\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=2.8cm\pageannouncements={usual}
                                      \date{{\large November 28, 2000}}
                                      \lastversion=2
                                      \input exam.perm4
                                      \title{{\large\bf Math 119
                                          Calculus, Exam 3}}
\comment{\large \hspace*{-1cm}
\begin{minipage}[l]{5.8in}%
  This Examination contains {\bf 16} problems, worth a total of
  {\bf 100} points, on {\bf 11} sheets of paper including the
  front cover.  The first {\bf 12} problems are multiple choice
  with no partial credit, and each is worth {\bf 5} points.
  Please cross \fbox{$\times $} the correct answers for the
  multiple choice questions 1--12.  The last {\bf 4} problems are
  partial credit problems worth {\bf 10} points each. For these
  problems, {\bf show} your computations and {\bf clearly} mark
  your answers on the page.  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{7}
%\noprofessor 
\nosection \vertmultchoice=1.8in \multiplechoiceskip=0.55cm
\let\ansborder=\boxborder\boxwidth=1cm\boxdepth=0.5cm

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%1
\problem  $\lim_{x \to \infty} \frac{\sqrt{x^4
+ x^2 + 1} + \cos (x^2 + \pi)}{x^2} = $ 


\correct  $1$
\wrong    $2$
\wrong    $3$
\wrong    $0$
\wrong    $\infty$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem
If the second derivative of a function $f$ is
given by
$$
f{''}(x) = \frac{x (x - 2)(x - 4)(x - 6)}{(x^2 +1)}
$$
then the number of inflection points of the graph of
$f$ is

\correct  $4$
\wrong 3
\wrong 2
\wrong 5
\wrong $1$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem
The shortest distance from the origin to the
line $2 x + y = 4$ is 

\correct  $\frac{4}{5}\sqrt{5}$
\wrong 16/5
\wrong 4
\wrong 2
\wrong $1$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem
If $f{'}(x) = x^{-2}, f(1) = -1$ then $f(2)= ?$ 

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

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

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

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

\problem
Find all functions $f$ such that $f{''}(x)
= \sin x + \cos x$ ($A, B$ are constants)


\correct  $- \sin x - \cos x + Ax + B$
\wrong  $- \sin x - \cos x + A + B$
\wrong $\sin x - \cos x + A + B$
\wrong $\sin x + \cos x + Ax + B$
\wrong $\sin x + \cos x + A + B$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem Find the area under the graph of the function
$$
f(x) = \sqrt x
$$
from $x =0$ to $x=4$.
\correct ${{16}\over3}$
\wrong 1
\wrong ${{14}\over3}$
\wrong ${1\over4}$
\wrong ${{11}\over2}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem
Let $f(x) = 8x^3 - x^4$.  Notice that 
$$
f'(x) = 4x^2 (6-x)
$$ 
$$
f''(x) = 12x(4-x)
$$  
Which of the following describes the graph of $f(x)$?
\correct
local maximum at $x=6$; inflection points only at $x=0$ and $x=4$
\wrong
local minimum at $x=0$ and $x=6$; inflection point only at $x=4$
\wrong
local minimum at $x=0$; local maximum at $x=6$; no inflection points
\wrong
local minimum at $x=0$; local maximum at $x=6$; inflection points only at $x=0$ and
$x=4$
\wrong
local maximum at $x=6$; inflection point only at $x=4$
and $x=2$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem
Find all the horizontal and vertical asymptotes of
$y={{x+1}\over{{\sqrt {x-1}}}}$.
\correct
No horizontal asymptote; $x=1$
\wrong
$y=1$; $x=1$
\wrong
$y=-1$; $x=-1$
\wrong
$y=0$; $x=-1$
\wrong
$y=-1$; no vertical asymptote

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem
Evaluate the integral $\int_{-1}^1 (1-|x|)\, dx$
\correct $1$
\wrong $-1$
\wrong ${1\over 2}$
\wrong $-{1\over 2}$
\wrong $0$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\problem
Evaluate the integral $\int_{0}^1 (x+1)^3\, dx$
\correct $\frac{15}{4}$
\wrong $\frac{15}{16}$
\wrong  $\frac{1}{4}$
\wrong $7$
\wrong $\frac{7}{3}$

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Part II: Partial Credit Problems.

\problem
%{\bf Part II: Partial Credit Problems.}

Let $f(x) = \frac{x}{x^2 - 1}$. Then
$f{'}(x) = - \frac{x^2 + 1}{(x^2 - 1)^2}$ and $f{''}(x) = 
\frac{2x (x^2 + 3)}{(x^2 - 1)^3}$.


(a) Find all horizontal and vertical asymptotes.
\vspace{3.4cm}


(b) Find all critical points.
\vspace{3.4cm}


(c) Find the intervals on which the function is
increasing or decreasing.
\vspace{3.4cm}

(d) Sketch the graph of $f$. Indicate  the intervals on which the function is
concave up or down.

%\begin{center}
%%  \setlength{\unitlength}{15mm}
%\begin{picture}(10,8)(0,0)
%% \put(0,4){\vector(1,0){10}}
%% \put(5,0){\vector(0,1){8}}
%% \put(5.2,7.8){\makebox(0,0){$y$}}
%% \put(9.7,3.7){\makebox(0,0){$x$}}
% \multiput(0.8,3.9)(0.6,0){15}{\line(0,1){0.2}}
% \multiput(4.9,0.4)(0,0.6){13}{\line(1,0){0.2}}
%\end{picture}
%\end{center}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
Evaluate the following integrals:

\medskip
$(a) \int 3x^2
\sqrt{x^3 + 1} \, dx =$
\vspace{8cm}


(b) $\int \frac{\sin x}{\cos^3 x}\,dx =$
\vspace{8cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
A box with a square base and open top must have a volume of
$32,000\, {\rm cm}^3$. Find the dimensions of the box that
minimize the amount of material used.
\vspace{14cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\problem
Use Part 1 of the Fundamental Theorem of Calculus to find the
derivative $g'(x)$ of the function
$$
g(x)=\int_0^x \sqrt{1+2t}\, dt
$$

\endtest


