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\preface
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\title
\noindent
{\bf Math 105: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl September 18, 1997}\hfill
Time of MWF class:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Calculators are not allowed.  Do not remove this answer page-- you will return
the whole exam.  There are  16 multiple choice questions, worth 5 points
each, and
3 partial credit questions worth a total of 20 points.  Record your answers
to the
multiple choice problems by  placing an $\times$ through one letter for each
problem on this answer sheet.  On problems 17--19, write your answer
clearly in the
place indicated, but also show all necessary work.

\vskip .25cm \noindent {\bf
You are taking this exam under the honor code.}

\def\answerskip{\vskip0.25in}

\def\problemskip{\vskip1in}

\note %1
\problem
If $\displaystyle f(x) = {{x^2 + x} \over {x+1}}$ and $\displaystyle g(x) = {1
\over x}$, find $\displaystyle f(x) + g(x)$.
\correct $\displaystyle {{x^3 + x^2 + x + 1} \over {x^2 + x}}$
\wrong $\displaystyle {{x^2 + x + 1} \over {2x+1}}$
\wrong $\displaystyle {{x^3 + x^2 + x + 1} \over {x+1}}$
\wrong $\displaystyle {{x^3 + x^2 +  1} \over {2x+1}}$
\wrong $\displaystyle {{x^2 + x + 1} \over {x^2 + x}}$

\note %2
\problem
Let
$$
f(x) = \cases{x^2 + 1, &if $x < 0$;\cr
              3, &if $0 \leq x \leq 1$;\cr
              2x+1, &if $x>1$.}
$$
Which of the following is true?

\correct $f(x)$ is continuous everywhere except at $x=0$.
\wrong $f(x)$ is continuous everywhere except at $x=1$.
\wrong $f(x)$ is continuous everywhere except at $x=0$ and $x=1$.
\wrong $f(x)$ is continuous everywhere.
\wrong $f(x)$ is not a function.

\note %3
\problem
The cost of receiving a certain operation at Mercy Hospital is \$3,000, and the
cost for a semiprivate room is \$500 per day.  (Ignore all other costs.)  Bob
received a bill of \$5,000 for receiving the operation and staying $x$ days in
the hospital.  What is $x$?

\correct 4
\wrong 2
\wrong 6
\wrong 5
\wrong 3

\note %4
\problem
If $g(t) = t^2 + 1$ and $h(t) = \sqrt{t}$, find $h(g(t))$.

\correct $\sqrt{t^2 + 1}$
\wrong $t+1$
\wrong $(\sqrt{t} + 1)^2$
\wrong $t+2$
\wrong $t^4 + 1$

\note %5
\problem
Find all the points of intersection of the curves $y = x^2 -3x+2$ and $y =
-8x +8$.

\correct $(-6, 56)$ and $(1, 0)$
\wrong $(-6, 0)$ and $(1,0)$
\wrong $(1,0)$ and $(2,0)$
\wrong $(1,0)$ and $(2, -8)$
\wrong $(1,0)$

\note %6
\problem
Which of the following is equal to $\displaystyle \left ( {1 \over {27}} \right
)^{-2/3}$?

\correct 9
\wrong $\displaystyle {1 \over 9}$
\wrong 81
\wrong $\displaystyle {1 \over {81}}$
\wrong 18

\note %7
\problem
Find $\displaystyle \lim_{h \rightarrow 0} {{\sqrt{25+h} - 5} \over h}$, if it
exists.

\correct $\displaystyle {1 \over {10}}$
\wrong 10
\wrong $\displaystyle {1 \over {5}}$
\wrong 5
\wrong the limit does not exist


\note %8
\problem
Find the equation of the line through the point $(-2, 3)$ and perpendicular
to the
line $x - 2y = 5$.  Make sure your answer is in the ``slope-intercept''
form, $y =
mx + b$.

\correct $y = -2x - 1$
\wrong $\displaystyle y = -{1 \over 2} x +2$
\wrong $\displaystyle y = {1 \over 2} x +2$
\wrong $y = -2x -4$
\wrong $y = -{1 \over 2} x -1$

\note %9
\problem
Find the slope of the curve $y = x^3 - x^2$ at $x = 2$.

\correct 8
\wrong 4
\wrong 1
\wrong 16
\wrong 2

\note %10
\problem
Find the equation of the tangent line to the curve $y = x^2 - {1 \over x}$
at $x =
1$.

\correct $y = 3x - 3$
\wrong $y = 3x$
\wrong $y = x - 1$
\wrong $y = x + 1$
\wrong $y = 2x + 1$

\note %11
\problem
Let $f(x) = \sqrt{x^2 + x}$.  Find $f'(2)$.

\correct $\displaystyle {5 \over {2 \sqrt{6}}}$
\wrong $\displaystyle {1 \over {2 \sqrt{6}}}$
\wrong $\displaystyle {{5\sqrt{6}} \over {2 }}$
\wrong $\sqrt{6}$
\wrong $\displaystyle {2 \over { \sqrt{6}}}$

\note %12
\problem
Find $\displaystyle \lim_{x \rightarrow 3} {{x^2 - 9} \over {3-x}}$, if it
exists

\correct $-6$
\wrong 0
\wrong 6
\wrong $-1$
\wrong the limit does not exist

\note %13
\problem
Find the $x$-coordinate of the point on the graph of $y = \sqrt{x}$ where the
tangent line has slope $\displaystyle {1 \over 4}$.

\correct 4
\wrong $\sqrt{2}$
\wrong 8
\wrong 16
\wrong $\sqrt{8}$

\note %14
\problem
For the graph of $y = x^2 - x$, which of the following represents the slope
of the
secant line between the point on the graph where $x=2$ and the point on the
graph
where $x=2+h$?

\correct $3+h$
\wrong $4+h$
\wrong $2+h$
\wrong $5+h$
\wrong $\displaystyle {{2+h} \over h}$

\note %15
\problem
Find the domain of the function $\displaystyle f(x) = {{1 \over \sqrt{x+1}}}$

\correct $(-1,\infty)$
\wrong $[-1,\infty)$
\wrong $(1, \infty)$
\wrong $(-\infty, 1)$
\wrong $(-\infty, -1)$

\note %16
\problem
Which of the following is true about the differentiability of the function
$$
f(x) = \cases{x^2 + 1, &if $x < 0$;\cr
              1-x^2, &if $0 \leq x \leq 1$;\cr
              -2x+2, &if $x>1$.}
$$

\correct $f(x)$ is differentiable at both $x=0$ and $x=1$
\wrong $f(x)$ is differentiable at $x=0$ but not at $x=1$
\wrong $f(x)$ is differentiable at $x=1$ but not at $x=0$
\wrong $f(x)$ is not differentiable at either $x=0$ or $x=1$
\wrong $f(x)$ is not a function


\noindent {\bf Partial credit Part}

\bigskip

\vbox{
\note % partial credit
\noindent\llap{17.\ }A farmer wants to enclose his livestock in a rectangular
corral with two partitions, forming three sections of equal size as illustated
below.

\vskip 1in
\hskip 1.3in \corral

\medskip
\noindent Suppose that the total area of the corral (i.e.\ all three sections
combined) is 600 square feet.  Write your answers to the following questions in
the box provided, and show all necessary work in the remaining space.

\bigskip

\noindent (a) (3 points) Express $y$ as a function of $x$.

\vskip .3in
\hskip 3.5in \answer

\vskip .5in
\noindent (b) (4 points) Express the total length of fencing as a \hfil\break
function of $x$.

\vskip .2in
\hskip 3.5in \answer
}\vfil\goodbreak

\bigskip

\vbox{
\note % partial credit
\noindent\llap{18.\ }There is a point $P$ on the graph of $f(x) = x^2 + x$
where
the tangent line to the graph at $P$ has equation $y = -5x + b$, for some
constant
$b$.

\bigskip

\noindent (a) (4 points) What is the $x$-coordinate of $P$?

\vskip .3in
\hskip 3.5in \answer
\vskip .5in

\noindent (b) (4 points) What is the value of $b$?

\vskip .3in
\hskip 3.5in \answer
}\vfil\goodbreak

\bigskip

\vbox{
\note %partial credit
\noindent\llap{19.\ } (5 points) Sketch a line on the graph below which is
tangent to the  graph and which has slope ${1 \over 2}$.

\vskip 2.8in
\hskip 1in \graphftn
}\vfil\goodbreak

%That's all folks!

\end
