
%%%%exam2 math 111
\documentstyle[12pt]{article}

\hoffset=-1in
\textwidth=7in
\textheight=8in


\begin{document}
\parindent=0mm

\begin{center}
{\bf\underline{MATH 111. TEST II}}
\end{center}

\vspace{5mm}
{\bf NAME:}

\vspace{5mm}
Directions: To receive full credit you must show all your work. No notes, 
books, etc are to be used for the test. Calculators are only allowed to be 
used in performing simple operations (you are not allowed to use their 
graphic and programing functions, memory, etc). 

\vspace{5mm}
1.(10 points) Given that $a$ and $b$ are constants, find
$$\frac{d}{dt}\left(at^3+a^2\sqrt{t}+b\right)\;.$$

\vspace{40mm}
2.(10 points) Find the second derivatives of the following functions:

\vspace{5mm}
$(i)\;\;f(x)=x^5+3x^4-x+2\;,\hspace{10mm}f''(x)=$


\vspace{40mm}
$(ii)\;\;g(x)=(2x+1)^3\;,\hspace{10mm}g''(x)=$

\newpage
3.(10 points) Let $f(x)=x^2-x+1$.

(i) Find the average rate of change of $f$ over the interval from 1 to 3.

(ii) Find the instantaneous rate of change of $f$ at $x=2$.

\vspace{80mm}
4.(10 points) Sketch an acurate graph of the function $f(x)=-2x^2+4x+1$. 

\newpage
5.(10 points) The position function of an object moving on a straight line is 
given by $s(t)=t^3-2t^2+t+2$ ($s$ in feet, $t$ in seconds). 

(i) Find the velocity of the object at the time $t=2$ seconds.

(ii) Find the acceleration of the object at the time $t=3$ seconds.

\vspace{80mm}
6.(10 points) Find all the relative maximum and minimum points of the 
function $f(x)=x^4-2x^2-2$.

\newpage
7.(10 points) Sketch the graph of a function $f$ defined near $x=1$ with 
the following properties: $f(1)=1$, $f'(1)=0$, $f''(1)>0$.


\vspace{60mm}
8.(10 points) Find the dimensions of the rectangular garden of greatest 
area that can be surrounded (all four sides) with 
400 yards of fencing.

\newpage
9.(10 points) The cost function of producing $x$ units of a product is 
$C(x)=0.01x^3-2x$. Find the marginal cost at production level $x=100$.

\vspace{50mm}
10.(10 points) Which of the following graphs best represents the graph of 
$f(x)=x^3-3x$ ? Justify your answer.


\end{document}

