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\preface
\magnification=1200

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\title
\noindent
{\bf Math 120: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\hfill
Tutorial Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl February 21, 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
Let $\displaystyle f(x) = \int_2^{3x^2} {1 \over {1 + t^2}} dt$.  What is $f'(x)$?
\correct $\displaystyle {{6x} \over {1 + 9x^4}}$
\wrong $\displaystyle {{3x^2} \over {1 + 9x^4}}$
\wrong $\displaystyle {{x^3} \over {1 + x^2}}$
\wrong $\displaystyle {{3x^2} \over {1 + x^2}}$
\wrong $\displaystyle {{x^3} \over {1 + 9x^4}}$

\note %2
\problem Find the following antiderivative:  $\displaystyle \int x^3 \sqrt{1 +
x^4} dx$
\correct $\displaystyle {1 \over {6}} (1+ x^4 )^{3/2} + C$
\wrong $\displaystyle {8 \over 3} (1+x^4 )^{3/2} + C$
\wrong $\displaystyle {1 \over 4} x^4 + {1 \over 6} x^6 + C$
\wrong $\displaystyle \pm {1 \over 4} x^4 \pm {1 \over 6} x^6 +C$
\wrong $\displaystyle {1 \over {6}} x^4 (1+ x^4 )^{3/2} + C$

\note %3
\problem Find the following antiderivative: $\displaystyle \int {{\sin x} \over
{\cos^2 x}} dx$
\correct $\displaystyle {1  \over {\cos x}}$
\wrong $\displaystyle - {1  \over {\cos x}}$
\wrong $\displaystyle {1 \over 2} \sin^2 x$
\wrong $\sec^3 x$
\wrong $\displaystyle - \ln (\cos x)$

\note %4
\problem Find the volume of the solid obtained by rotating the region bounded by
the curves $\displaystyle y = 2 x$, $x = 0$, $x = 3$ and $y = 0$
about the $x$-axis.
\correct $36 \pi$
\wrong $18 \pi$
\wrong $9 \pi$
\wrong $18 \pi^2$
\wrong $4 \pi$

\note %5
\problem Consider the region bounded above by the curve $y = x^2$, below by $y =
0$ and to the left and right respectively by $x=0$ and
$x = 2$.  Which of the following integrals represents the volume of the solid
obtained by rotating this region about the line $x = 2$?
\correct $\displaystyle \pi \int_0^4 \left ( 2 - \sqrt{y} \right )^2 dy$
\wrong $\displaystyle \pi \int_0^4 \left [ (2)^2 - (\sqrt{y} )^2 \right ] dy$
\wrong $\displaystyle \pi \int_0^2 \left ( 2- x^2 \right )^2 dx$
\wrong $\displaystyle \pi \int_0^4 \left ( \sqrt{y} \right )^2  dy$
\wrong $\displaystyle \pi \int_0^2  \left [ (4)^2 - (x^2 )^2 \right ] dx$


\note %6  
\problem  The natural length of a certain spring is 10 cm.  The work required to
stretch it from its natural length to a length of 12 cm is 6 joules.  How much
work is required to stretch it from 12 cm to 14 cm?
\correct 18 joules
\wrong 6 joules
\wrong 78 joules
\wrong $\displaystyle {{78} \over {11}}$ joules
\wrong $\displaystyle 24$ joules

\note %7
\problem  Evaluate the following definite integral:
$\displaystyle \int_{-1}^1 x(x^2 - 3)^6 dx$.
\correct 0
\wrong $\displaystyle {1 \over 7}$
\wrong 128
\wrong $-386$
\wrong 6

\note %8
\problem If
$\displaystyle 
y = {{\sqrt{x^2 +3}} \over {e^x \cos^3 x}}
$
find $\displaystyle {{dy} \over {dx}}$ using logarithmic differentiation.
\correct
$\displaystyle {{dy} \over {dx}} = y \cdot \left [ {x \over {x^2 +3}} -1 + {{3
\sin x} \over {\cos x}} \right ]$
\wrong 
$\displaystyle {{dy} \over {dx}} = y \cdot \left [ {x \over {x^2 +3}} -x + {{3
\sin x} \over {\cos x}} \right ]$
\wrong 
$\displaystyle {{dy} \over {dx}} = y \cdot \left [ {x \over {x^2 +3}} + {{3
\sin x} \over {\cos x}} \right ]$
\wrong 
$\displaystyle {{dy} \over {dx}} = y \cdot \left [ {2x \over {x^2 +3}} -x - {{3
\sin x} \over {\cos x}} \right ]$
\wrong 
$\displaystyle {{dy} \over {dx}} = y \cdot \left [ {{2x} \over {x^2 +3}} -1 - {{3
\sin x} \over {\cos x}} \right ]$

\note %9
\problem
If $\displaystyle f(x) = e^{x^2}$ find $f'(3)$.
\correct $6e^9$
\wrong $2e^6$
\wrong $2e^9$
\wrong $e^9$
\wrong $6e^6$

\note %10
\problem
A tank has the shape of an inverted circular cone with height 9 m and base radius
3 m.  It is filled with water to a height of 5 m.  (See picture.)  Which of the
following integrals represents the work required to empty the tank by pumping all
of the water to the top of the tank?  (The density of water is 1000 kg/m$^3$ and
the acceleration due to gravity is 9.8 m/sec$^2$.  Notice that the answers don't
all have the same limits of integration.)
\vskip 1.8in
\hskip 1.3in\workprob
\bigskip
\correct 
$\displaystyle \int_0^5 9800 \left [ \pi \left ( {y \over 3} \right )^2 \right ]
(9-y) dy$
\wrong
$\displaystyle \int_0^9 9800 \left [ \pi \left ( 3y \right )^2 \right ]
(9-y) dy$
\wrong
$\displaystyle \int_0^5 9800 \left [ \pi \left ( 3y \right )^2 \right ]
(9-y) dy$
\wrong
$\displaystyle \int_5^9 9800 \left [ \pi \left ( {y \over 3} \right )^2 \right ]
(5-y) dy$
\wrong
$\displaystyle \int_5^9 9800 \left [ \pi \left ( {3y} \right )^2 \right ]
(5-y) dy$

\note %11
\problem
Find the average value of the function $f(x) = x^2 -1$  over the interval $[-3,
3]$.
\correct 2
\wrong 12
\wrong 0
\wrong 3
\wrong 1

\note %12
\problem Simplify (the variable $z$ represents a positive real number):
$\displaystyle {{z^2 z^{1/2}} \over {z^{-2}}}$
\correct $\displaystyle z^{9/2}$
\wrong $\displaystyle z^{3}$
\wrong $\displaystyle z^{-1}$
\wrong $\displaystyle z^{1/2}$
\wrong $\displaystyle z^{2}$

\note %13
\problem 
What is the domain of the function $\displaystyle e^x \ln (1-x^2 )$?
\correct $(-1,1)$
\wrong $(-\infty ,-1 ) \cup (1,\infty )$
\wrong $[-1,1]$
\wrong $[0,1]$
\wrong $(0, 1)$

\note %14
\problem Which of the following is equal to $\ e^{{1\over2}\ln x}\>?$
\correct $\sqrt x$
\wrong ${x\over2}$
\wrong $\big({1\over2}\big)^x$
\wrong ${1\over{x^2}}$
\wrong $e^{\sqrt x}$

\note %15
\problem
Solve for $x$:  $2\log_9 (x) = 1$.
\correct $x = 3$
\wrong $x = \pm 3$
\wrong $x = 9$
\wrong $x = 81$
\wrong $x = \pm 81$

\note %16 
\problem Let $\displaystyle f(x) = e^x + e^{-x}$.  Where is $f(x)$ increasing and
where is it concave up?
\correct increasing on $[0,\infty ]$; concave up everywhere
\wrong increasing on $[-\infty ,0]$; concave up everywhere
\wrong increasing everywhere; concave up everywhere
\wrong increasing everywhere; concave up on $[0,\infty]$
\wrong increasing on $[0,\infty ]$; concave up on $[0,\infty]$


\note %17
\problem Find $\displaystyle \int_0^1 {1\over{2x+1}}\>dx$
\correct $\displaystyle \ln(\sqrt3)$
\wrong $\displaystyle {1\over 2}$
\wrong $\ln 3$
\wrong $\displaystyle \ln {1 \over 3}$
\wrong $\displaystyle {2\over 9}$

\note %18
\problem
Find the derivative of the function $\displaystyle f(x) = \ln \left ( \ln (3x)
\right )$.
\correct $\displaystyle {1 \over {x \ln (3x)}}$
\wrong $\displaystyle {1 \over {3x \ln (3x)}}$
\wrong $\displaystyle {3 \over {x \ln (3x)}}$
\wrong $\displaystyle {1 \over {x \ln x}}$
\wrong $\displaystyle {3 \over { x \ln x}}$

\note %19
\problem
Find $\displaystyle \int e^x \left ( 1 + e^x \right )^8 dx$
\correct $\displaystyle {1 \over 9} \left ( 1 + e^x \right )^9 + C$
\wrong $\displaystyle \left ( 1 + e^x \right )^9 + C$
\wrong $\displaystyle {1 \over 8} \left ( 1 + e^x \right )^8 + C$
\wrong $\displaystyle e^x + e^{9x} + C$
\wrong $\displaystyle {1 \over 9} e^x \left ( 1+e^x \right )^9 + C$

\appendix
%That's all folks!

\end
