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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:} \vrule height .4pt depth 0 pt width 1.4 in
\bigskip
\bigskip
\bigskip

\centerline{\bf {Math 119, Calculus}}
\centerline{\bf Fall Semester 2000}
\centerline{\bf  Exam 1 }
\centerline{\bf {Thursday, Sept 21}}
\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:

% Blank answer sheet


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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 Problems}}\vskip .2in

\noindent{\bf 1.} (5 pts.) 1. Which of the following lines
 is parallel to the line $y={3\over 2} x + 5$ ?
\vskip .2in

a)\;\; $y = 3$
\qquad\qquad
b)\;\; $2y + 3x - 5 = 0$
\qquad\qquad
c)\;\; $3y - 2x - 5 = 0$

\vskip.1in\qquad\qquad
d)\;\; $3y + 2x - 11 = 0$
\qquad\qquad
e)\;\; $2y - 3x - 7 = 0$
\vskip 1in
\noindent {\bf 2.} (5 pts.) What is the domain of the function
$f(x)=\root 4\of{8-4x^2} $ ?
\vskip .2in

a) \;\;$-{2}<x<{2}$\qquad\quad
b) \;\;$x\leq -\sqrt 2$ or $x\geq \sqrt 2$\qquad\quad
c) \;\;$- \sqrt 2\leq x\leq  \sqrt 2$

\vskip .1in

\qquad \qquad d) \;\;All real numbers\qquad
e) \;\;$x\leq  - 2$ or $x\geq  2$.

\vskip 1in

\noindent{\bf 3.} (5 pts.) Find the radius of the circle
$$ 2x^2 + 12x + 2y^2 + 4y = -16 $$
\vskip .2in

a)\;\; $0$
\qquad\qquad
b)\;\; $2$
\qquad\qquad
c)\;\; $\sqrt{-16}$
\qquad\qquad
d)\;\; $4$
  \qquad\qquad
e)\;\; $\sqrt{2}$.
\vskip 1in

\noindent {\bf 4.} (5 pts.) Which of the following is  the value of the limit
$$ \lim_{x\rightarrow 4}\frac{x^2-16}{\sqrt x-2}\quad ?$$

\vskip .2in

a)\;\;$32$\qquad\quad(b)\;\;$16$\qquad\quad(c)\;\;$8$\qquad\quad(d)\;\;
$2$\qquad\quad
(e)\;\;$1$.

\newpage

%\noindent {\bf 3.} (5 pts.)  Which of the following is  the
 %value of the limit
%$$ \lim_{x\rightarrow 3}\frac{x^2-4x+3}{ x^2-2x-3}\quad ?$$
%\vskip .2in
%
%a)\;\;$\frac{1}{2}$\qquad\quad b)\;\;$2$\qquad\quad c)\;\;$\frac{1}{3}$\qquad\quad
%d)\;\;$4$\qquad\quad
%e)\;\;The limit does not exist.

\newpage

\noindent {\bf 5.} (5 pts.) The height $f(t)$ in feet of a ball $t$ seconds
after it is
thrown vertically  into the air is given by the formula $f(t)=64t-16t^2$.
Find the average
velocity (in feet per second)  of the ball
over the time period from $t=1$ to $t=2$.


\vskip .2in

a)\;\;$64$\qquad\quad b)\;\;$-32$\qquad\quad c)\;\;$48$\qquad\quad d)\;\;$16$
\qquad\quad e)\;\;$-48.$

\vskip 1.2in
\noindent
{\bf 6.} (5 pts.)  Let $$ f(x)=\cases  4x^2 - c,&\text{if $ x < 1$}\\
                        cx, &\text{if $x\geq 1$}\endcases$$
where $c$ is a constant. What value of $c$ makes $f(x)$ continuous?


\vskip .2in
a)\;\; -2
\qquad\qquad
b)\;\; 0
 \qquad\qquad
c)\;\; 1
\qquad\qquad
d)\;\; 2\vskip .1in
\qquad\qquad
e)\;\; No value of $c$ makes $f(x)$ continuous.

\vskip 1.2in

\noindent {\bf 7.} (5 pts.) A particle moves so that its position at time
$t$ is given by
$$s(t)=2t^2+\frac{4}{\root 3\of {t^2}}.$$ Which of the following is an
expression for the
instantaneous velocity of the particle at time $t$?

\vskip .2in

a)\;\;$4t+12t^{\frac{1}{3}}$\qquad\qquad
b)\;\;$4t-\frac{8}{3}t^{-\frac{5}{3}}$\qquad\qquad
c)\;\;$4t+\frac{8}{3}t^{-\frac{5}{3}}$

\vskip .1in

\qquad\qquad d)\;\;$4t-12t^{\frac{5}{3}}$\qquad\qquad
e)\;\;$4t-12t^{-\frac{1}{3}}$.

\newpage
{\bf 8.} (5 pts.) If $f(x) = {1\over {\sqrt{x-1}}}$, which of the following
equals
${{f(3+h)-f(3)}\over h}$?

\vskip .2in
a)\;\;${{-1}\over {(\sqrt{3} +
\sqrt{3+h})\sqrt{3}\sqrt{3+h}}}$
\qquad\qquad
b)\;\;${{-1}\over {(\sqrt{2} + \sqrt{2+h})\sqrt{2}\sqrt{2+h}}}$
\vskip .1in
c)\;\;${{-1}\over {(\sqrt{2} - \sqrt{2+h})\sqrt{2}\sqrt{2+h}}}$
\qquad\qquad
d)\;\; ${{-\sqrt{2+h}}\over {(\sqrt{2} + \sqrt{2+h})\sqrt{2}}}$
\vskip .1in
\qquad\qquad\qquad\qquad
e)\;\; ${1\over {(\sqrt{2} + \sqrt{2+h})\sqrt{2}\sqrt{2+h}}}$

\vskip 1.2in

\noindent {\bf 9.} (5 pts.) If $h(x)=\frac{x-1}{x^2+1}$, find $h'(x)$.

\vskip .2in

a) \;\;$ \frac{x^2+2x+1}{x^2+1}$\qquad\qquad
b) \;\;$ \frac{x^2+2x+1}{(x^2+1)^2}$\qquad\qquad
c) \;\;$ \frac{-x^2+2x+1}{(x^2+1)^2}$

\vskip .15in

\qquad\qquad d) \;\;$ \frac{-x^2+2x+1}{x^2+1}$\qquad\qquad
e) \;\;$ \frac{3x^2+2x+1}{(x^2+1)^2}$

\vskip 1.2in

%\vskip .5in
%
%\noindent {\bf 7.} (5 pts.) Find the equation of the tangent
 %line to the graph of
%$y=2\sqrt x-x +3$ at the point on the curve where $x=1$.
%
%\vskip .2in
%
%a)\;\; $y=4$\qquad\quad b) $y-4=\frac{1}{2}(x-1)$\qquad\quad c)\;\;$y-4=2(x-1)$
%\vskip.1in \qquad\qquad d)\;\; $x=1$\qquad\qquade)\;\;$y-4=-\frac{1}{2}(x-1)$.

\noindent {\bf 10.} (5 pts.) Let $f(x)$ denote any differentiable function
of $x$. Which of
the following is an expression for the derivative $g'(x)$ of the function
$g(x)=x^3f(x)$?

\vskip .2in




a)\;\;$x^3f'(x) +3x^2 f(x)$\qquad b)\;\;$3x^2f'(x) +x^3
f(x)$\qquad c)\;\;$x^3f'(x) +x^2 f(x)$\vskip .1in \qquad\qquad
d)\;\;$x^2f'(x) +x^3
f(x)$\qquad\qquad  e)\;\;$x^3f'(x) +x^3 f(x)$.

\newpage
\noindent{\bf 11.} (5 pts.)
  If $\sin (\theta) = {3\over 4}$, and ${{\pi}/2}<\theta < \pi$,
what is $\cos (\theta)$?\vskip .2in


a)\;\; $-{1\over 4}$
\qquad\qquad
b)\;\; ${{\sqrt 7}\over 4}$
\qquad\qquad
c)\;\; $-{{\sqrt 7}\over 4}$
\qquad\qquad
d)\;\; ${1\over 4}$\vskip .1in
\qquad\qquad\qquad\qquad
e)\;\; Cannot be determined

\vskip 1in
\noindent
{\bf 12.} (5 pts.)  Let the number $x$ of heartbeats of a mouse after $t$
seconds
be given by the following
values:

$$\vbox{\offinterlineskip
\halign{
\strut \vrule # & \vrule # & \vrule # & \vrule #
& \vrule # & \vrule # & \vrule # & \vrule  # & \vrule #   height .2 in
depth .1 in
width .3pt
\cr
\noalign{\hrule}
$\ \ t$ &\ \  17 &\ \  {18} &\ \ 19 &\ \ {20} &\ \ {21} & \ \ 22 &\ \ 23 &\cr
\noalign{\hrule}
$\ \ x$  &  \ \ 25&\ \ 29\ \  &\ \ 34
&\ \  40&\ \ 48\,\,& \ \ 57 &\ \ 68&\cr \noalign{\hrule}
}}$$




Which of the following is the best estimate for the rate of heartbeats of
the mouse when $t=20$?\vskip .2in

a)\;\; 4 beats/second
\qquad\qquad
b)\;\;7 beats/second
\qquad\qquad
c)\;\; 11 beats/second\vskip .1in
\qquad\qquad
d)\;\;  40  beats/second
\qquad\qquad
e)\;\;  0 beats/second

\newpage {\bf \centerline{Part B: Partial Credit Problems}}\vskip .2in
 \noindent{\bf 13.} (10 pts.) Consider the graph of the function $f(x)$
pictured below.
\vskip 2in For
each of the following, either give the value of the limit, or indicate why
it does  not exist.
\vskip .2in
\noindent a)\;\; ${\lim}_{x\to 5^+}\, f(x)$
\vskip 1in
\noindent b)\;\; ${\lim}_{x\to 5^-}\, f(x)$
\vskip 1in
\noindent c)\;\; ${\lim}_{x\to 5}\, f(x)$
\vskip 1in
\noindent d)\;\;
${\lim}_{x\to 2} f(x)$




\newpage




\noindent {\bf 14.} (10 pts.)\; Let $f(x)=\frac{1}{x+2}$. \vskip .2in
\noindent a) Find the slope $m(h)$ of the secant line joining the points
$$(1,f(1))\quad  \text{\rm and}\quad (1+h,f(1+h))$$ on the graph of $y=f(x)$.
\vskip 2in
\noindent{\bf Answer:}
\vskip .2in \noindent b) Using your answer to (a), compute  the value of
the limit
$$\lim_{h\rightarrow 0} m(h)$$ using only algebra and the limit laws.
\vskip 2in \noindent{\bf Answer:}
%\vskip .2in (c) Using your anser to (b), find the equation of
 %the tangent line to $y=f(x)$
%at the point $(1,f(1))$.
%\vskip 1.5in Answer:
%



\newpage


\noindent{\bf 15.} (10 pts.) Let $f(x)=\vert\, x\,\vert -\vert\, 1-x\,\vert$.

\vskip .2in

\noindent a)\;\; Complete the following table of values of the function $f(x)$.
 \vskip .2in

$$\vbox{\offinterlineskip
\halign{
\strut \vrule # & \vrule # &\vrule # &\vrule #
&\vrule # &\vrule #\vrule  height .2 in depth .1 in width .3pt \cr
\noalign{\hrule}
$\ \ x$ &\ \  -2 &\ \  {-1} &\ \ 0 &\ \ {1} &\ \ {2\,\,} \cr
\noalign{\hrule}
$\ \ f(x)$  &\ \ &\ \ \  &\ \
&\ \  &\ \ {\ \ \,\,\,}\cr \noalign{\hrule}
}}$$

\vskip 0.2in \noindent  b)\;\;  Graph $y=f(x)$ on the interval $-2\leq
x\leq 2$. Make sure to
include a scale on the $x$ and $y$ axes.




%\vskip 2.5in
%c) Determine the value of the derivative $f'(\frac{1}{2})$ of $f$ at$x=\frac{1}{2}$.\vskip
%1in Answer:

\newpage

 \noindent {\bf 16.} (10 pts.)  Find the equation of the tangent line to
$y=\sqrt{x-1}$ at
the point $(5,2)$.\vskip 6in
\noindent{\bf Answer:}





% \noindent {\bf 11.} (10 pts.)\; Let $f(x)=x^2-2x$.\vskip .2in
 % a) Write down the limit which
% represents the derivative $f'(a)$ of $f(x)$ at $x=a$.
%
% \vskip 1.5in Answer:
% \vskip .2in
% \;b) Determine by any means the value of the derivative $f'(a)$ in (a).
%
% \vskip 1.5in
% Answer: \vskip .2 in c) Find the  point on the graph of $y=f(x)$ at which
% the tangent line is perpendicular to the line $y=2x-1$.
% \vskip 1.5in Answer;


\end


%3. For the function whose graph is shown below, find
%$${lim}_{x\to 2^-} f(x) $$
%\vskip 2cm
%\qquad\qquad
%a. $1$
%\qquad\qquad
%b. $4$
%\qquad\qquad
%c. $2$
%\qquad\qquad
%d. $-2$
%\qquad\qquad
%e. $-1$





\newpage

\bigskip
\noindent
\newpage

\bigskip
\noindent
%8. Let $$f(x)={1\over {x+1}}$$
%\qquad\qquad
%(a) find $${{f(2+h) - f(2)}\over h}$$
%\vskip 4cm
%(b) Compute $f'(2)$ (by any means you wish).


10. Find the equation of the tangent line to $y=\sqrt{x_1}$ at the
point $(5,2)$.

