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\noindent {\bf 21-121} \hfill {\bf IM/Econ Calculus I} \hfill {\bf Fall 2000} 

\centerline{\bf Review Problems for Test 1} 

\vspace{.25in} 

\begin{enumerate} 

\item Find the indicated limit or explain why it does not exist. 

$$
\begin{array}{llll} 
{\rm (a)} \  {\displaystyle \lim_{x \rightarrow 2}} {\displaystyle
\frac{x^3+x-2}{x-1}} & {\rm (b)} \ {\displaystyle \lim_{x \rightarrow
1}} {\displaystyle \frac{x^2-3x+2}{x^3-1}} \\
\\
{\rm (c)}\ {\displaystyle \lim_{x \rightarrow 3^+}} {\displaystyle
\frac{|3-x|}{9-x^2}} & {\rm (d)}\ {\displaystyle \lim_{x \rightarrow 3}}
{\displaystyle \frac{|3-x|}{9-x^2}} \end{array}
$$


\item Consider the function $f$ defined by 

$$f(x) = \left\{ \begin{array}{lll}
-1 & x < 0\\
x^2 & 0 \leq x < 1\\
2 & x = 1\\
x & 1 < x < 2\\
3-x & 2 \leq x \end{array} \right. $$

\begin{enumerate}

\item Sketch the graph of $y=f(x)$. 

\item Evaluate ${\displaystyle \lim_{x \rightarrow 1}} f(x)$ or explain
why it does not exist. 

\item Where is $f$ continuous? 

\end{enumerate}

\item Use the definition of derivative to find $f'(x)$ for each of the
following: 

(a) $f(x) = 1+3x -x^2$ \hspace{.75in} (b) $f(x) = {\displaystyle
\frac{2}{x+1}}$ 

(c) $f(x) = \sqrt{x-1}$

\item Show that the equation 

$$x^{10} + {\displaystyle \frac{1}{\sqrt{x+3}}} - 10x + 2 = 0$$

has at least one (real) root. 

\item Find the tangent line to the curve at the point $P$. 

\begin{enumerate} 

\item $y = {\displaystyle \frac{x}{x^2+1}}$ \hspace{.75in} $P = \left(1,
{\displaystyle \frac{1}{2}} \right)$. 

\item $y = x^4 - 3x^2 + 5$ \hspace{.75in} $P = \left(-1,3\right)$

\end{enumerate} 

\newpage

\item Find $f'(x)$ for each of the following:

\begin{enumerate} 

\item $f(x) = \left( x^3 - {\displaystyle \frac{1}{x}}\right) \left(
e^{3x} -2e^{5x} \right)$ 

\item $f(x) = {\displaystyle \frac{4}{x^5 - 2x + 7}}$ 

\item $f(x) = \sqrt{x}e^{-x}$ 

\item $f(x) = \left(x^2+1\right)^3 \left(x^2-1\right)$ 

\item $f(x) = \sqrt{x} + {\displaystyle \frac{1}{x}} + {\displaystyle
\frac{1}{x^2}} $ 

\end{enumerate}

\item Suppose that the demand for a monopolists product is given by $p =
2+ {\displaystyle \frac{200}{\sqrt{q}}}$, where $p$ is the price per
unit and $q$ is the number of units, and that the average cost function
is given by $\overline{c} = 3 + {\displaystyle \frac{100}{q}}$. 

\begin{enumerate} 

\item Find the marginal revenue function. 

\item Find the marginal cost function. 

\item Find an expression for the profit $P$ as a function of $q$. 

\item Find all values of $q$ at which ${\displaystyle \frac{dP}{dq}} = 0$. 

\end{enumerate} 

\item Suppose that the demand for a monopolists product is given by $p =
F(q)$ where $F$ is a function satisfying $F(100) = 3$ and $F'(100) =
-.01$.  Find the marginal revenue when $q=100$.  

\item Find a line that passes through the point $\left(1, {\displaystyle
\frac{11}{4}}\right)$ and is tangent to $y =x^2 +2x$ somewhere. 

\item Evaluate the following limit or explain why it does not exist: 

$${\displaystyle \lim_{x \rightarrow 0^+}} \left({\displaystyle
\frac{1}{\sqrt{x^2+x}}} - {\displaystyle \frac{1}{\sqrt{x}}} \right). $$

\item Assume that $f:(0,\infty) \rightarrow {\Bbb R}$ satisfies $f(s) -
f(t) = f({\displaystyle \frac{s}{t}})$ for all $s,\ t > 0$ and
${\displaystyle \lim _{h\rightarrow 0} \frac{f(1+h)}{h}} = 3$.  Show
that $f$ is differentiable on $(0,\infty)$ and find a formula for
$f^{\prime} (x)$.
\end{enumerate} 

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