\documentstyle{letter}
\textwidth 6.2in
\textheight 8.7in  \parindent 0cm
\topmargin -1cm
\oddsidemargin 0.7cm

\begin{document}

\baselineskip 0.5cm
\everyproblem={\mathsurround=4pt}
\normalafterproblemskip=2.5cm
\pageannouncements={usual}

\input final.perm1

\title{{\large\bf MATH 105, FINAL EXAMINATION}}
\comment{ \large 
\begin{minipage}[c]{6.2in}
\begin{center}{\bf READ THESE DIRECTIONS}\end{center}
\vspace*{1cm}
1. This exam consists of 30 multiple choice problems worth 5 points
each with no partial credit.
\\
\\
2. For each multiple choice problem mark an {\bf X} through the one 
correct answer \\
{\bf on this page}.
\\
\\
%3. {\bf On all partial credit problems, show your work, indicating clearly how you
%arrived at your answer. The points you receive for problems 14 and 15 will depend on the
%extent to which the work you show convinces the grader (at the time of grading) that you
%did all or part of the problem correctly.}
%\\ 
%\\
 3. This test will be exactly  120 minutes in
length.   When you are told to begin, but not before, glance 
through the entire test.  No calculators allowed. After you finished 
return the answer sheet only.
\end{minipage}} 
\answersheetfootline{\large\bf 
\fbox{\rule[-2mm]{0mm}{8mm}Sign your name:\hspace{6cm}}
\hspace{6cm}}
\twocolumn{16} 
\date{{\large December 12, 1992}}

\lastversion=4 

\vertmultchoice=2.5in
\multiplechoiceskip=0.5cm \let\ansborder=\boxborder \boxwidth=5mm
\boxdepth=0.2cm

\pagenumberstyle=1



 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\note 

%{{\bf MULTIPLE CHOICE SECTION} (5 points each)}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm 

Find the $x$-intercept and the $y$-intercept of the line $$5x +
12y =120.$$



\correct (24, 0) and (0,10)
\wrong (5, 0)  and  (0, 12)
\wrong (12, 0)  and  (0, 5)
\wrong (0, 0)  and    (0, 0)
\wrong (17, 0)   and  (0, 7)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm
The lines  $2x - y = 0$  and  $x - 2y + 3 = 0$  intersect at the point




\correct (1, 2)
\wrong (0, 0)
\wrong  (2, -1)
\wrong  (3, 0)
\wrong  the lines have no point in common.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm

Which of the lines below is parallel to the line   $y = 2x + 3$?


\correct $10x - 5y = 14$ 
\wrong $11x + 6y = 0$
\wrong $y = -2 x$
\wrong $ x=-1$
\wrong $ y = 3$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm

Find the slope of a line, which passes through the points (-1, 7) and (2, -2).

\correct -3
\wrong 3
\wrong -7
\wrong 2
\wrong $-\frac{2}{7}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm

  Find the equation of the line tangent to the curve  $y = x^3 - 2x + 1$  at the
point (2, 5).


\correct $y = 10 x - 15$
\wrong $y = 3x^2 - 2$
\wrong $x = 2$
\wrong $y = 7x + 1$
\wrong $y = 5x$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm

 Find the limit

			$$\lim_{x \to \infty} \frac {2x^3 + 2x^2 + x + 3}{x^3 - 3x^2 + 2x + 1}.$$



\afterproblemskip = 1.cm
\correct 2
\wrong  $\frac{2}{3}$
\wrong 3
\wrong $ \frac{1}{2}$
\wrong the limit does not exist
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\problem 
\afterproblemskip = 2.cm

 Find the limit $$\lim_{x \to 3} \frac {x^2 -2x - 3}{x^2 - 9}.$$


\correct $\frac{2}{3}$
\wrong $\frac{1}{3}$
\wrong 3
\wrong 2
\wrong the limit does not exist



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm

  Compute the following logarithm    $$\ln \frac{e^3}{\sqrt{e^5}}.$$


\correct $\frac{1}{2}$
\wrong  0
\wrong $\frac{6}{5}$
\wrong $\ln \frac{6}{5}$
\wrong $e^{\frac{6}{5}}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm

Simplify the expression    $e^{2+2\ln2}.$




\correct $4e^2$
\wrong $e^4$
\wrong 0
\wrong $2^{\ln 2}$
\wrong 4

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm

 In which interval does the function  $f(x)= e^x - x$   increase?




\correct $(0, \infty)$
\wrong $(-\infty, e)$
\wrong $(-e, e)$
\wrong $(-\infty, 0)$
\wrong the function decreases everywhere 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm

Determine in which interval is the graph of the function  $f(x) = x^3 -
3x^2 + x - 5$ concave up.

\correct $(1, \infty)$
\wrong $ (-\infty, 0)$
\wrong $(-\infty, 1)$ 
\wrong $(0, \infty)$
\wrong $(-1, 1)$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm

What are the inflection points of the function  
$f(x) = x^4 - 6x^2 + x + 1$.


\correct (1, -3) and (-1, -5)
\wrong (0, 0) only
\wrong (3, 1) and (3, -1)
\wrong (0, 1) and (1, 0)
\wrong the function has no inflection points

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2.cm

Find the largest value (absolute maximum) of the function  $f(x) = x^4 - 2x^2$  
in the interval  [-2, 2].


\correct 8
\wrong 1
\wrong 5
\wrong -1
\wrong 10


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

\problem
\afterproblemskip = 2.cm

Find the smallest value (absolute minimum) of the function  $f(x) = x^4 -
2x^2$   in the interval  $[-2, 2].$



\correct -1
\wrong -4
\wrong 0
\wrong 2
\wrong 1

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm

Find the dimensions of a closed rectangular box with square base and volume 1000
cubic inches that can be constructed with the least amount of material.

\correct  $(10, 10, 10)$
\wrong $(5, 5, 40)$
\wrong $(8, 8, \frac {125} 8)$
\wrong $(12, 12,\frac {125}{18})$ 
\wrong $(20, 20, 2.50$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm
 
Find the approximate value of $\sqrt {4.4}$ using differential approximation.

\correct 2.1
\wrong  2.4
\wrong  2.2
\wrong  2.5
\wrong  2.05

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm


The derivative of $f(x) = (x^4 - 1)(x^3 + 2)$ at $x = 1$ equals

\correct 12
\wrong -2
\wrong 5
\wrong 4
\wrong -3

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm

Find the value of the derivative of the function $e^x \ln x$ at the point
$x=2$

\correct $e^2 \ln 2 + \frac 1 2 e^2$
\wrong $\frac{e^2}{\ln 2}$
\wrong 4
\wrong $\frac{e^2}2$
\wrong $1 + \ln x$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm

Find the derivative of $\ln (x^4 + e^{5x}).$


\correct $\frac {4x^3 + 5e^{5x}}{x^4 + e^{5x}}$
\wrong $\ln (x^4 + e^{5x}) (4x^3 + 5e^{5x})$
\wrong $4x^3 + 5e^{5x}$
\wrong $\ln (4x^3 + 5e^{5x})$
\wrong $\frac {x^4 + e^{5x}}{4x^3 + 5e^{5x}}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm

The function $f(x) = 2e^{{(x+1)}^2}$ has a relative minimum with $x$-coordinate

\correct -1
\wrong 0
\wrong 1
\wrong 2
\wrong -2

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm

Find the derivative of the function  $$\ln \frac{(x^2+2x)\root 3
\of{x^3+1}}{(x^4+2x^3+x)^3}$$


\correct $\frac {2x+2}{x^2+2x} + \frac{x^2}{x^3+1} - 3 \frac{4x^3 +
6e^2 + 1}{x^4+2x^3+x}$
\wrong $\frac {1}{x^2+2x} + \frac{1}{x^3+1} - \frac{1}{x^4+2x^3+x}$
\wrong $\frac{(2x+2)+x}{(x^4+2x^3+x)^6}$
\wrong $\frac{(x^4+2x^3+x)^3}{(x^2+2x)\root 3\of{x^3+1}}$	
\wrong $\frac{1}{x^2+2x} + \frac {1}{(x^2+2x) \root 3
\of{x^3+1}} + 3(x^4+2x^3+x)^3$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\problem
\afterproblemskip = 2cm

An object moving on a straight line travels $s(t)=6t+t^2$ meters in $t$
minutes. After how many minutes will its velocity reach 8 meters per minute.


\correct 1
\wrong $\sqrt 6$
\wrong 2
\wrong $\sqrt 2$
\wrong $\frac 1 3$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm
  

 The half life of Carbon 14 is 5,750
years. How much of the initial 50 gram of Carbon 14 will remain after 2000 years?

\correct  $50e^{-\frac {2000\ln 2}{5750}}$
\wrong $50 \frac {2000}{5750}$
\wrong $\frac {\ln 2}{5750}$
\wrong 0
\wrong 25

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm

The function $y = f(x)$ satisfies the conditions $y^\prime = 3y,\ \ \  f(0) =
2$. Find the value $f(5).$

\correct $2e^{15}$
\wrong $3e^{10}$
\wrong $5e^6$
\wrong $2\ln 3$
\wrong 30

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm

Find the value at $x = 2$ of the second derivative ot the function $3x^3 + 2x^2
+ x.$

\correct 40
\wrong 34
\wrong 2
\wrong 6
\wrong 24

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

\problem
\afterproblemskip = 2cm

The velocity of a rock at time t after it was thrown straight up from a tower 5
meters high is given by formula $v(t)=-9.8t+20$ meters per second. Find the
formula for the height $s(t)$ [in meters] above ground level.

\correct $-4.9 t^2 + 20t + 5$
\wrong $-9.8 t^2 + 20t$
\wrong $-9.8t+20 + 5$
\wrong $-\frac {9.8}3 t^3 + 5t^2 + 20$
\wrong $5t^2 + 20t + 9.8$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm

Knowing that $F^\prime = 3\sqrt{x} - 2$ and $F(0)=-2$ find $F(4)$

\correct 6
\wrong 0
\wrong -2
\wrong 7
\wrong 8

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm

Find the integral $\int (x^4 + x^2 + e^{2x})dx$.


\correct $\frac {1}{5} x^5 + \frac {1}{3} x^3 + \frac {1}{2} e^{2x} + C$
\wrong $4x^3 + 2 + 2e^{2x}$
\wrong $\frac1 5 x^5 + \frac 1 3 x^3 + \frac 1 {2x+1} e^{2x+1}$
\wrong $5 \ln x + 3\ln x + 2x$  
\wrong $x^5+x^3+xe^{2x}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm

Calculate the integral $$\int_1^3 \frac {2}{x} dx.$$

\correct  2$\ln 3$
\wrong $\frac {2}{3}$
\wrong $2e^3$
\wrong $3e^2$
\wrong $\ln 8$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
\afterproblemskip = 2cm

Find the area under the curve $y = 3x^2$ for $x$ between 1 and 2.

\correct 7
\wrong 3
\wrong 6
\wrong 8 
\wrong 9


\endtest



