
\documentstyle{latest}
\textwidth 6.3in\textheight 8.5in\parindent 0cm
\topmargin -1cm\oddsidemargin 0.7cm

\begin{document}
\baselineskip 0.5cm\everyproblem={\mathsurround=4pt}
\normalafterproblemskip=2.5cm\pageannouncements={usual}
                                         \input exam.perm 
                                         \lastversion=3
                                         \date{{\large May 6, 1999}}
                                         \title{{\large\bf Math 105, Final Exam}}
\comment{\large 
\begin{minipage}[c]{6in}
This exam is worth 150 points.  There are 20 multiple choice questions, each worth
5 points, and 5 partial credit questions.  You will have 2 hours to complete the
exam.  Calculators are allowed.  Record your answers to the multiple choice problems
by placing an $\times$ in the appropriate box.  Answers to problems in the partial credit 
section may be written in terms of $\ln x$ or $e^x$, and need not be simplified. 
Remember that you are taking this examination under the honor code. 
Good Luck.  



\answersheetfootline{\bf\fbox{\rule[-2mm]{0mm}{6mm}Sign your name-\hspace{10cm}}
\hspace{6cm}}\twocolumn{11}\noprofessor \nosection \vertmultchoice=1in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=.7cm
\boxdepth=0.4cm\pagenumberstyle=1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem %#1% 

What is the equation of the line through the points $(2,3)$ and $(-1,9)$?


\wrong $y=-\frac{1}{2}x+7$
\wrong $y=-\frac{1}{2}x+4$
\correct $y=-2x+7$
\wrong $y=-2x-3$
\wrong $y=-2x-4$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#10%

Let $f(x)=x^4-2x^3+3x+5.$  What is $f''(x)$?


\wrong $7x^2 -8x$
\wrong $12x^2-12x+8$
\wrong $4x^3-6x^2+3$
\correct $12x^2-12x$
\wrong $4x^3-6x^2+5$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#18%

If you use linear approximation to estimate $\frac{1}{(1.00136)^5}$, what answer
do you get?

\correct $1-5(0.00136)$
\wrong $1+6(0.00136)$
\wrong $1.00136-5(0.00136)^{-6}$
\wrong $\frac{1}{5}(0.00136)+1$
\wrong $\frac{1}{5}(0.00136)+1.00136$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#14%

What is the equation for the slope of the tangent line to the graph of
$f(x)=x^3\ln{x}$?


\wrong $3x$
\correct $x^2(1+3\ln{x})$
\wrong $x^2\ln{x}$
\wrong $x(x\ln{x}+3)$
\wrong $3x^2\ln{x}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#3%

Which of the following is true about the function
$$f(x)=\frac{2x^2+3x-7}{x^2-1}?$$

\correct $f(x)$ has vertical asymptotes at $x=1$ and $x=-1$ and a horizontal
asymptote at $y=2.$
\wrong $f(x)$ has vertical asymptotes at $x=1$ and $x=-1$ and a horizontal
asymptote at $y=0.$
\wrong $f(x)$ has a vertical asymptote at $x=1$  and no horizontal
asymptotes.
\wrong $f(x)$ has a vertical asymptote at $x=1$ and horizontal
asymptotes at $y=2$ and $y=1.$
\wrong $f(x)$ has no vertical asymptotes and no horizontal asymptotes.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#16%

The volume of a sphere is expanding.  The volume of a sphere of radius $r$ is
$V=\frac{4}{3}\pi r^3$.  The rate of expansion of the radius ($\frac{dr}{dt}$) is $5$
in/sec when $r=2.$  What is $\frac{dV}{dt}$ when $r=2$?


\wrong $16\pi$
\wrong $\frac{32}{3}\pi$
\wrong $\frac{32}{15}\pi$
\correct $80\pi$
\wrong $23\pi$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#7%

If an investor opens an account with \$10,000 where interest is 4\% compounded
continuously, how long will it take for the investment to triple?


\wrong $\frac{-\ln{3}}{0.04}$
\wrong $\frac{\ln{30000}}{\ln{10000}}$
\correct $\frac{\ln{3}}{0.04}$
\wrong $\frac{\ln{0.04}}{3}$
\wrong $\frac{-\ln{0.04}}{3}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem %#2% 

A ball is thrown into the air at a velocity of 96 ft/sec from the top of a
16 foot tall platform.  The equation for the height of the object at any
time $t$ is then given by $h(t)=-16t^2+96t+16.$  When does the ball hit the
ground?


\wrong $48+16\sqrt{10}$ seconds
\wrong $3$ seconds
\wrong $6$ seconds
\wrong $2$ seconds
\correct $3+\sqrt{10}$ seconds

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#20%

What are the critical points of the function $f(x)=e^{x^3-3x}$?


\wrong $x=1$ 
\wrong $x=0$
\wrong $x=1, x=-1$ and $x=0$
\correct $x=1$ and $x=-1$
\wrong There are no critical points

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#9%

If $f(x)=\sqrt{x^2+1}$, which of the following limits equals $f'(x)$?

\correct $\lim_{h\to0}\frac{\sqrt{(x+h)^2+1}-\sqrt{x^2+1}}{h}$
\wrong $\lim_{h\to0}\frac{\sqrt{x^2+1}+h-\sqrt{x^2+1}}{h}$
\wrong $\lim_{h\to0}\frac{\sqrt{x^2+h+1}-\sqrt{x^2+1}}{h}$
\wrong $\lim_{h\to0}\frac{\sqrt{x^2+h^2+1}+\sqrt{x^2+1}}{h}$
\wrong $\lim_{h\to0}\frac{\sqrt{(x+h)^2+1}+\sqrt{x^2+1}}{h}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#6%

Which of the following is an equivalent expression for $\ln{(3e^{2x})}$?


\wrong $2x\ln{3}$
\correct $\ln{3}+2x$
\wrong $6x$
\wrong $2x+3$
\wrong $3\ln{2x}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#19%

Suppose $f(x)$ is a function which satisfies the following:

$\bullet$ The only critical points are $x=-1$ and $x=2$.

$\bullet$ $f'(-3)=2, f'(0)=-1, f'(3)=-2.$

Which of the following must be true?

\correct $f(x)$ has a relative maximum at $x=-1$
\wrong $f(x)$ has a relative minimum at $x=-3$
\wrong $f(x)$ is increasing on $(-\infty,0)$
\wrong $f(x)$ is concave up on $(-1,2)$
\wrong $f(x)$ has a relative minimum at $x=2$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#4%

A clock company finds that it can sell each clock for \$20 apiece.  If the cost 
of manufacturing $x$ clocks is $C(x)=10x+1300$, what is the company's \underline{profit} 
function?


\wrong $P(x)=30x+1300$
\wrong $P(x)=200x+26000$
\correct  $P(x)=10x-1300$
\wrong $P(x)=10x+1280$
\wrong $P(x)=-10x+1300$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#13%

Compute the derivative of 
$\frac{e^{3x}}{x^2}.$


\wrong $\frac{xe^{3x}(2-3x)}{x^4}$
\wrong $\frac{3e^{3x}}{2x}$
\wrong $\frac{3e^{3x}(x^2)}{2xe^{3x}}$
\wrong $\frac{2xe^{3x}}{3x^2e^{3x}}$
\correct $\frac{xe^{3x}(3x-2)}{x^4}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#8%

Over the past ten years, a certain stock has yielded an annual interest rate
of 8\%, compounded \underline{monthly}.  If you bought \$3000 worth of this stock 10 years
ago, how much would that investment be worth now?

\correct $3000(1+\frac{0.08}{12})^{120}$
\wrong $3000(1+0.08)^{120}$
\wrong $3000(1+\frac{0.08}{12})^{10}$
\wrong $3000e^{0.08}$
\wrong $3000e^{120}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem  %#11%

What is the equation of the tangent line to $f(x)=e^x+1$ at the point $(0,2)$?


\wrong $y=2x+3$
\correct $y=x+2$
\wrong $y=\frac{1}{2}x+2$
\wrong $y=\frac{1}{e}(x-2)$
\wrong $y=x+5$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#17%

A hard-boiled egg of $150^{\circ}$F is set to cool in a room of $60^{\circ}$F.
If after 1 minute the egg has cooled to $140^{\circ}$F, then what is the equation
for the temperature of the egg at any time t?  Recall that Newton's Law states that
the temperature of the egg at any time t is given by the equation 
$H(t)=60+Ae^{kt}.$

\correct $H(t)=60+90e^{\ln{(8/9)}t}$
\wrong $H(t)=60+90e^{\ln{(4/9)}t}$
\wrong $H(t)=60+90e^{\ln{(-1/9)}t}$
\wrong $H(t)=60+10e^{\ln{(8/9)}t}$
\wrong $H(t)=60+10e^{\ln{(-1/9)}t}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#12%

Compute the derivative of $\ln{(2x)}-3x^2+2$.


\wrong $\frac{1}{2x}-6x$
\wrong $\frac{1}{x}-6$
\wrong $\frac{1}{2}+\frac{1}{x}-x$
\correct $\frac{1}{x}-6x$
\wrong $\frac{1}{2+x}-6x$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#5%

Let 
$$ f(x)=\cases{
  -x^2+3,&if $x<0$;\cr
-2x+2,&if $0\leq x < 2$;\cr
x^2-6,&if  $x\geq2$.\cr}$$
Where is $f(x)$ discontinuous?


\wrong $x=0$ and $x=2$
\wrong $x=2$
\correct $x=0$
\wrong $x=3$
\wrong nowhere


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem  %#15%

If $f(x)=(x^3-4x^2+6)^{1/2}$, then what is $f'(x)$?

\correct $\frac{1}{2}(x^3-4x^2+6)^{-1/2}(3x^2-8x)$
\wrong $\frac{1}{2}(x^3-4x^2+6)^{-1/2}$
\wrong $\frac{1}{2}(3x^2-8x)^{-1/2}$
\wrong $\frac{3}{2}x^{1/2}-4$
\wrong $\frac{3}{2}x^{1/2}-4+\frac{1}{2\sqrt{6}}$

\pagebreak

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem
(10 points)

A photographer knows that she can sell 90 reproductions of one of her prints
for \$20 each.  For each \$1 increase in price, she will sell three fewer prints.
How much should she charge per print in order to maximize revenue? 
\hfil\break
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problem
(10 points)

Where is the function $f(x)=x^4-2x^2+1$ concave up and where is it concave down? 
\hfil\break
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
(8 points)

Consider the graph of the function $g(x).$
%%    see file final-pic.ps for picture in postscript format.
\vskip 4.5in
Where is this function increasing, and where is it decreasing?
\vskip .7in
Does this function have relative maximum or minimum points, and if so where?
\vskip .7in
Where does the function appear to be concave up and concave down?
\vskip .7in
What are the horizontal and vertical asymptotes of this function?
\hfil\break
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
(12 points)
Suppose that $f(x)$ is a function which satisfies the following list of properties:

$\bullet f(-1)=3, f(0)=4, f(1)=0, f(2)=-2,$ and $f(3)=0.$ 

$\bullet$ The only solutions to $f'(x)=0$ are $x=0$ and $x=2$.

\hskip 1in $\bullet f'(x)>0$ on $(-\infty,0)$ and $(2,\infty)$

\hskip 1in $\bullet f'(x)<0$ on $(0,2)$

$\bullet$  The solutions to $f''(x)=0$ are $x=-1, x=1$ and $x=3$.

\hskip 1in  $\bullet f''(x)>0$ on $(-\infty,-1)$ and $(1,3)$

\hskip 1in $\bullet f''(x)<0$ on $(-1,1)$ and $(3,\infty)$

$\bullet f(x)$ has no vertical asymptotes.

$\bullet f(x)$ has a horizontal asymptote at $y=2.$


Using this information, sketch the graph of $f(x).$
\hfil\break
\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
(10 points)

Consider the function
$$g(x)=\frac{x^2+3x-1}{x-2}.$$
The function $g(x)$ has critical points at $x=5$ and $x=-1$. Given that the
second derivative is 
$$g''(x)=\frac{18}{(x-2)^3},$$
determine whether the critical points are relative maxima or relative minima. 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\endtest


