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\topmatter

\vfil\eject\output={}
\vbox{{\noindent
{\bf Name:}  \vrule height .4pt depth 0 pt width 1.4 in
\hskip .5 in {\bf Instructor/Section:} \vrule height .4pt depth 0 pt width  1
in
\bigskip
\bigskip
\bigskip

\centerline{\bf {Math 119, Calculus}}
\centerline{\bf Fall Semester 2000}
\centerline{\bf  Exam 2 }
\centerline{\bf {Thursday, Oct. 26}}
\bigskip
\hrule
\bigskip
This Examination contains {\bf 16} problems, worth a total of  {\bf 100}
points,
on {\bf 9} sheets of paper including the front cover.
The first {\bf 12} problems (Section A) are multiple choice with no partial
credit, and
each is worth {\bf 5} points. Record your answers to these problems by placing
an $\times$ through one letter for each problem below:

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\bigskip
\bigskip
The last {\bf 4} problems (Section B) are partial credit problems worth
{\bf 10} points each. For these problems, {\bf show} your
computations and {\bf clearly} mark your answers on the page.
Books and notes are not allowed. You may use your calculator.
\bigskip
\hrule
\bigskip
\bigskip
\bigskip


{\bf Sign the pledge:}  \lq \lq {On my honor,
I have neither given nor received unauthorized aid on this Exam}":
\bigskip
\hskip .7 in\vrule height .10pt depth 0 pt width 2 in
\bigskip
\bigskip
\bigskip
\centerline{\bf GOOD LUCK}

\newpage
{\bf\centerline{Part A: Multiple Choice or short answer Problems}}\vskip .2in

\noindent{\bf 1.} (5 pts.) 1. Which of the following is the value of the
limit
$$\lim_{t\rightarrow 0} \frac{\sin (4t)}{\tan (5t)}\quad?$$ 
\vskip .2in

a)\;\; $\frac{5}{4}$
\qquad\qquad
b)\;\; $\frac{4}{5}$
\qquad\qquad
c)\;\; $0$
\qquad\qquad
d)\;\; undefined
\qquad\qquad
e)\;\; $\frac{5 \cos(5t)}{4 \cos(4t)}$


\vskip 1in
\noindent {\bf 2.} (5 pts.) Suppose 
$$xy=x^2 + xy^3 \quad.$$ Find $\frac{dy}{dx}$.
\vskip .2in


\newpage

\noindent{\bf 3.} (5 pts.) 
Let $f(x)=x^3+x, 0\le x\le 2$. For which value
of $c$  is it true that $0\le c\le 2$ and 
$$ f'(c)=\frac {f(2)-f(0)}{2-0} \quad ?$$
\vskip .2in

a)\;\; $-\frac{2}{\sqrt{3}}$
\qquad\qquad
b)\;\; $-1$
\qquad\qquad
c)\;\; $0$
\qquad\qquad
d)\;\; $2$
  \qquad\qquad
e)\;\; $\frac{2}{\sqrt{3}}$
\vskip 1in

\noindent {\bf 4.} (5 pts.) Let $f(x)=x+\cos(x)$. For which of
the following values of $x$ does the graph of $y=f(x)$ have
a horizontal tangent?

\vskip .2in

a)\;\;$\frac{3\pi}{2}$\qquad\quad
b)\;\;$2\pi$\qquad\quad
c)\;\;$\frac{\pi}{2}$\qquad\quad
d)\;\;$1$\qquad\quad
e)\;\;$\pi$.

\newpage



\noindent {\bf 5.} (5 pts.) The force of attraction between two
giant frogs is given by the formula 
$$F(r) = \frac{2}{r^2 + 1}$$
newtons, where $r$ is the distance between the two frogs.
What is the rate of \newline
 change in the force
with respect to $r$ when $r=2$?
 \vskip 2in

\noindent
{\bf 6.} (5 pts.)  
The position of a particle is given by the function $f(t)=\frac{t^2+1}{t}$,
where $t$ is the time measured in seconds. What is the acceleration of the
particle after 2 seconds?
\vskip .2in


\newpage

\noindent
{\bf 7.} (5 pts.)  
Find the derivative of $f(x)=\tan^2(3x)$
\vskip .2in


\vskip 1.2in
\vskip .4in

\noindent
{\bf 8.} (5 pts.)  
Find the derivative of $f(x)=\sqrt{x^3+x^2-4x}$.
\vskip .2in

\vskip 1.2in

\newpage

\noindent
{\bf 9.} (5 pts.)  
Suppose that $f=g \circ h$, $g(0)=1, h(0)=1, g(2)=4,  h(2)= 3,$
$g'(0)=3, h'(0)=2, g'(1)=3, h'(1)=11, g'(2)=4$, and $h'(2)=6$. 
What is $f'(0)$?

\vskip .2in
a)\;\; $33$
\qquad\qquad
b)\;\; $6$
\qquad\qquad
c)\;\; $8$
\qquad\qquad
d)\;\; $4$
\qquad\qquad
e)\;\; $0$. 
 
\vskip 1.2in

\noindent
{\bf 10.} (5 pts.)
Suppose a function $f(x)$ has $f'(1)=0$ and $f''(1)=86.5$.
Which of the following are true?

\vskip .2in
a)\;\; $f(x)$ has a local maximum at $x=1$.
\vskip 1cm
b)\;\; $f(x)$ has neither a local maximum nor a local minimum at $x=1$.
\vskip 1cm
c)\;\; $f(x)$ has a local minimum at $x=1$.
\vskip 1cm
d)\;\; $f(x)=(x-1)^2$.
\vskip 1cm
e)\;\; The slope of the tangent line to the graph of $f(x)$ is equal to $3$
when $x=1$.
 

\newpage
\vskip 1.2in 
\noindent
{\bf 11.} (5 pts.)  
A particle moves along the curve $y=\sqrt{x^2 + 5}$. As it reaches the point
$(2,3)$, the $y$-coordinate is increasing at a rate of $4$ cm/sec.
How fast is the $x$-coordinate of the point changing at that instant?


\vskip 1.2in
\noindent
{\bf 12.} (5 pts.)  
Find the equation of the tangent line to the curve $f(x)=\sin(2x) + \cos(x)$
 at 
the point $(\frac{\pi}{4}, 1+\frac{\sqrt{2}}{2})$.



\newpage {\bf \centerline{Part B: Partial Credit Problems}}\vskip .2in
{\bf 13.} (10 pts.) 
Suppose George is five meter north of a soccer ball and Jennifer is four meter
east of  the soccer ball that is motionless on the ground.
 Suppose George starts to run toward the ball at a
speed of
one meter per second, and Jennifer does not move.
 How quickly is the distance between George and Jennifer 
changing after one second?


\vskip 2cm


\newpage
\noindent {\bf 14.} (10 pts.)\; Find the absolute maximum and absolute 
minimum values of 
$$f(x) = x^3 -12x\/,\qquad x\in [-3, 4]$$.


\vskip 2cm


\newpage
\noindent {\bf 15.} (10 pts.)\; Compute the first and second derivatives of
$$f(x)=x\sin(x)$$.



\vskip 2cm


\newpage
\noindent {\bf 16.} (10 pts.)\; 
Consider the function $f(x)=\frac{x}{x^2+4} $.
\vskip .2in

a) \; \; Find the set of critical numbers of $f$.
\vskip 7truecm
b)\;\; Find local maxima and minima of $f$.

\vfill

\end
