\item $f(x) = \frac{4}{\sqrt{1-x}}$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Find the derivative of the function $f(x)$ using the definition of derivative.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $f(x) = 4 + 8x - 5x^2$;
			\item $f(x) = 1/\sqrt{x}$;
			\item $f(x) = x^4$;
			\item $f(x) = \frac{1-2x}{3 + x}$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Show that the function $f(x) = |x-6|$ is not differentiable at $6$. Find a formula for $f'$ and sketch its graph.
	\end{problem}

	\begin{problem}
		Sketch the graph of the function $f(x) = x |x|$, and find the values of $x$ where $f$ is differentiable. Find a formula for $f'$.
	\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}