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\title
\noindent
{\bf Math 119: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam III}\hfill
Tutorial Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl December 6, 1994}\hfill
Tutorial Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Calculators are not allowed.  Hand in this answer page only.  Record your answers 
to the multiple choice problems by  placing an $\times$ through one letter for
each problem on this answer sheet. There are  19 multiple choice questions, worth 5
points each.  An additional 5 points will be given for your correct tutorial
section number.  \vskip .25cm \noindent {\bf You are taking this exam under the
honor code.}

\def\answerskip{\vskip0.25in}

\def\problemskip{\vskip1in}


\note %1
\problem
Let $\displaystyle f(x) = {{\cos^2 x} \over {\sin x}}$.  Which of
the following statements is true about the limits 
\smallskip
\noindent as $\displaystyle x \rightarrow
\pi$ from the left and from the right?
\smallskip

\correct
$\displaystyle \lim_{x \rightarrow \pi -} f(x) = + \infty$, 
$\displaystyle \lim_{x \rightarrow \pi +} f(x) = - \infty$.
\wrong
$\displaystyle \lim_{x \rightarrow \pi -} f(x) = + \infty$, 
$\displaystyle \lim_{x \rightarrow \pi +} f(x) = + \infty$.
\wrong
$\displaystyle \lim_{x \rightarrow \pi -} f(x) = - \infty$, 
$\displaystyle \lim_{x \rightarrow \pi +} f(x) = - \infty$.
\wrong
$\displaystyle \lim_{x \rightarrow \pi -} f(x) = - \infty$, 
$\displaystyle \lim_{x \rightarrow \pi +} f(x) = + \infty$.
\wrong
$\displaystyle \lim_{x \rightarrow \pi -} f(x) = 1$, 
$\displaystyle \lim_{x \rightarrow \pi +} f(x) = - 1$.

\note %2
\problem
Let $$ f(x) = \cases{x^2 + 1, &if $0 \leq x \leq 1$;\cr
                   2x,& if $1 \leq x \leq 2$.\cr}
$$
Find $A_0^x (f)$ for $x$ satisfying $1 \leq x \leq 2$.
\correct
$\displaystyle A_0^x (f) = x^2 + {1 \over 3}$
\wrong 
$\displaystyle A_0^x (f) = x^2$
\wrong 
$\displaystyle A_0^x (f) = x^2 + {4 \over 3}$
\wrong
$\displaystyle A_0^x (f) = {1 \over 3} x^3 + x$
\wrong
$\displaystyle A_0^x (f) = {1 \over 3} x^3 + x + {4 \over 3}$

\note %3 
\problem
Consider the function $$f(x) = 6x^2 + 1$$  Find the area under the graph of $f(x)$
between
$x = -1$ and $x = 2$.
\correct
21
\wrong
18
\wrong
15
\wrong
9
\wrong
36

\note %4
\problem
Consider the function $$f(x) = x + 1$$  Find ${\bf U}_0^4 (P,f)$ (the upper sum)
if
$P$ partitions the interval $[0,4]$ into four equal segments.
\correct 
14
\wrong
10
\wrong
6
\wrong
18
\wrong
12

\note %5
\problem
Let $f(x)$ be a function such that $f(x)$, $f'(x)$ and $f''(x)$  exist for all
$x$, and such that
\hfil\break$\displaystyle f''(x) = {{x^2 - 4}
\over {\sqrt{x^4 + 1}}}$.  You {\it don't} have to figure out what $f(x)$ is.  But
which of the following is true about the graph of $f(x)$ at $x=2$?
\correct
It {\it must} have an inflection point at $x=2$.
\wrong
It {\it must} have a local maximum at $x=2$.
\wrong
It {\it must} have a local minimum at $x=2$.
\wrong
At $x=2$ it may have a local minimum or maximum, but it cannot have an inflection
point.
\wrong
From the given information it is impossible to tell if it has a local maximum,
local \hfil\break minimum or inflection point at $x=2$.

\note %6
\problem
Let  $\displaystyle f(x) = {1 \over x} + {1 \over 4}x + 2$.  Note that
$\displaystyle f'(x) = {{-1} \over {x^2}} + {1 \over 4}$ and $\displaystyle
f''(x) = {2
\over {x^3}}$.  On what \hfil\break \smallskip \noindent interval(s) is
$f(x)$ decreasing?  \smallskip
\correct
$\displaystyle -2\leq x < 0$ and $\displaystyle 0 <x\leq 2$
\wrong
$\displaystyle -\infty < x \leq -2$ and $\displaystyle 2 \leq
x <
\infty$
\wrong
$\displaystyle -\infty < x \leq -2$ and $\displaystyle 0 <x\leq 2$
\wrong
$\displaystyle -2 \leq x < 0$ and $\displaystyle 2 \leq x <
\infty$
\wrong
$\displaystyle -\infty < x < 0$

\note %7
\problem
Suppose $f(x) = 1+x^2$ for $-3 \leq x \leq 2$.  Take the partition $P$ that
divides the interval $[-3,2]$ into five equal segments, as shown in the diagram
below.  (Note that the diagram is not drawn perfectly to scale.)  Find the lower
sum
${\bf L}_{-3}^2 (P,f)$.

\vskip 2.5in
\hskip 1in\parabola2

\correct 
11
\wrong
24
\wrong
6
\wrong
19
\wrong
9

\note %8
\problem
Let $\displaystyle g(x) = {{\sqrt{x^2 + 9}} \over {2x+7}}$. 
Find the horizontal asymptote(s) of $g(x)$, if there are any.
\correct
$\displaystyle y = -{1 \over 2}$ and $\displaystyle y = {1 \over 2}$
\wrong
Only $y = 0$
\wrong
Only $\displaystyle y = {1 \over 2}$
\wrong
$g(x)$ has no horizontal asymptotes
\wrong
Only $\displaystyle y = {3 \over 7}$

\note %9
\problem
A ball is thrown upward with a velocity of 16 ft/sec from a height of 32
ft.  After how \hfil\break many seconds does it hit the ground?  (Remember that the
acceleration due to gravity is \hfil\break
$-32$ ft/$\hbox{sec}^2$.)
\correct
2 seconds
\wrong
$\displaystyle {1 \over 2}$ second
\wrong
1 second
\wrong
4 seconds
\wrong 
3 seconds

\note %10
\problem
Let $f(x) = \sin (x^2 )$.  Which of the following statements is true about
$f(x)$?  (Only one of the statements is true.)
\correct
$f(x)$ is symmetric about the $y$-axis.
\wrong
$f(x)$ is symmetric about the origin.
\wrong
$f(x)$ is periodic with period $4 \pi^2$.
\wrong
$f(x) \geq 0$ for all $x$.
\wrong
$\displaystyle \lim_{x \rightarrow \infty} f(x) = 0$.

\note %11
\problem
Let $\displaystyle g(x) = {{x-1} \over {x+1}}$.  Find all the asymptotes of
$g(x)$.
\correct
vertical asymptote $x = -1$; horizontal asymptote $y = 1$
\wrong 
vertical asymptote $x = -1$; horizontal asymptote $y = -1$
\wrong 
vertical asymptote $x = 1$; horizontal asymptote $y = 1$
\wrong 
vertical asymptote $x = 1$; horizontal asymptote $y = -1$
\wrong 
vertical asymptotes $x = -1$ and $x = 1$; horizontal asymptote $y = 1$

\note %12
\problem
Let $g(x) = x^4 - 8x^2$.  Note that $$g'(x) = 4x(x^2 - 4)$$ $$g''(x) = 4(3x^2 -
4)$$  At which
$x$, if any, does
$g(x)$ have a local maximum?
\correct
Only at $x=0$.
\wrong
$g(x)$ has no local maximum.
\wrong
At $x=-2$ and $x=2$.
\wrong
At $x= -2$, $x=0$ and $x=2$.
\wrong
At $\displaystyle x = -{2 \over {\sqrt{3}}}$ and $\displaystyle x = {2 \over
{\sqrt{3}}}$

\note %13
\problem
Let $\displaystyle h(x) = x - 3 x^{1/3}$.  Note that $$h'(x) = 1 - x^{-2/3}$$
$$h''(x) = {2 \over 3} x^{-5/3}$$ On which interval(s) is
$h(x)$ concave up?
\correct
$0 < x < + \infty$
\wrong
$-\infty < x < 0$
\wrong
$-\infty < x < 0$ and $0 < x < +\infty$
\wrong
$-1 < x < 0$ and $0 < x < 1$
\wrong $-\infty < x < -1$ and $1 < x < +\infty$

\note %14
\problem
Find the point on the line $y = 3x+2$ that is closest to the point $(5,7)$.
\correct
$(2,8)$
\wrong
$ \left ( {5 \over 2} , {19 \over 2} \right )$
\wrong
$(0,2)$
\wrong
$(1,5)$
\wrong
$(3,11)$

\note %15
\problem
A box with a square base and open top must have a volume of 4 $\hbox{ft}^3$. 
Find the length $x$ of one side of the base the box if you are to minimize the
amount of material used to construct it.  (Note you're finding $x$, not $y$.)

\vskip 1.1in
\hskip 1.8in\boxprob 

\correct
$x=2$
\wrong
$x=1$
\wrong
$x=4$
\wrong
$\displaystyle x = {1 \over 2}$
\wrong
$x = \root 3 \of 4$

\note %16
\problem
Find an antiderivative of $f(x) = \sin x + \sqrt{x}$.
\correct
$\displaystyle -\cos x + {2 \over 3} x^{3/2}$
\wrong
$\displaystyle \cos x + {2 \over 3} x^{3/2}$
\wrong
$\displaystyle -\cos x - {1 \over 2} x^{-1/2}$
\wrong
$\displaystyle \cos x - {1 \over 2} x^{-1/2}$
\wrong
$\displaystyle -\cos x - {2 \over 3} x^{3/2}$

\note %17
\problem
Let $f(x) = x^4 - 4x^3$.  Notice that $$f'(x) = 4x^2 (x-3)$$ $$f''(x) =
12x(x-2)$$  Which of the following describes the graph of $f(x)$?
\correct
local minimum at $x=3$; inflection points at $x=0$ and $x=2$
\wrong
local minimum at $x=0$ and $x=3$; inflection point at $x=2$
\wrong
local maximum at $x=0$; local minimum at $x=2$; no inflection points
\wrong
local maximum at $x=0$; local minimum at $x=2$; inflection point at $x=3$
\wrong
local maximum at $x=0$; local minimum at $x=3$; inflection points at $x=0$
and $x=2$

\note %18
\problem
A particle is moving along a straight line with position function $s(t)$.  
(Length is measured in  feet and time is in seconds.)  Its
acceleration function is given by
$a(t) = 4-6t$.  You are also given that its initial velocity is $v(0) = 5$ and its
initial position is $s(0) = 7$.  Find the position after 1 second (i.e.\ find
$s(1)$).
\correct
13 feet
\wrong
6 feet
\wrong
1 foot
\wrong
$-2$ feet
\wrong
5 feet

\note %19
\problem
Let $g(x) = 12-3x^2$.  Find the area of the portion of the graph of $g(x)$ which is
above the $x$-axis.  (Hint: you'll have to figure out where the graph crosses the
$x$-axis.)

\vskip .7in
\hskip 1.8in\parab

\correct
32
\wrong
16
\wrong
64
\wrong
48
\wrong
24

\appendix
%That's all folks!

 \end{document}
