% fin Version 1, Thu Dec  4 16:47:57 1997


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% Preface

\magnification=1200

% Problems

\vbox{
%3

\noindent\llap{1.\ }Which of the following is true about the continuity of
the function
$$
f(x) = \cases{x^2 - 1, &if $x < 0$; \cr
 x^2 &if $0 \leq x \leq 1$; \cr
 2x - 1 &if $x > 1$?}
$$

\beginanswers
{(a)~$f(x)$ is continuous everywhere except at $x = 0$.\answerspace}
{(b)~$f(x)$ is not continuous anywhere.\answerspace}
{(c)~$f(x)$ is continuous everywhere except at $x = 0,1$.\answerspace}
{(d)~$f(x)$ is continuous everywhere.\answerspace}
{(e)~$f(x)$ is continuous everywhere except at $x = 1$.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%7

\noindent\llap{2.\ }Let $h(x) = f(x)\cdot g(x)$.  Suppose $f(1) = 2$,
$f'(1) = 3$,
$f'(-1) = 4$, $g(1) = -1$, $g'(1) = -2$, and $g'(-1) = -3$. {\bf Find} $h'(1)$.

\beginanswers
{(a)~$-3$\answerspace}
{(b)~$-6$\answerspace}
{(c)~$-4$\answerspace}
{(d)~$-7$\answerspace}
{(e)~$-8$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%2

\noindent\llap{3.\ }Let $f(x) = \ln(x)$.  Which of the following is equal
to $f'(1)$?

\beginanswers
{(a)~$\displaystyle{\lim_{h\to 0} {\ln(h) \over h}}$\answerspace}
{(b)~$\displaystyle{\lim_{h\to 0} \ln(h)}$\answerspace}
{(c)~$\displaystyle{\lim_{h\to 0} {\ln(1 + h) \over h}}$\answerspace}
{(d)~$f'(1)$ does not exist.\answerspace}
{(e)~$\displaystyle{\lim_{h\to 0} {{\ln(h) -1}\over h}}$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%18

\noindent\llap{4.\ }Find all solutions to \ \ $\ln(x + 2) + \ln(x - 2) = 0$.

\beginanswers
{(a)~$x = -5, 4$\answerspace}
{(b)~$x = \sqrt{5}$\answerspace}
{(c)~$x = 2, 5$\answerspace}
{(d)~$x = \sqrt{7}$\answerspace}
{(e)~$x = 5$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%21

\noindent\llap{5.\ }A certain investment earns continuously compounded
interest at a rate
which results in the investment doubling every 10 years. If \$1000 is
invested today, in how many years will there be \$3000 in the account?

\beginanswers
{(a)~$15$\answerspace}
{(b)~$3$\answerspace}
{(c)~${\ln(2) \over 10}$\answerspace}
{(d)~$100\cdot\ln\left({3 \over 2}\right)$\answerspace}
{(e)~${\ln(3) \over \ln(2)}10$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%13

\noindent\llap{6.\ }Let $f(x) = xe^x$. Identify the $x$-value(s) of the
inflection point(s) of $f(x)$.

\beginanswers
{(a)~$x = -1,1$\answerspace}
{(b)~$x = 1,2$\answerspace}
{(c)~$x = 1$\answerspace}
{(d)~$x = 0$\answerspace}
{(e)~$x = -2$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%9

\noindent\llap{7.\ }The cost of producing $x$ (in thousands) toys at the
North Pole is
$C(x) = {(x-2)^3 \over (x-1)^4}$. For what level(s) of production $x$ will the
marginal cost be zero?

\beginanswers
{(a)~$x = 3$\answerspace}
{(b)~$x = 1,5$\answerspace}
{(c)~$x = 1$\answerspace}
{(d)~$x = 3,4$\answerspace}
{(e)~$x = 2, 5$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%4

\noindent\llap{8.\ }Find the slope of the curve $y = xe^{x^2 - 1}$ at the point $(1,1)$.\beginanswers{(a)~$2$\answerspace}{(b)~$2 + e$\answerspace}{(c
)~$e$\answerspace}
{(d)~$3$\answerspace}
{(e)~$1 + e$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%26

\noindent\llap{9.\ }Find the area of the region in the plane between the
$x$-axis and the curve
$y = {1 \over x}$ from $x = 1$ to $x = 5$.

\beginanswers
{(a)~${4 \over 5}$\answerspace}
{(b)~${1 \over 5}$\answerspace}
{(c)~$\ln(5)$\answerspace}
{(d)~$\ln(4)$\answerspace}
{(e)~$\ln\left({1 \over 5}\right)$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%12

\noindent\llap{10.\ }Let $f(x) = {1 \over 3}x^3 - 2x^2 - 12x - 19$. On
which of the following
intervals is $f(x)$ always decreasing?

\beginanswers
{(a)~$(-5,-3)$\answerspace}
{(b)~$(-2,6)$\answerspace}
{(c)~$(-\infty,-2)$\answerspace}
{(d)~$(1,12)$\answerspace}
{(e)~$(4,\infty)$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%23

\noindent\llap{11.\ }A ball is thrown upwards from the top of a $1200$ foot
tall building with an
initial speed of $40$ feet per second so that the velocity at any time $t$ is
given by $v(t) = -32t + 40$.  How many feet {\bf above the ground} is the ball
after $10$ seconds?

\beginanswers
{(a)~400 feet\answerspace}
{(b)~800 feet\answerspace}
{(c)~1200 feet\answerspace}
{(d)~1600 feet\answerspace}
{(e)~0 feet\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%16

\noindent\llap{12.\ }Santa has just received a load of feed for his reindeer.
The feed is being dumped out of a truck at the constant rate of $16\pi$
cubic feet
per second. The feed is forming a {\bf conical} pile in a such a way that
the height
is  always equal to the radius ({\bf see picture}).  How fast is the radius of
the pile increasing when the radius is $2$ feet?
\medskip
\noindent {\bf Hint:} The volume of such a cone is  $V = {1 \over 3}\pi r^3$,
where $r$ is the radius.
\medskip
\beginanswers
{(a)~4 ft/s\answerspace}
{(b)~2 ft/s\answerspace}
{(c)~$-3$ ft/s\answerspace}
{(d)~$3\pi$ ft/s\answerspace}
{(e)~6 ft/s\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%24

\noindent\llap{13.\ }If a car starts from rest with an acceleration of $8$
feet per second per second,
how far has it traveled by the time it reaches a speed of $88$ feet per second?

\beginanswers
{(a)~88 feet\answerspace}
{(b)~484 feet\answerspace}
{(c)~11 feet\answerspace}
{(d)~196 feet\answerspace}
{(e)~312 feet\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%22

\noindent\llap{14.\ }Compute the integral $\int (7x^{1\over 2} + 2x -1)dx$.

\beginanswers
{(a)~$7x^{3 \over 2} + {1 \over 2}x^2 - x + C$\answerspace}
{(b)~${14\over 3}x^{-{1 \over 2}} + x^2 + C$\answerspace}
{(c)~${14\over 3}x^{3 \over 2} + x^2 - x + C$\answerspace}
{(d)~${7 \over 2}x^{3 \over 2} + x^2 - x + C$\answerspace}
{(e)~${7 \over 2}x^{-{1 \over 2}} + 2 + C$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%20

\noindent\llap{15.\ }The number of subscribers to a certain magazine is
growing exponentially.
There were initially 100 subscribers in 1960 and this had increased to 1000
in 1980.  How many subscribers will there be in the year 2000?

\beginanswers
{(a)~10,000\answerspace}
{(b)~2000\answerspace}
{(c)~5000\answerspace}
{(d)~1900\answerspace}
{(e)~20,000\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%8

\noindent\llap{16.\ }Let $y = \ln(u + 2)$ and $u = x^3 - 3x$.  Find ${dy
\over dx}$.

\beginanswers
{(a)~$\ln(x + 2)$\answerspace}
{(b)~$\displaystyle{3x^2 - 3}$\answerspace}
{(c)~$\displaystyle{1 \over x^3 - 3x + 2}$\answerspace}
{(d)~$\displaystyle{\ln(3x^2 - 3) \over x^3 - 3x + 2}$\answerspace}
{(e)~$\displaystyle{3x^2 - 3 \over x^3 - 3x + 2}$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%25

\noindent\llap{17.\ }Compute the integral $\int_1^4{{x - 1} \over {x^{1
\over 2}}}dx$.

\beginanswers
{(a)~${8 \over 3}$\answerspace}
{(b)~${11 \over 3}$\answerspace}
{(c)~${3 \over 2}$\answerspace}
{(d)~$5$\answerspace}
{(e)~$4$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%5

\noindent\llap{18.\ }Differentiate the function $f(x) = x^{1997} - 2x^{1
\over 3}$.

\beginanswers
{(a)~$x^{1996} -{2\over 3}x^{-{2\over 3}}$\answerspace}
{(b)~$1997x -x^{1 \over 3}$\answerspace}
{(c)~$1997x^{1996} -{2 \over 3}x^{-{2\over 3}}$\answerspace}
{(d)~${1 \over 1998}x^{1998} -{3 \over 2}x^{4 \over 3}$\answerspace}
{(e)~$1997x^{1996} - x^{-{1\over 3}}$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%14

\noindent\llap{19.\ }Santa wants to build a {\bf rectangular} pen along
side of a river for his
reindeer. He will fence in only {\bf three} of the sides, leaving the side
by the river open.  If he has 80 yards of material to use, what is the
maximum area of such a pen which can be built?

\beginanswers
{(a)~800 square yards\answerspace}
{(b)~400 square yards\answerspace}
{(c)~1200 square yards\answerspace}
{(d)~600 square yards\answerspace}
{(e)~1000 square yards\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%6

\noindent\llap{20.\ }If $f(x) = \sqrt{x^2 + x}$, find $f'(1)$.

\beginanswers
{(a)~$\sqrt{3}$\answerspace}
{(b)~${3 \over \sqrt{2}}$\answerspace}
{(c)~${1 \over 2\sqrt{2}}$\answerspace}
{(d)~${3 \over 2\sqrt{2}}$\answerspace}
{(e)~$\sqrt{2}$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%15

\noindent\llap{21.\ }Find the equation of the tangent line to the curve
$y^3 - x^3 + xy = 1$ at
the point $(1,1)$.

\beginanswers
{(a)~$y = x + {1\over 2}$\answerspace}
{(b)~$y = {1\over 2}x + 1$\answerspace}
{(c)~\ \ $y = x$\answerspace}
{(d)~$y = {1\over 2}x + {1 \over 2}$\answerspace}
{(e)~$y = 2x - 1$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%17

\noindent\llap{22.\ }Simplify $e^{2\ln(x) - 1}$.

\beginanswers
{(a)~${x^2 \over e}$\answerspace}
{(b)~$2e$\answerspace}
{(c)~$\ln(2x) - 1$\answerspace}
{(d)~$x^2 + e$\answerspace}
{(e)~$x^2$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%19

\noindent\llap{23.\ }Differentiate $f(x) = x^{3x}$.

\beginanswers
{(a)~$x\ln(3x)$\answerspace}
{(b)~$3x^{3x}$\answerspace}
{(c)~$x^{3x}\ln(3x)$\answerspace}
{(d)~$3x^{3x}(\ln(x) + 1)$\answerspace}
{(e)~$e^{3x}$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%1

\noindent\llap{24.\ }Compute the limit\ \ \ $\displaystyle{\lim_{x \to 2}
{{x^2+5x -
14}\over{x-2}}}$\ \ \ if it exists.

\beginanswers
{(a)~2\answerspace}
{(b)~9\answerspace}
{(c)~0\answerspace}
{(d)~The limit does not exist.\answerspace}
{(e)~$-14$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%10

\noindent\llap{25.\ }Let $P(t)$ be the number of presents Santa has
delivered on Christmas Eve
as a function of time in {\bf minutes}, starting at midnight.  If $P'(60) =
50$,
approximately how many presents will Santa deliver between 1:00 a.m.\ and 1:03
a.m.?

\beginanswers
{(a)~60\answerspace}
{(b)~150\answerspace}
{(c)~3\answerspace}
{(d)~200\answerspace}
{(e)~50\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%11

\noindent\llap{26.\ }Let $f(x) = \ln\left(\sqrt{9 - x^2}\right)$ for $- 3 <
x < 3$.  Find
the $x$-value(s) of the relative extreme point(s) of $f(x)$.

\beginanswers
{(a)~$x = -2, 2$\answerspace}
{(b)~$x = 1$\answerspace}
{(c)~$x = -\sqrt{3}, \sqrt{3}$\answerspace}
{(d)~$x = 1, 2$\answerspace}
{(e)~$x = 0$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\noindent {\bf Partial Credit Part}
\vskip0.25in
\noindent You {\bf MUST} show your work in
order to receive full credit for these problems.

\medskip
% 27
\noindent{27.\ }Consider the definite integral $\displaystyle{\int_{-1}^2(2x
- 2)dx}$.

\medskip
\noindent a) (2 points) Sketch the graph of the curve $y = 2x - 2$.

\vskip 2in
\noindent b) (3 points) Compute the integral {\bf geometrically} via an area
computation.

\vskip 2in
\noindent c) (5 points) Compute the integral by using {\bf antiderivatives}.

\vskip 2in

% 28
\noindent{28.\ }(10 points) Let $f(x)$ be a function which satisfies the
following
list of properties:
\item{$\bullet$} The only $x$-intercept is $x = 4$ and the $y$-intercept is
$y =
3$.
\item{$\bullet$} $f(-2) = 1$ and $f(2) = 5$.
\item{$\bullet$} The only solutions to $f'(x) = 0$ are $x = -2, 2$.
\itemitem{$\bullet$} $f'(x) > 0$ on the interval $(-2,2)$.
\itemitem{$\bullet$} $f'(x) < 0$ on the intervals $(-\infty,-2), (2,\infty)$.
\item{$\bullet$} The only solution to $f''(x) = 0$ is $x = 0$.
\itemitem{$\bullet$} $f''(x) > 0$ on the interval $(-\infty,0)$.
\itemitem{$\bullet$} $f''(x) < 0$ on the interval $(0,\infty)$.

\medskip
\noindent a) Sketch the graph of $f(x)$.

\vskip4.5in
\noindent b) In your graph of part (a), label all the extreme points and
inflection points.
%That's all folks!

% Appendix

{}

% Blank answer sheet

\vfil\eject\output={}
\vbox{{\noindent
{\bf Math 105: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl December 16, 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  26 multiple choice questions, worth 5 points
each, and
2 partial credit questions worth 10 points each.  Record your answers to the
multiple choice problems by  placing an $\times$ through one letter for each
problem on this answer sheet.  For the partial credit problems (27 and 28),
{\bf show} all necessary work and {\bf clearly} identify your answers. {\bf
Good
Luck!} and {\bf Happy Holidays!}

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

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}}}


% Marked answer sheet

\vfil\eject
\vbox{\vbox to0pt{\vskip-22.5pt\hbox to \hsize{\hfil\ansbox{70pt}{\ Answer
Key 1\ }\hfil}\vss}
\prevdepth=-1000pt
{\noindent
{\bf Math 105: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl December 16, 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  26 multiple choice questions, worth 5 points
each, and
2 partial credit questions worth 10 points each.  Record your answers to the
multiple choice problems by  placing an $\times$ through one letter for each
problem on this answer sheet.  For the partial credit problems (27 and 28),
{\bf
show} all necessary work and {\bf clearly} identify your answers.  {\bf Good
Luck!} and {\bf Happy Holidays!}


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

\def\answerskip{\vskip0.15in}

\def\problemskip{\vskip0.1in}


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}}

\end


