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

\preface
\magnification=1200

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    \hrule width #1 height 0pt depth 0pt
    \vfill
    \special{picture #3} % this is the low-level interface
    }
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\def\volumeone{\scaledpicture .5in by 0in (volume1 scaled 700)}
\def\volumetwo{\scaledpicture .5in by 0in (volume2 scaled 700)}
\def\tank{\scaledpicture .5in by 0in (tank scaled 700)}


\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 13, 1996}\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 $A$ be the region in the plane bounded above by the curve $y =
2-x^2$, below by the curve $y = x^2$ and to the left by the $y$-axis.  (This
region is given in the diagram below.)  Find the volume of the solid obtained by
revolving this region about the $y$-axis.  You don't have to find the actual
volume-- just express it in terms of integrals.

\vskip 1.2in

\hskip 1.8in  \volumeone
\medskip

\correct $\displaystyle \pi \int_0^1 \left ( \sqrt{y} \right )^2  dy + 
\pi \int_1^2 \left ( \sqrt{2-y} \right )^2  dy$

\wrong $\displaystyle \pi \int_0^1 \left ( 2-x^2 \right )^2  dx - 
\pi \int_0^1 \left ( \sqrt{x^2} \right )^2  dx$

\wrong $\displaystyle \pi \int_0^1 \left ( x^2 \right )^2  dx + 
\pi \int_1^2 \left ( 2-x^2 \right )^2  dx$

\wrong $\displaystyle \pi \int_0^{\sqrt{2}} \left ( 2-x^2 \right )^2  dx - 
\pi \int_0^{\sqrt{2}} \left ( x^2 \right )^2  dy$

\wrong $\displaystyle \pi \int_0^2 \left ( \sqrt{2-y} \right )^2  dy - 
\pi \int_0^1 \left ( \sqrt{y} \right )^2  dy$

\note %2
\problem Let $\displaystyle F(x) = \int_{\pi / 2}^x \sin^2 t \ dt$.  Find
$F'(x)$.
\correct $\displaystyle \sin^2 x$
\wrong $\displaystyle 2 \sin x \cos x$
\wrong $\displaystyle -2 \sin x \cos x$
\wrong $\displaystyle \sin^2 x - 1$
\wrong $\displaystyle 2 \sin x \cos x - 1$

\note %3
\problem Let $\displaystyle F(x) = \int_2^{x^3} t^2 \ dt$.  Find $F'(x)$.

\correct $\displaystyle 3x^8$
\wrong $\displaystyle x^6$
\wrong $\displaystyle 2x^3$
\wrong $\displaystyle 6x^5$
\wrong $\displaystyle 3x^7$

\note %4 
\problem Let $\displaystyle F(x) = \int_{x^2}^{x^3} \sin t \ dt$.  Find $F'(x)$.

\correct $\displaystyle 3x^2 \sin (x^3 )- 2x \sin (x^2 )$

\wrong $\displaystyle 3x^2 \sin (x^3 )+ 2x \sin (x^2 )$

\wrong $\displaystyle \sin (x^3 )- \sin (x^2 )$

\wrong $\displaystyle -\cos (x^3) + \cos (x^2)$

\wrong $\displaystyle \cos (x^3) - \cos (x^2)$

\note %5
\problem Let $B$ be the region bounded above by the line $y=2$, 
below by the curve $y = x^2$ and to the left by the $y$-axis.  (This
region is given in the diagram below.)  Find the volume of the solid obtained by
revolving this region about the line $y=2$.  You don't have to find the actual
volume-- just express it in terms of integrals.

\vskip 1.2in

\hskip 1.8in  \volumetwo

\medskip

\correct $\displaystyle \pi \int_0^{\sqrt{2}} (2-x^2 )^2 dx$

\wrong $\displaystyle \pi \int_0^{\sqrt{2}} (x^2 )^2 dx$

\wrong $\displaystyle \pi \int_0^2 (\sqrt{y} )^2 dy$

\wrong $\displaystyle \pi \int_0^2 (2-x^2 )^2 dx$

\wrong $\displaystyle \pi \int_0^2 (2-\sqrt{y} )^2 dy$

\note %6
\problem Evaluate $\displaystyle \int_{-2}^1 \sqrt{x+3} \ dx$.

\correct $\displaystyle {{14} \over 3}$

\wrong $\displaystyle 7$

\wrong $\displaystyle {2 \over 3}$

\wrong $\displaystyle {{16} \over 3}$

\wrong This integral cannot be evaluated.

\note %7  
\problem Evaluate the following indefinite integral: 
$\displaystyle \int {x \over {\sqrt{1+x^2}}} \ dx$

\correct $\displaystyle \sqrt{x^2 +1} +C$

\wrong $\displaystyle 2 \sqrt{x^2 +1} +C$

\wrong $\displaystyle {1 \over 2}\sqrt{x^2 +1} +C$

\wrong $\displaystyle {1 \over {2 \sqrt{x^2 +1}}} +C$

\wrong $\displaystyle {1 \over {\sqrt{x^2 +1}}}+C$

\note %8 
\problem Evaluate the following indefinite integral: 
$\displaystyle \int \sec^2 3\theta \ d \theta$

\correct $\displaystyle {1 \over 3} \tan 3\theta +C$

\wrong $\displaystyle {1 \over 9} \sec^3 3\theta +C$

\wrong $\displaystyle {1 \over 9} \tan^3 3\theta +C$

\wrong $\displaystyle 3 \tan 3\theta +C$

\wrong $\displaystyle {1 \over 3} \sec^3 3\theta +C$

\note %9
\problem The velocity function (in meters per second) for a particle
moving along a line is given by $v(t) = t^2 - 2t$.  Find the {\it displacement}
during the time interval $0 \leq t \leq 3$.

\correct 0 m

\wrong 3 m

\wrong $\displaystyle {8 \over 3}$ m

\wrong 4 m

\wrong $-6$ m

\note %10
\problem A tank has the shape illustrated in the diagram below.  The sides are
isosceles triangles, \break and a magnified side view is also provided.  It is
filled with water to a height of 1 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 1.6in

\hskip .3in \tank

\medskip

\correct $\displaystyle \int_2^3 9800 \cdot 10 \left ( {{2(3-x)} \over 3} \right )
x dx$

\wrong $\displaystyle \int_0^2 9800 \cdot 10 \left ( {{2(1-x)} \over 3} \right )
x dx$

\wrong $\displaystyle \int_2^3 9800 \cdot 10 \left ( {{2} \over 3} \right )
x dx$

\wrong $\displaystyle \int_2^3 9800 \cdot 10 \left ( {{2(1-x)} \over 3} \right )
(2+x) dx$

\wrong $\displaystyle \int_0^2 9800 \cdot 10 \left ( {{2(3-x)} \over 3} \right )
x dx$

\note %11
\problem  Suppose that 16 J of work are needed to stretch a spring from its
natural length of 10 cm to a length of 14 cm.  How much work is needed to stretch
it from 12 cm to 15 cm?

\correct 21 J

\wrong $\displaystyle {{27} \over 2}$ J

\wrong $\displaystyle {{21} \over 4}$ J

\wrong $\displaystyle {{81} \over 4}$ J

\wrong 11 J

\note %12
\problem Find the average value of the function $f(x) = x^3$ on the interval
$[1,3]$.

\correct 10

\wrong 20

\wrong $\displaystyle {{20} \over 3}$

\wrong 80

\wrong 40

\note %13
\problem Find the average value of the function $f(x) = \cos^2 x \sin x$ on the
interval $[0,\pi]$.

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

\wrong 0

\wrong $\displaystyle {2 \over 3}$

\wrong $\displaystyle -{1 \over {3\pi}}$

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

\note %14
\problem The velocity function (in meters per second) for a particle
moving along a line is given by $v(t) = t^2 - 2t$.  Find the {\it total distance
travelled} during the time interval $0 \leq t \leq 3$.

\correct $\displaystyle {8 \over 3}$ m

\wrong 3 m

\wrong 0 m

\wrong 4 m

\wrong $6$ m

\note %15
\problem Find the volume of the solid obtained by rotating the following region
about the $x$-axis.  The region is bounded by the curves $y = \sqrt{2x-1}$, the
$x$-axis, the line $x=1$ and the line $x = 5$.

\correct $20 \pi$

\wrong $\displaystyle {{26\pi} \over 3}$

\wrong $\displaystyle {{13} \over 6}$

\wrong $\displaystyle {{\pi} \over 3} \left [ 5^{2/3} - 1 \right ]$

\wrong $\displaystyle {{\pi} \over 6} \left [ 5^{2/3} - 1 \right ]$

\note %16
\problem A cable weighing 4 lbs/ft is used to lift a 500 lb piano 200 ft up the
side of a building.  Which of the following represents the amount of
work done?

\correct $\displaystyle \int_0^{200} (4x+500)dx$

\wrong $\displaystyle 500 + \int_0^{200} 4xdx$

\wrong $\displaystyle \int_0^{200} [4(200-x)+500]dx$

\wrong $\displaystyle 500 + \int_0^{200} 4(200-x)dx$

\wrong $\displaystyle \int_0^{200} 504xdx$

\note %17
\problem A certain experiment involves heating a metal rod, which is  3 m long.  It
is discovered that the temperature of the rod (in $^\circ$C), at a distance of $x$
meters from one end, is $4x+100$.  What is the average temperature of the rod?

\correct $106^\circ$

\wrong $318^\circ$

\wrong $102^\circ$

\wrong $112^\circ$

\wrong $104^\circ$

\note %18
\problem Evaluate the following indefinite integral: $\displaystyle \int {{\sin
\sqrt{x}} \over {\sqrt{x}}} dx$

\correct $-2 \cos \sqrt{x} + C$

\wrong $2 \cos \sqrt{x} + C$

\wrong $\displaystyle -{1 \over 2} \cos \sqrt{x} + C$

\wrong $\displaystyle {1 \over 2} \cos \sqrt{x} + C$

\wrong $\cos \sqrt{x} +C$

\note %19
\problem An animal population is increasing at a rate of $100 + 20t$ per year,
where $t$ is measured in years.  (Note that this formula represents the rate, not
the population itself.)  By how much does the population increase between the
second and fifth years?

\correct 510

\wrong 60

\wrong 210

\wrong 750

\wrong 240

\appendix
%That's all folks!

\end
