\input ignore.tex %This line causes the mce macros to be ignored.

\preface
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\def\problemskip{\vskip0.5in}

\title
\noindent
{\bf Math 119}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam III}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl April 14, 1997}\hfill
Section:\hbox to 2.5truein{\hrulefill}


\def\answerskip{\vskip0.25in}


\note %1
\problem
Where on the interval $(0,2\pi)$ is the graph of 
$$f(x) = x^2 + \sin x$$
concave up?
\correct on $(0, 2\pi)$
\wrong nowhere
\wrong on $(0,\pi)$
\wrong on $(0,{\pi\over2})$ and $(\pi,{{3\pi}\over2})$
\wrong on $({\pi\over2},\pi)$ and $({{3\pi}\over2},2\pi)$


\note %2
\problem
Which of the following  is true?
\correct
if $f(-x)=f(x)$, then $f$ is symmetric around $y$-axis
\wrong
if $f$ has an absolute minimum value at $c$, then $f'(c)=0$
\wrong
if $f''(c)=0$, then $(c,f(c))$ is an inflection point of the
curve $y=f(x)$
\wrong
if $f'(x)=g'(x)$ for $0<x<1$, then $f(x)=g(x)$ for $0<x<1$.
\wrong
$\sum_{i=1}^{n} a_i b_i 
=\big( \sum_{i=1}^{n} a_i\big) \big(\sum_{i=1}^{n} b_i\big) $


\note %3
\problem
How many real roots does the quartic equation 
$x^4 + 10 x^2 +21$ have?
\correct none
\wrong 1
\wrong 2
\wrong 3
\wrong 4




\note %4
\problem
Find $f(x)$ if $f'(x)=\sec x\tan x$ and $f(0)=0$.
\correct
$\sec x -1$
\wrong
$-\sec x$
\wrong
$\tan x\sec x$
\wrong
$\tan x\sec x +1$
\wrong
$\tan x$


\note %5
\problem
Find $\lim_{x\rightarrow \infty}({\sqrt {x^2 +x+1}}-{\sqrt {x^2-x}})$.
\correct
1
\wrong
-1
\wrong
0
\wrong
2
\wrong
does not exist



\note %6
\problem
Find all the points on the hyperbola $y^2 - x^2=4$ which are closest 
to the point $(2,0)$.
\correct
$(1,\pm \sqrt 3)$
\wrong
$(1,\sqrt 3)$
\wrong
$(0,1)$
\wrong
$({\sqrt 2}, 1)$
\wrong
$(-{\sqrt 3}, 1)$


\note %7
\problem
Given that the graph of $f$ passes through the point $(1,2)$
and the slope of its tangent line at $(x, f(x))$ is $2x-1$,
find $f(0)$.
\correct
$2$
\wrong
$-2$
\wrong
$0$
\wrong
${1\over 2}$
\wrong
$-{1\over 2}$


\note %8
\problem
Which of the following lines are vertical asymptotes of the graph of
$y = \tan 2x$?
\correct $x = -{\pi\over4}$ and $x = {\pi\over4}$
\wrong $y = 0$
\wrong $x = -{\pi\over2}$ and $x = {\pi\over2}$
\wrong $x = 0$
\wrong $x = -{\pi}$ and $x = \pi$


\note %9
\problem
An open-top box with a square base is to be made in such a way that its volume equals 4 cubic feet. Let $S$ stand for the total surface area of the box.
Which of the following statements is true?
\correct There is a minimum possible value for $S$, but not a maximum.
\wrong There is a maximum possible value for $S$, but not a minimum.
\wrong There are both minimum and maximum possible values for $S$.
\wrong There is neither a minimum nor a maximum possible value for $S$.
\wrong More information is needed to determine if there are minimum and/or
 maximum possible values for $S$.


\note %10
\problem
Calculate $\sum_{k=0}^{4} \cos {k\pi \over 2}$
\correct
$1$
\wrong
$1+{{\sqrt 3}\over 2}$
\wrong
$-1$
\wrong
$0$
\wrong
${1\over 2}$

\note %11
\problem
Find $\lim_{x\rightarrow \infty} x\sin (1/x)$.
\correct
$1$
\wrong
$0$
\wrong
$-1$
\wrong
undefined because it approaches $\infty$
\wrong
undefined because it approaches $-\infty$



\note %12
\problem
Find all the horizontal and vertical asymptotes of 
$y={{\sqrt {x^2 -9}}\over {2x-6}}$.
\correct
$y=\pm {1\over 2}, x=3$
\wrong
$y={1\over 2}, x=-3$
\wrong
$y=-{1\over 2}, x=-3$
\wrong
no horizontal asymptote, $x=3$
\wrong
no horizontal or vertical asymptotes

\note %13
\problem
Find a description which 
is {\bf false} for a curve $y=f(x)$ having the following 
characteristic (assume the domain of $f$ is $(-\infty ,\infty)$) :
$$\displaylines{
f'(2)=f'(-2)=0\cr
f'(x)>0 \quad {\rm for}\quad |x|>2\cr
f'(x)< 0 \quad{\rm for}\quad |x|<2\cr
f''(x)<0 \quad{\rm for}\quad x<0,\quad f''(x)>0 \quad{\rm for} \quad 
x>0\cr
f(-2)=8, f(0)=4, f(2)=0\cr}
$$
\correct
increasing on $[-2, 2]$ 
\wrong
$x=0, y=4$ is an inflection point of $f$
\wrong
CD on $(-\infty , 0)$
\wrong
CU on $(0,\infty)$
\wrong
$x=\pm 2$ are critical points of $f$

\note %14
\problem
What is the general antiderivative of $f(x)={{x^2+x}\over{x^{3/2}}}$.
\correct
${2\over 3}x^{3/2} +2x^{1/2}+C$
\wrong
${3\over 2}x^{3/2} +2x^{1/2}+C$
\wrong
${2\over 3}x^{3/2} -2x^{1/2}+C$
\wrong
$x^{3/2} - 2x^{1/2}+C$
\wrong
${2\over 3}x^{1/2} +2x^{-1/2}+C$


\note %15
\problem {\bf Partial Credit}
\hskip .5cm
An athletic field is to be built in the shape of a rectangle with a semi-circle
attached to the east and west ends, as shown in the picture.  The outer
boundary is to be used as a race-track and is to be 2 miles long. What is the
largest possible area of the rectangular part
(shaded in the picture)? 
\smallskip
{\bf Note:} If your answer involves a rational number, such as  $\pi$, or
$\sqrt 2$, leave it in that form.  
\vskip 3cm

\note %16
\problem {\bf Partial Credit}
\hskip .5cm
Sketch the graph of the curve $y=|x^2 -3|$.
\vskip 2cm

\appendix
That's all folks!

\end
