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\begin{document}
	
	\mbox{}
	\title{Problem Set 8: Rates of Change and Differentials}
	
	\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}
		A bacteria culture initially contains $100$ cells and grows at a rate proportional to its size. After an hour the population has increased to $420$.
		\begin{enumerate}
			\item Find an expression for the number of bacteria after $t$ hours.
			\item Find the number of bacteria after $3$ hours.
			\item Find the rate of growth after $3$ hours.
			\item When will the population reach $10000$?
		\end{enumerate}.
	\end{problem}

	\begin{problem}
		Strontium-$90$ has a half-life of $28$ days.
		\begin{enumerate}
			\item A sample has a mass of $50$ mg initially. Find a formula for the mass remaining after $t$ days.
			\item Find the mass remaining after $40$ days.
			\item How long does it take the sample to decay to a mass of $2$ mg?
		\end{enumerate}
	\end{problem}


	\begin{problem}
		A plane flying horizontally at an altitude of $1 mi$ and a speed of $500$ \emph{mi/h} passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 \emph{mi} away from the station.
	\end{problem}
	
	\begin{problem}
		If a snowball melts so that its surface area decreases at a rate of $1$~\emph{cm}$^2/$\emph{min}, find the rate at which the diameter decreases when the diameter is $10$ \emph{cm}.
	\end{problem}

	\begin{problem}
		A street light is mounted at the top of a $15$-\emph{ft}-tall pole. A man $6$ \emph{ft} tall walks away from the pole with a speed of $5$ \emph{ft/s} along a straight path. How fast is the tip of his shadow moving when he is $40$ \emph{ft} from the pole?
	\end{problem}
	
	\begin{problem}
		At noon, ship A is $150$ \emph{km} west of ship B. Ship A is sailing east at $35$ \emph{km/h} and ship B is sailing north at $25$ \emph{km/h}. How fast is the distance between the ships changing at $4:00$ \emph{pm}?
	\end{problem}

%	\begin{problem}
%		Two people start from the same point. One walks east at $3$ \emph{mi/h} and the other walks northeast at $2$ \emph{mi/h}. How fast is the distance between the people changing after 15 minutes?
%	\end{problem}

	\begin{problem}
		Find the linearization $L(x)$ of the functions
		\[
			f(x) = \sin x \text{ at } \pi/6 \quad \text{and} \quad g(x) = \sqrt{x} \text{ at } 4. 
		\]
	\end{problem}

	\begin{problem}
		Find the differential of the functions $f(x) = \ln(\sin x)$ and $g(x) = \frac{e^x}{1 - e^x}$.
	\end{problem}
	
	\begin{problem}
		 Estimate $\sqrt[4]{16.1}$ and $\cos 29.5^\circ$. 
	\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}