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\title[Problem Sets 1 \& 2]{Problem Sets 1 \& 2.  Due Thursday 7 September}

\begin{document}

\maketitle



\centerline{\sc Problem Set 1.  Problems from Lecture 1.}

\noindent {\bf Reading.} {\em Quick Calculus}, Chapter 1.

\noindent {\bf Supplementary reading.} Simmons, Chapter 1.

\begin{enumerate}

\item Given a quadratic equation of the from $ax^2+bx+c=0$, we can solve for $x$ using the formula
$$
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.
$$
\noindent Using the above formula, solve the equation $4x^2-5x-6=0$ for $x$.

\medskip

\item Find $f(x)$ if $f(x+1)=x^2-5x+3$.
\medskip


\item Graph the following functions, and give their domain and range.
\begin{enumerate}
\item $y=3x-2$.
\medskip
\item $y=4x^2+3$.
\end{enumerate}
\medskip


\item Graph the following functions, and give their domain and range.
\begin{enumerate}
\item $y=2^x$.
\medskip
\item $y=2^{x+3}$.
\end{enumerate}
\medskip

\item Graph the following functions, and give their domain and range.
\begin{enumerate}
\item $y=log_2(x)$.
\medskip
\item $y=log_2(x)+5$.
\end{enumerate}
\medskip

\item Graph the following functions, and give their domain and range.
\begin{enumerate}
\item $y=\sin(x)$.
\medskip
\item $y=\sin(2x)$.
\end{enumerate}
\medskip

\item Graph the following functions, and give their domain and range.
\begin{enumerate}
\item $y=\tan(x)$.
\medskip
\item $y=4\tan(x)$.
\end{enumerate}
\medskip

\item Simplify the following expressions.
\begin{enumerate}
\item $\log_{10}(\frac{x+y}{z})$.
\medskip
\item $25^{\log_{25}(x+y)+\log_5(\frac{x}{y})}$.
\end{enumerate}

\medskip

\item Make the following computations using right triangles.
\begin{enumerate}
\item For $\theta=\frac{\pi}{4}=45^{\circ}$, compute $\sin(\theta)$,  $\cos(\theta)$, and $\tan(\theta)$.
\medskip

\item For $\theta=\frac{\pi}{6}=30^{\circ}$, compute $\sec(\theta)$,  $\csc(\theta)$, and $\cot(\theta)$.
\end{enumerate}

\medskip


\item Simplify the following expressions (i.e. write them in terms of
elementary trig functions $\sin(\phi)$, $\sin(\theta)$, etc.).
\begin{enumerate}
\item $\sin(\theta+\phi)$.
\medskip
\item $\cos(3\theta)$.
\end{enumerate}

\medskip









\end{enumerate}


\vfill
\pagebreak





\centerline{\sc Problem Set 2.  Problems from Lecture 2.}

\noindent {\bf Reading.} {\em Quick Calculus}, pp. 50--97.

\noindent {\bf Supplementary reading.} Simmons, Chapter 2, sections
2.1--2.5.  Read section 2.6 if you are interested in some applications 
of the derivative.

\begin{enumerate}

\item  Compute the following limits.

\begin{enumerate}
\item $\lim_{\theta\to 0}\frac{\sin(5\theta)}{\theta}$
\item $\lim_{\theta\to 0}\frac{\sin(3\theta)}{\sin(4\theta)}$
\item $\lim_{x\to\infty}\frac{x}{x^2+1}$
\item $\lim_{x\to\infty}\frac{2x^2+3x}{3x^2-2}$
\end{enumerate}

\item Where are the following functions discontinuous?

\begin{enumerate}
\item $\frac{x}{x^2+1}$
\item $\frac{1}{x^2+x-6}$
\item $\frac{x^3+x}{x^2+1}$
\item $\frac{x^2+2x}{x^3+2x^2-x-2}$
\end{enumerate}

\item Use the definition of the derivative to show that for
$f(x)=ax^2+bx+c$, for constants $a,b,c\in\R$, the derivative is
$f'(x)=2ax+b$.

\item Use the definition of the derivative to find the derivative of
the function $f(x)=\frac{x}{x+1}$.

\item Use the definition of the derivative and the double angle
formula to compute the derivative of $f(\theta)=\cos(\theta)$.

\item Sketch the graph of the following two functions.  For each,
state where it is {\bf not} differentiable.

\begin{enumerate}
\item $f(x)=\sqrt{|x|}$.
\item $f(x)=|x^2-9|$.
\end{enumerate}


\item Let
$\begin{array}{lll}
f(x) & = & \left\{\begin{array}{ll}
			x^2  & \mbox{ if } x\leq -1,\\
			mx+b & \mbox{ if } x>-1.
		   \end{array}\right.
\end{array}
$
What must $m$ and $b$ be for $f(x)$ to be differentiable at all points?


\item \label{previous} A penny is dropped off a ledge on the World
Trade Center in New York City.  The ledge is $1024$ feet above the
ground.  The penny falls a distance of $s=16t^2$ feet in $t$ seconds.
\begin{enumerate}
\item How long does the penny fall before it hits the ground?
\item What is the average velocity at which the penny falls during the 
first three seconds?
\end{enumerate}

\item With the same situation as in Problem~\cite{previous}, answer
the following questions.
\begin{enumerate}
\item What is the average velocity at which the penny falls during the 
last four seconds?
\item What is the instantaneouse velocity of the penny when it hits
the ground?
\end{enumerate}


\item An oil tank is to be drained for cleaning.  There are $V$
gallons of oil left in the tank after $t$ minutes of draining, where
$V=50(40-t)^2$.
\begin{enumerate}
\item What is the average rate at which oil drains out of the tank
during the first $20$ minutes?
\item What is the rate at which oil is flowing out of the tank $20$
minutes after draining begins?
\end{enumerate}



\end{enumerate}



\end{document}