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\noindent

MATH 104 \hspace*{.2in} REVIEW QUESTIONS: HOUR 1




\noindent  1. Find $\frac{dy}{dx}$  (You need not simplify):\\
   (a) $\ds y = \frac{x^3-6x}{1-4x}\;\;$(b) $\;\;\ds y=6x^{\frac{8}{3}}-\frac{3}{x^4} +x$\\
 \noindent  2. Let $\ds h(x)=\frac{3x+1}{2x-1}$ for $x\neq \frac{1}{2}$.  Compute:\\
\noindent(a)  $h(1+h(1)) =\;$ ; (b) $  h'(x) = \;$ ; (c) $ h'(2)=\;$\\
\noindent 3.  Find the value $k$ so that the function $y=f(x)=$
\left\{
\begin{array}{ll}
               $kx +3$  &  \mbox{if $x \leq 2 $}\\
               $x^2-2$  &\mbox{if $x >2$} \\
                 \end{array}$\\ is continuous for all $x$.\\
 \noindent 4. Find an equation of the tangent line to the graph
of the curve $y=f(x) =6x-2x^2-7$ at the point $(1,-3)$.\\
\noindent  5.  Let $y=f(x)=$ \left\{
\begin{array}{ll}
               \frac{4x^2+x-5}{x-1}  &  \mbox{if $x \neq 1 $}\\
               $8$  &\mbox{if $x =1$} \\
                 \end{array}\]\\

\noindent (a)  Evaluate $\ds\lim_{x\rightarrow
1}\frac{4x^2+x-5}{x-1};$ \noindent (b) Is $\;f(x)$ a continuous
function at $x=1$?  Why or why not?\\
\noindent 6.  Find the \underline{average rate of change} of $y$
with respect to $x$ of the function $y=f(x)=2x^2-x$ on the interval
$[1,3]$.\\
\noindent 7.  Evaluate the limits:\\

\noindent (a) $\ds\lim_{\Delta x \rightarrow 0}\frac{(3x-\Delta x)^2
-9x^2}{7\Delta x}$\\
\noindent (b) $\ds\lim_{x \rightarrow 9}\frac{\sqrt{x}-3}{x-9}$\\

\noindent  8.  Suppose that the supply equation for a certain
commodity is $p=S(x)=5+.3x$ dollars and the demand equation is
$p=D(x)=40-.2x$ dollars.  Find the equilibrium point
$(x_0,p_0)$.\\
\noindent  9. (a)  Find all points $(x,y)$ at which the graph of
$\ds y=f(x)=\frac{x^2}{x-2}$ has a horizontal tangent line.  Hint:
First find $f'(x)$.\\
(b) Find an equation of the tangent line to the graph of the
function in part (a) at $x=1$.\\

\noindent  10.  The total cost of producing $x$ units of a
certain product is $C(x)=800+24x-.1x^2$ dollars.\\
\noindent (a) Find the \underline{marginal cost} function.\\
\noindent (b) At what \underline{production level} $x$ does the
marginal cost equal $14$ dollars?\\
\noindent (c) Find the \underline{marginal cost} when $x = 5$
units.\\
\noindent (d)  Find the \underline{exact cost} of the 6th unit.\\

\noindent 11.  Suppose that the cost function for a certain
commodity is given by $\ds C(x) = \frac{4x^2}{x+1}+75$.\\
\noindent (a) What is the marginal cost function? \\
\noindent (b) What is the {\bf{average cost function?}} \\
\noindent (c) What is the {\bf{marginal average cost function}}?\\

\noindent 12. A company which produces widgets has an initial
investment of \$10000.00.  If each widget costs \$21.50 to produce
and can be
sold at a price of \$30.65 find:\\
\noindent (a) the equation for the total cost $C(x)$ and the total revenue $R(x)$; \\
\noindent (b) the break even point (the intersection of the cost
and revenue functions); (Round off to the closest integer).\\
\noindent (c) how many widgets must be sold to yield a profit of
\$8000.00? (Round off to the closest integer).\\




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