\preface
\magnification = 1200
\title
\noindent
{\bf Math 120: Calculus B}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl September 27, 2001}\hfill
Time of MWF class:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Calculators are not allowed.  Do not remove this answer page-- you will return
the whole exam.  There are  12 multiple choice questions, worth 5 points
each, and
4 partial credit questions worth 10 points each.  Record your answers  to the
multiple choice problems by  placing an $\times$ through one letter for each
problem on this answer sheet.  On problems 13--16, show all necessary work
and be
sure to mark your answers.

\vskip .25cm \noindent {\bf
You are taking this exam under the honor code.}

\def\answerskip{\vskip0.25in}

\def\problemskip{\vskip1in}

\note %1
\problem
The area of the region bounded by the curves $y = -x^3$ and $y = x^2$ is

\correct $\displaystyle {{1}\over {12}}$
\wrong 2
\wrong $\displaystyle {{3} \over {4}}$
\wrong $-3$
\wrong $\displaystyle - {{5}\over {2}}$

\note %2
\problem
The area of the region bounded by $y = 0$, $y = x$, $y = \ln{x}$ and $y=2$ is
given by which of the following?

\correct  $\displaystyle \int^2_0 (e^y - y)\,{\rm d}y$
\wrong  $\displaystyle \int^4_0 (y - \ln{y})\,{\rm d}y$
\wrong  $\displaystyle \int^2_0 (y - \ln{y})\,{\rm d}y$
\wrong  $\displaystyle \int^2_0 (y - e^y)\,{\rm d}y$
\wrong  $\displaystyle \int^2_0 (\ln{y} - y) {\rm d}y$


\note %3
\problem
What is the volume of the solid obtained by revolving the region bounded by
$x = 1$, $y = x-1$ and $y = 3$  around the y-axis?

\correct $18\pi$
\wrong  $2\pi$
\wrong  0
\wrong  $9\pi$
\wrong  $-3\pi$


\note %4
\problem
 Use the method of cylindrical shells ({\bf not the disk method}) to set
up an  integral which represents the volume of a right circular cone  of
height 6
and base radius 3.

\correct $2\pi \int^3_0 (6x - 2x^2)\,{\rm d}x$
\wrong $\displaystyle \pi \int^6_0 (3 - {{1}\over {2}} y)^2\,{\rm d}y$
\wrong  $\pi \int^3_0 (6 - 2x)^2\,{\rm d}x$
\wrong  $2\pi \int^3_0 (x^2)\, {\rm d}x$
\wrong  $\pi \int^6_0 ((3-x)x)\,{\rm d}y$


\note %5
\problem
 A spring has a natural length of 80 cm. If a force of 40 N is required to
hold it stretched at a length of 90 cm, how  much work is done in
stretching the spring from its natural length to 130 cm?

\correct 50 J
\wrong  100 J
\wrong  30 ft-lb
\wrong  0 J
\wrong  40 J


\note %6
\problem
Find $\displaystyle \lim_{x \rightarrow -\infty} {{3e^{2x}} \over
{1+2e^{2x}}}$

\correct $0$
\wrong  $1$
\wrong  $\infty$
\wrong  $\displaystyle {{2}\over {3}}$
\wrong  $-\infty$


\note %7
\problem
Find the average value of the function $f(x) = e^{3x}$ on the interval $[0,5]$.

\correct ${{1} \over {15}}(e^{15} -1)$
\wrong  ${{1}\over {15}}e^{15}$
\wrong  ${{1}\over {3}} e^{15}$
\wrong  ${{1}\over {3}} (e^{15}-1)$
\wrong ${{1}\over {5}} e^{15}$


\note %8
\problem
Let $f(x) = e^x + 3e^{-x}$.  Find $(f^{-1})'(4)$.

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

\note %9
\problem
Find a formula for the inverse function for $f(x) = x^3-5$.

\correct $f^{-1}(x) = \root 3 \of {x+5}$
\wrong  $f^{-1}(x) = \root 3 \of {x-5}$
\wrong  $f^{-1}(x) = x+5$
\wrong  $f^{-1}(x) = \root 3 \of {x^3+5}$
\wrong  $f^{-1}(x) = {x^3+5}$


\note %10
\problem
Find the derivative of the function $f(x) = xe^{x^2 +1}$.

\correct $e^{x^2 +1} (2x^2 +1)$
\wrong  $e^{x^2 +1}$
\wrong  $2xe^{x^2 +1} $
\wrong  $xe^{x^2 +1} $
\wrong  $e^{x^2 +1} (x +1)$


\note %11
\problem
Solve for $x$: $\log_4 (\log_3 x) = 1$

\correct $x=81$
\wrong  $x = 256$
\wrong  $x = 3$
\wrong  $x = 12$
\wrong $x = {{4}\over {3}}$


\note %12
\problem
Simplify: $\displaystyle {{\ln 2 + \ln 72 - \ln 9} \over {2}}$

\correct $\ln 4$
\wrong  $\ln 5$
\wrong  $\ln 3$
\wrong  $\sqrt{10}$
\wrong  10


\appendix

13. A tank in the shape of a rectangular box is two-thirds of the way full
with
water. If it is 3 meters high,  1 meter wide and 5 meters long, how much
work is
done in emptying the tank out the top? (The density of water is  (1000
$kg/m^3$).
{\bf Set up the integral but don't evaluate it.}

\vskip 1in


14. Sketch the region bounded by the curves $y = e^x$, $y = x+1$ and $x = 2$.
Find the volume of the solid obtained by revolving the region around the
x-axis.

\vskip 1in



15. Find the value of $x$ where the tangent line to the graph of $f(x) =
e^{x+1} +
e^{-x}$ is horizontal.  Hint: $\displaystyle e^{-x} = {{1}\over {e^x}}$.

\vskip 1in



16. Let $f(x) = 6-3x^2$.  Find the value of $b$ for which the average value
of $f(x)$ is 2 on the interval $[0,b]$.






\end
