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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 November 30, 1995}\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  Which of the following is equal to
$\displaystyle
\sum_{i=1}^n (4-3i)
$ \ \ ?
\correct $\displaystyle 4n - {{3n(n+1)} \over 2}$
\wrong $\displaystyle 4n - {{3n(n+1)(2n+1)} \over 6}$
\wrong $\displaystyle 4n - {{n(n+1)} \over 2}$
\wrong $\displaystyle 4n - {{3(n+1)(n+2)} \over 2}$
\wrong  $\displaystyle 4n - {{3n(n+1)(2n+1)} \over 2}$

\note %2 
\problem  Which of the following is equal to
$\displaystyle 
\sum_{i=4}^{100} i
$ \ \ ?  (Note that it says $i=4$, not $i=1$.)
\correct 5044
\wrong 10094
\wrong 96
\wrong 10100
\wrong 97

\note %3
\problem Find $\displaystyle \lim_{x \rightarrow \infty} ( \sqrt{x^2 +1} - x)$
\correct 0
\wrong 1
\wrong $\infty$
\wrong $-\infty$
\wrong $-1$

\note %4
\problem Find all the asymptotes of the function $\displaystyle g(x) = {{3x+1}
\over {2x-1}}$.
\correct horizontal asymptote at $\displaystyle y = {3 \over 2}$; vertical
asymptote at $\displaystyle x = {1 \over 2}$
\wrong horizontal asymptotes at $\displaystyle y = {3 \over 2}$ and $\displaystyle
y = -{3 \over 2}$; vertical asymptote at $\displaystyle x = {1 \over 2}$
\wrong horizontal asymptotes at $\displaystyle y = {3 \over 2}$ and $\displaystyle
y = -{3 \over 2}$; vertical asymptote at $\displaystyle x = -{1 \over 3}$
\wrong horizontal asymptote at $\displaystyle y = {3 \over 2}$; vertical
asymptote at $\displaystyle x = {2}$
\wrong horizontal asymptote at $\displaystyle
y = {3 \over 2}$; vertical asymptote at $\displaystyle x = -{1 \over 3}$

\note %5
\problem Let 
$$
f(x) = \cases{2-x &if $0 \leq x \leq 1$; \cr
              x &if $1 \leq x \leq 2$.}
$$
Find $A_0^x (f)$ (the area function).
\correct $\displaystyle 
A_0^x (f) = \cases{2x-{1 \over 2} x^2 &if $0 \leq x \leq 1$; \cr
              {1 \over 2} x^2 + 1 &if $1 \leq x \leq 2$}$
\wrong $\displaystyle 
A_0^x (f) = \cases{2x-{1 \over 2} x^2 &if $0 \leq x \leq 1$; \cr
              {1 \over 2} x^2  &if $1 \leq x \leq 2$}$
\wrong $\displaystyle 
A_0^x (f) = \cases{2x-{1 \over 2} x^2 &if $0 \leq x \leq 1$; \cr
              {1 \over 2} x^2 + {1 \over 2} &if $1 \leq x \leq 2$}$
\wrong $\displaystyle 
A_0^x (f) = \cases{2x-{1 \over 2} x^2 &if $0 \leq x \leq 1$; \cr
              {1 \over 2} x^2 - 1 &if $1 \leq x \leq 2$}$
\wrong $\displaystyle 
A_0^x (f) = \cases{2x-{1 \over 2} x^2 &if $0 \leq x \leq 1$; \cr
              {1 \over 2} x^2 - {1 \over 2} &if $1 \leq x \leq 2$}$

\note %6
\problem Let $\displaystyle f(x) = x^3 + x + 1$.  Which of the following statements
are true about this function?  

\bigskip

\itemitem{I.} $f$ has no critical numbers.
\itemitem{II.} $f$ has no inflection points.
\itemitem{III.} $f$ has no asymptotes (either horizontal or vertical).

\bigskip

(Recall that a critical number is a value $c$ such that either $f'(c) = 0$ or
$f'(c)$ does not exist.  An inflection point is a place where $f$ changes from
being concave up to being concave down, or vice versa.)

\correct I.\ and III.
\wrong only I.
\wrong only II.
\wrong only III.
\wrong II. and III.

\note %7
\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 $x=2$

\note%8
\problem A farmer wants to build a rectangular enclosure for his pigs along the side
of his barn, which is 40 ft.\ long.  He has 100 ft.\ of fencing, and he wants to
enclose as large an area as possible.  Of course he only has to use the fencing for
 three sides of the enclosure since the barn serves as the fourth side, but that
means that the barn side of the enclosure (and hence also the side opposite it) is
forced to be {\it at most} 40 ft.\ (it could be less).  What is the largest area
that he can enclose under these conditions?

\vskip 1.5in
\hskip 1.3in \fence

\vskip .35in

\correct 1200 ft.
\wrong 1250 ft.
\wrong 2400 ft.
\wrong 800 ft.
\wrong 1111.11 ft.

\note %9
\problem Find an antiderivative of $\displaystyle f(x) = \cos x - {\root 3 \of
x}$.
\correct $\displaystyle \sin x - {3 \over 4} x^{4/3}$ 
\wrong $\displaystyle -\sin x - {3 \over 4} x^{4/3}$ 
\wrong $\displaystyle -\sin x + {1 \over 3} x^{-2/3}$ 
\wrong $\displaystyle \sin x + {3 \over 4} x^{4/3}$ 
\wrong $\displaystyle \sin x - {1 \over 3} x^{-2/3}$ 

\note %10
\problem A particle moves 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
$a(t) = 2-12t$.  You are also given that its initial velocity is $v(0) = 3$ and its
initial position is $s(0) = 2$.  Find the position after 1 second (i.e.\ find
$s(1)$).
\correct 4 ft.
\wrong 3 ft.
\wrong 2 ft.
\wrong 1 ft
\wrong 5 ft.

\note %11
\problem Let $f(x) = x+2$ on the interval $[1,4]$.  If the interval is divided
into 3 equal subintervals (i.e.\ $n=3$) and $x_i^* =$ left endpoint, find the sum
of the areas of the corresponding approximating rectangles.
\correct 12
\wrong 15
\wrong 8
\wrong 13
\wrong $\displaystyle {{27} \over 2}$

\note %12
\problem The following limit represents the area of the region below the graph of
a certain function $f(x)$ and above a certain interval $[a,b]$:
$$
\lim_{n \rightarrow \infty} \sum_{i=1}^n {4 \over n} \sqrt{\cos(1+{{4i} \over n})}
$$
Find $f(x)$ and $[a,b]$.
\correct $f(x) = \sqrt{\cos x}$,\ \ \ $[a,b] = [1,5]$
\wrong $f(x) = 4\sqrt{\cos x}$,\ \ \ $[a,b] = [1,2]$
\wrong $f(x) = 4\sqrt{\cos (1+x)}$,\ \ \ $[a,b] = [1,5]$
\wrong $f(x) = \sqrt{\cos x}$,\ \ \ $[a,b] = [0,4]$
\wrong $f(x) = \sqrt{\cos (1+x)}$,\ \ \ $[a,b] = [1,5]$

\note %13
\problem Let $x$ and $y$ be two numbers (not necessarily positive) whose
difference is 100.  If the product of $x$ and $y$ is the
minimum possible, what is the sum of $x$ and $y$?
\correct 0
\wrong 100
\wrong $-100$
\wrong 102
\wrong 200

\note %14
\problem Find the area of the region below the graph of $y = \sin x$ and above
the $x$-axis, from $x=0$ to $x = \pi$.
\vskip 1in
\hskip 1.5in \sine

\correct 2
\wrong 1
\wrong $\pi$
\wrong $\displaystyle {{\pi} \over 2}$
\wrong $\displaystyle {{\pi} \over 3}$

\note %15
\problem Let $\displaystyle f(x) = {{x+1} \over {|{x-1}}|}$.  Which of the
following statements is true about the limits as $x \rightarrow 1$ from the left
and from the right?
\correct $\displaystyle \lim_{x \rightarrow 1-} f(x) = +\infty$; $\displaystyle
\lim_{x \rightarrow 1+} f(x) = +\infty$
\wrong $\displaystyle \lim_{x \rightarrow 1-} f(x) = +\infty$; $\displaystyle
\lim_{x \rightarrow 1+} f(x) = -\infty$
\wrong $\displaystyle \lim_{x \rightarrow 1-} f(x) = -\infty$; $\displaystyle
\lim_{x \rightarrow 1+} f(x) = +\infty$
\wrong $\displaystyle \lim_{x \rightarrow 1-} f(x) = -\infty$; $\displaystyle
\lim_{x \rightarrow 1+} f(x) = -\infty$
\wrong $\displaystyle \lim_{x \rightarrow 1-} f(x)=1$;
$\displaystyle
\lim_{x \rightarrow 1+} f(x)=1$

\note %16
\problem Let $f(x)$ be a function such that $f(x)$ and $f'(x)$ both exist for all
x, and such that $$
 f'(x) = {{4-x^2} \over {\sqrt{x^4 +1}}}
$$  
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 a local maximum at $x=2$.
\wrong It {\it must} have an absolute maximum at $x=2$.
\wrong It {\it must} have a local minimum at $x=2$.
\wrong It {\it must} have an absolute minimum at $x=2$.
\wrong It {\it must} have an inflection point at $x=2$.

\note %17
\problem Given that the graph of $f$ passes through the point $(1,4)$ and that the
slope of the tangent line at $(x,f(x))$ is $4x-1$, find $f(2)$.
\correct 9
\wrong 6
\wrong 4
\wrong 7
\wrong 2

\note %18
\problem A box with an open top is to be constructed from a square piece of
cardboard, 6 ft wide, by cutting out a square from each of the four corners and
bending up the sides (along the dashed lines in the picture below).  Find the
largest volume that such a box can have.  

\vskip 1.6in
\hskip 1.5in \boxprob

\medskip

\correct 16 $\hbox{ft}^3$
\wrong 32 $\hbox{ft}^3$
\wrong 8 $\hbox{ft}^3$
\wrong 4 $\hbox{ft}^3$
\wrong 12 $\hbox{ft}^3$

\note %19
\problem  Find
$\displaystyle 
\lim_{x \rightarrow \infty} {{1 - x^2} \over {2 + x^2}}
$
\correct $-1$
\wrong 1
\wrong $\infty$
\wrong $-\infty$
\wrong ${1 \over 2}$ 

\appendix
%That's all folks!

 \end{document}