\input /afs/nd.edu/user6/rosen/Public/LaTest/LaTest.tex
\documentstyle{latest} \textwidth 6.4in\textheight
8.5in\parindent 0cm\topmargin -1cm \oddsidemargin 0.7cm

\begin{document}
\baselineskip 0.5cm\everyproblem={\mathsurround=4pt}\pagenumberstyle=1
\normalafterproblemskip=2.2cm\pageannouncements={usual}
                                      \date{{\large December 15, 2000}}
                                      \lastversion=3
                                      \input exam.perm3
                                      \title{{\large\bf Math 119
                                          Calculus, Final}}
\comment{\large \hspace*{-1cm}
\begin{minipage}[l]{5.8in}%
  This Examination contains {\bf 25} problems, worth a total of
  {\bf 150} points, on {\bf 11} sheets of paper including the
  front cover.  Each Problem is worth {\bf 6} points.
  Please cross \fbox{$\times $} the correct answers for the
  multiple choice questions.  Books and notes are not allowed. You
  may use your calculator.  \vspace{1mm}
\end{minipage}} 
\answersheetfootline{\large\bf\fbox{\rule[-2mm]{0mm}{8mm}Sign
    your name:\hspace{6cm}} \hspace{6cm}}\twocolumn{16}
%\noprofessor 
\nosection \vertmultchoice=2.9cm \multiplechoiceskip=0.53cm
\let\ansborder=\boxborder\boxwidth=1cm\boxdepth=0.5cm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Pit-Mann
%

\problem Let $f(x)$ be a differentiable function of $x$. Which of
the following expressions is the derivative of the function $g(x)
= \frac{f(x)}{x^2}$ ?

\correct $\frac{x^2 f{'}(x) - 2x f(x)}{x^4}$
\wrong $\frac{2x f(x) - x^2 f{'}(x)}{x^4}$
\wrong $- 2 x^{-3} f^{'}(x) + x^{-2} f(x)$
\wrong $2x f(x) + \frac{f{'}(x)}{x^3}$
\wrong $2x f(x) - \frac{f{'}(x)}{x^3}$

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

\problem Assume $x\in \left[0,\frac{\pi}{2}\right]$. If $\sin x =
\frac{3}{5}$ then $\tan x = $

\correct $\frac{3}{4}$
\wrong $\frac{4}{5}$
\wrong $\frac{4}{3}$
\wrong $\frac{5}{4}$
\wrong $\frac{5}{3}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

\problem Suppose that $f = g \circ h + h \circ g$, $g(0) = 1$,
$h(0) = 2$, $g{'}(0) = 3$, $g{'}(2) = 5$, $h{'}(0) = 6$, $h{'}(1)
= 7$ then $f{'}(0) = $

\correct $51$
\wrong $41$
\wrong $31$
\wrong $21$
\wrong $11$

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

\problem Which of the following quantities is equal to
$$
\lim_{h \to 0} \frac{(2 + h)^3 -8}{h}
$$

\correct $f{'}(2), f(x) = x^3$
\wrong $f{'}(h), f(x) = x^2$
\wrong $3h^2$
\wrong $1$
\wrong $0$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

\problem The volume $V$ of a cube is increasing at a rate
of $300 {\rm cm}^3$ per minute. How fast is the length $L$ of
each side increasing where $L = 10$ cm ?

\correct $1 {\rm cm/min}$
\wrong $3 {\rm cm/min}$
\wrong $10 {\rm cm/min}$
\wrong $30 {\rm cm/min}$
\wrong $100 {\rm cm/min}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

\problem Compute the indefinite integral
$$
\int \frac{2x}{(x^2 + 1)^5}dx = 
$$

\correct $- \frac{1}{4 (x^2 + 1)^{4}} + C$
\wrong $\frac{1}{4 (x^2 + 1)^{4}} + C$
\wrong $\frac{(x^2 + 1)^{4}}{4 } + C$
\wrong $- \frac{(x^2 + 1)^{4}}{4 } + C$
\wrong $\frac{1}{x^4}+C$

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

\problem The function
$f(x) = \frac{x^2}{x^2 - 1}$ is increasing on the
interval(s):

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


\problem Which of the following is $\frac{f(x+h)-f(x)}{h}$ if
$f(x)=x^2 + 7x$? Note that you are not being asked to compute
the limit as $h\to 0$.

\correct $2x + 7 + h$
\wrong $2x + 7$
\wrong $2x + 7x + h$
\wrong $2x + 7h$
\wrong $2xh + h^2$


\problem Let 
$$
f(x) = (\sin(2x))^3.
$$
Which of the following is $f'(x)$?

\correct $6(\sin(2x))^2 \cos(2x)$
\wrong $3(\sin(2x))^2 \cos(2x)$ 
\wrong $-6(\sin(2x))^2 \cos(2x)$
\wrong $ 3\sin(x)\cos(x)$
\wrong $(\sin(2x))^2 $

%Exam 3:

\problem Which of the following equals
$$ \int (\cos(x) + 5)^3 \sin(x) dx \ ?$$

\correct $-\frac{(\cos(x) + 5)^4}{4} + C$
\wrong  $\frac{(\cos(x) + 5)^4}{4} + C$ 
\wrong $-3(\cos(x) + 5)^2$
\wrong$-\frac{(\cos(x) + 5)^4}{5}$
\wrong$\frac{(\cos(x) + 5)^4}{4}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem
If $f(x)=x^2$,\ $0\leq x\leq 6$, find the Riemann sum with $n =3$
taking the sample points to be  \underline{Midpoints}.
\correct  $70$
\wrong  35
\wrong  18
\wrong  9
\wrong 74


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


\problem
Evaluate the integral $\int_{0}^{2\pi}
(\sin x - \cos x)\,dx$. The integral is equal to

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




\problem Find the area enclosed by the curves $y=x$ and
$y=x^3$, between $x=0$ and $x=1$.

\correct $\frac{1}{4}$
\wrong $-\frac{1}{4}$
\wrong $\frac{3}{4}$
\wrong $\frac{1}{2}$
\wrong $1$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Mara
% ad Chapter 2

\problem{ A particle is moving in a straight line with position
  function $s(t)=\frac{t^3}{6}$.  What is the average velocity
  over the time period $[1,3]$.}

\correct{$\frac{13}{6}$}
\wrong $13$
\wrong $4$
\wrong $\frac{9}{2}$
\wrong $\frac{13}{3}$

\problem{Find the equation of the tangent line to the curve
$x^2 +y^2=169$ at the point $(12,5)$. }
\correct{$y=-2.4x+33.8$}
\wrong{$y=\frac{5}{12}x$}
\wrong{$y=\frac{-5}{12}x$}
\wrong{$y=2.4x +23.8$}
\wrong{$y=-2x +13$}


% ad Chapter 4

\problem{ Find horizontal and vertical asymptotes to
$f(x)=\frac{1}{x}+\frac{1}{x+1}$.}
\correct{vertical asymptotes at $x=0$ and $x=-1$, horizontal asymptotes at
$y=0$}
\wrong{vertical asymptotes at $x=0$ and $x=1$, horizontal asymptotes at
$\infty$}
\wrong{vertical asymptotes at $x=0$, horizontal asymptotes at
$y=-1$}
\wrong{vertical asymptotes at $x=0$ and $x=-1$, horizontal asymptotes at
$y=1$}
\wrong{vertical asymptotes at $x=0$, horizontal asymptotes at
$y=0$}



% ad Chapter 5

\problem{Evaluate the integral $\int_0^1 \root{3}\of{x}(x-1)dx$.}
\correct{$\frac{-9}{28}$}
\wrong{$\frac{3}{7}$}
\wrong{$\frac{11}{28}$}
\wrong{$\frac{-3}{7}$}
\wrong{$-6$}

\problem{Evaluate the integral $\int (1-x)\sqrt{2x-x^2}dx$.}
\correct{$\frac{1}{3}(2x-x^2)^{\frac{3}{2}}+C$}
\wrong{$(2x-x^2)^{\frac{3}{2}}+C$}
\wrong{$(2x-x^2)^{-\frac{1}{2}}+C$}
\wrong{$-(2x-x^2)^{-\frac{1}{2}}+C$}
\wrong{$(2-x)^{\frac{3}{2}}+C$}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Matt

\problem Which of the following numbers  is
equal to  the value of the limit
$$
\lim_{x\rightarrow 4}{{\sqrt x-2}\over{x-4}}
$$ 
 \correct $1/4$
 \wrong $1/2$
 \wrong $1$
 \wrong $2$
 \wrong $4$

\problem  Which of the following limits is equal to the derivative of
$f(x)={1\over\sqrt{x+1}}$ at $x=3$?
\correct $f'(3)=\lim_{h\rightarrow 0}{{1\over\sqrt{4+h}}-{1\over 2}\over h}$
\wrong $f'(3)=\lim_{h\rightarrow 3}{{1\over\sqrt{4+h}}-{1\over 2}\over h}$
\wrong $f'(3)=\lim_{h\rightarrow 0}{{1\over\sqrt{1+h}}-{1\over 2}\over h}$
\wrong $f'(3)=\lim_{h\rightarrow 3}{{1\over\sqrt{1+h}}-{1\over 2}\over h}$
\wrong $f'(3)=\lim_{h\rightarrow 0}{{1\over\sqrt{1+x+h}}-{1\over 2}\over h}$


%Test 3

\problem
Which one of the following functions $F(x)$ satisfies $F'(x)=1/x$ for $x>0$?
\correct $F(x)=\int_1^x {1\over t}dt$
\wrong $F(x)=-{1\over{x^2}}$
\wrong  $F(x)=\int_1^x -{1\over {t}^2}dt$
\wrong $F(x)=\int_1^x {1\over {t+1}}dt-1$
\wrong $F(x)={1\over x^2}$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Joachim
\problem 
 The product of two positive numbers is $900$.  What is the minimum their
{\bf sum} can be?


\correct  $60$
\wrong    $-60$
\wrong    $0$
\wrong    $30$
\wrong    $-30$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem
Evaluate the integral $\int_{-1}^0 (x+2)^3\, dx$
\correct $\frac{15}{4}$
\wrong $\frac{15}{16}$
\wrong  $\frac{1}{4}$
\wrong $7$
\wrong $\frac{7}{3}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem A balloon is rising vertically from a point $A$.  It has
constant speed of  5m/sec.  A girl is watching from a position exactly
80m from $A$.  How fast is the distance between the girl and the balloon
increasing when the balloon is 60m high?

\correct  $3\frac{m}{s}$
\wrong    $5\frac{m}{s}$
\wrong    $7\frac{m}{s}$
\wrong    $9\frac{m}{s}$
\wrong    $11\frac{m}{s}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem
On what intervals is $\displaystyle{f(x) = \frac{1}{4x-7}
}$ concave up?

\correct $\displaystyle{\left (\frac{7}{4},  \infty\right ) }$
\wrong $\displaystyle{ \left (-\infty, \frac{7}{4} \right ) }$
\wrong $\displaystyle{\left (-\infty, \frac{7}{4}\right ) \cup
\left (\frac{7}{4},\infty\right ) }$
\wrong $(-\infty, \infty)$
\wrong $(0, \infty)$







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



\endtest


