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\preface
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\title
\noindent
{\bf Math 120: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\hfill
MWF Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl May 8, 1997}\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  25 multiple choice questions, worth
6 points each. \vskip .25cm \noindent {\bf You are taking this exam under the
honor code.}

\def\answerskip{\vskip0.25in}

\def\problemskip{\vskip1in}

\note % A
\problem  
What is the coefficient of $x^{8}y^2$ in the expansion of $(x+y)^{10}$?

\correct 45
\wrong 10
\wrong 8
\wrong 90
\wrong $10!$

\note % B
\problem
Suppose that 1/3 of the people in a certain population were born in
a month that ends in the letter ``r.''  Suppose that three people are chosen at
random.  What is the probability that exactly one of the three was born in a month
ending in the letter ``r?''

\correct 4/9
\wrong 1/3
\wrong 1
\wrong 2/3
\wrong 1/2


\note %1
\problem
Find the slope of the tangent line to the graph of $\displaystyle y = e^{\sin x}$
at $x = \pi /2$.

\correct 0
\wrong 1
\wrong $e$
\wrong $-1$
\wrong $1 / e$

\note %2
\problem 
On what interval(s) is the function $\displaystyle f(x) = x e^{x^2}$ increasing?

\correct $(-\infty ,\infty)$
\wrong $(0,\infty)$
\wrong $(-\infty, 0)$
\wrong It is never increasing
\wrong $(-\infty ,-\sqrt{2} ) \cup (\sqrt{2} ,\infty)$

\note %3
\problem
Simplify $\displaystyle \log_{2} 48 - \log_{2} 3$

\correct 4
\wrong $\log_{2} 45$
\wrong 16
\wrong 2
\wrong 8

\note %4
\problem
If $\displaystyle f(x) = {{\ln x} \over x}$, find $f'(e)$.

\correct 0
\wrong $\displaystyle {1 \over e}$
\wrong $\displaystyle {1 \over {e^2}}$
\wrong $\displaystyle -{1 \over e}$
\wrong $\displaystyle -{1 \over {e^2}}$

\note %5
\problem
Use logarithmic differentiation to find $\displaystyle {{dy} \over {dx}}$ for 
$\displaystyle y = {{e^{x^2} \sqrt{x^2 +1}} \over {x^5}}$

\correct $\displaystyle {{dy} \over {dx}} = {{e^{x^2} \sqrt{x^2 +1}} \over 
{x^5}} \left [ 2x + {x \over {x^2 + 1}} - {5 \over x} \right ]$
\wrong  $\displaystyle {{dy} \over {dx}} = {{e^{x^2} \sqrt{x^2 +1}} \over  {x^5}}
\left [ {{2x} \over {e^{x^2}}} + {x \over {x^2 + 1}} - {5 \over x} \right ]$
\wrong  $\displaystyle {{dy} \over {dx}} = {{e^{x^2} \sqrt{x^2 +1}} \over  {x^5}}
\left [ {{2x} \over {e^{x^2}}} + {2x \over {x^2 + 1}} - {5x^4} \right ]$
\wrong  $\displaystyle {{dy} \over {dx}} = {{e^{x^2} \sqrt{x^2 +1}} \over  {x^5}}
\left [ 2x + {2x \over {x^2 + 1}} - {5 \over x} \right ]$
\wrong  $\displaystyle {{dy} \over {dx}} = {{e^{x^2} \sqrt{x^2 +1}} \over   
{x^5}} \left [ 2x + {2x \over {x^2 + 1}} - {5x^4} \right ]$

\note %6
\problem
Find $\displaystyle \lim_{x \rightarrow 0} {{\cos x - 1} \over {e^x - 1}}$

\correct 0
\wrong 1
\wrong $-1$
\wrong The limit does not exist.
\wrong $\displaystyle -{1 \over 2}$

\note %7
\problem
Fred invests \$1000 in an account which offers continuous compounding.  After 
exactly 8 years, the amount of money in the account has tripled to \$3000.  What
annual interest rate is he getting in his account?

\correct $\displaystyle {1 \over 8} \ln 3$ 
\wrong $\displaystyle {8} \ln 3$
\wrong $\displaystyle {1 \over 3} \ln 8$
\wrong $\displaystyle {3} \ln 8$
\wrong $\displaystyle 24 \ln 2$

\note %8
\problem
A certain substance has a half-life of 4 hours.  If a sample has an initial mass 
of 20 mg, after how many hours will only 15 mg remain?

\correct $\displaystyle {{4 \ln ( 3/4 )} \over {\ln (1/2)}}$
\wrong $\displaystyle {{3 \ln ( 1/4 )} \over {\ln (1/2)}}$
\wrong $\displaystyle {{4 \ln ( 1/4 )} \over {\ln (1/2)}}$
\wrong $\displaystyle {{4 \ln ( 1/3 )} \over {\ln (1/2)}}$
\wrong $\displaystyle {{ \ln ( 4/3 )} \over {\ln (1/2)}}$

\note %9
\problem
Recall that $\int \ln x \ dx = x \ln x - x + C$.  Using this fact, find 
$\displaystyle \int_1^5 \log_5 x \ dx$.

\correct $\displaystyle 5 - {4 \over {\ln 5}}$
\wrong $\displaystyle 1$
\wrong $\displaystyle \ln 5 - 1$
\wrong $\displaystyle 4 \ln 5 - 1$
\wrong $\displaystyle 5 \ln 5 - 4$

\note %1
\problem  Let $R$ be the region bounded by $y = \sin x$ and the $x$-axis, from
$x=0$ to $x=\pi$.  Find the volume of the solid generated by rotating $R$ about
the $y$-axis.

\correct $\displaystyle 2\pi^2$
\wrong $\displaystyle {\pi^2}$
\wrong $\displaystyle \pi +2 $
\wrong $\displaystyle 2\pi^2 +\pi$
\wrong $\displaystyle \pi^2 - \pi$

\note %2
\problem Evaluate: $\displaystyle \int_0^{\pi / 2} \sin^3 x\ dx$.

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

\note %3
\problem Evaluate: $\displaystyle \int_2^3 {{2x^3} \over {x^2 -1}} dx$

\correct $\displaystyle 5 + \ln \left ( {8 \over 3} \right )$
\wrong $\displaystyle 5 + \ln \left ( {3 } \right )$
\wrong $\displaystyle 5 + \ln \left ( {9 \over 4} \right )$
\wrong $\displaystyle 3 + \ln \left ( {2 \over 3} \right )$
\wrong $\displaystyle 3 + \ln \left ( {7 \over 2} \right )$

\note %5
\problem Find the length of the curve $\displaystyle y = \int _0^x \sqrt
{t^2+4t+3}\  dt$ from $x=1$ to $x=3$.

\correct $\displaystyle 8$
\wrong $\displaystyle 2$
\wrong $\displaystyle 10$
\wrong $\displaystyle 6$
\wrong $\displaystyle 12$

\note %8 
\problem Solve: $\displaystyle \int x^3 {\it e}^{x^2} \ dx$

\correct $\displaystyle {1 \over 2} \left ( x^2{\it e}^{x^2} - {\it e}^{x^2}
\right ) +C$

\wrong $\displaystyle {1 \over 3} \left ( x^3{\it e}^{x^2} + 3{\it e}^{x^2}
\right ) +C$

\wrong $\displaystyle {1 \over 6} \left ( x^3{\it e}^{x^2}  - 3x^2{\it e}^{x^2} +
6x{\it e}^{x^2} - 6{\it e}^{x^2}\right ) +C$

\wrong $\displaystyle {1 \over 2} \left ( x^3{\it e}^{x^2} - 6{\it e}^{x^2}
\right ) +C$

\wrong $\displaystyle {1 \over 6} \left ( x^3{\it e}^{x^2} - 6x{\it e}^{x^2}
\right ) +C$

\note %10
\problem Let $f(x)=\sqrt {x+4}$. Find the second degree Taylor polynomial for
$f(x)$ and evaluate it at $x=1$ (i.e plug $x=1$ into the second degree Taylor
polynomial).

\correct $\displaystyle  {143 \over 64}$
\wrong $\displaystyle  {17 \over 8}$
\wrong $\displaystyle \sqrt {5} + {7 \over 8}$
\wrong $\displaystyle  {23 \over 8}$ 
\wrong $\displaystyle  {45 \over 16}$

\note %13
\problem Find the sum of the infinite series: $\displaystyle \ {4 \over 3} - {4 
\over 9} + {4 \over 27} - {4 \over 81} + {4 \over 243} - {4 \over 729 } + \dots$

\correct $\displaystyle 1$
\wrong $\displaystyle {5 \over 3}$
\wrong $\displaystyle {6 \over 5}$
\wrong $\displaystyle {5 \over 4}$
\wrong $\displaystyle {2 \over 3}$



\note %1.
\problem
 Find the area bounded by the graphs of the following equations
 $$
y={1\over x^2},\ y=-x^2,\ x=1,\ x=2
$$

\correct ${17/ 6}$

\wrong ${13/ 5}$

\wrong ${19/ 4}$

\wrong ${11/ 3}$

\wrong ${21/ 7}$


\note %3.
\problem
Let $R$ be the region bounded by the graphs of $y=x^3,$ and $y=x^2.$  Find the 
volume of the solid generated by rotating $R$ about the $x$-axis.

\correct $(2/ 35)\pi$

\wrong $(4/ 37)\pi$

\wrong $(39/121)\pi$

\wrong $(141/ 2)\pi$

\wrong $(3/2)\pi$


\note %5.
\problem
 A spring of natural length 10 in. stretches 1.5 in. under a force of 8 lb.  Find the work done in stretching the spring from
a length of 11 in. to a length of 13 in.

\correct ${64/ 3}$ in.-lb

\wrong ${59/ 3}$ in.-lb

\wrong ${58/ 3}$ in.-lb

\wrong ${61/ 3}$ in.-lb

\wrong ${63/ 3}$ in.-lb



\note %6.
\problem 
A fishtank has a rectangular base of width 2 ft and  length 4 ft.  The sides are rectangular and have height 3 ft.  If the tank is filled with water weighing 62.5 lb$/$ft$\null^3$, find the work required to pump all the water over the top of the tank.

\correct 2250 ft-lb

\wrong 1145 ft-lb

\wrong 1235 ft-lb

\wrong 3321 ft-lb

\wrong 2567 ft-lb



\note %7.
\problem 
Let $f(x)=3\sqrt{x+1}.$  Find the average value of $f(x)$ on the interval $[-1,8].$

\correct 6

\wrong 5.5

\wrong 6.5

\wrong 4

\wrong 4.5


\note %9.
\problem 
Let
$$
f(x)=\int_2^{x^4} {t\over \sqrt{t^3+2}} dt 
$$

Find $f^\prime(x).$


\correct ${4x^7/ \sqrt{x^{12}+2}}$


\wrong ${x/ \sqrt{x^{4}+2}}$

\wrong ${4x^7/ \sqrt{x^{7}+2}}$

\wrong ${x^4/ \sqrt{x^{4}+2}}$

\wrong ${x^4/ \sqrt{x^{12}+2}}$

\note %13.
\problem
 Let $R$ be the region bounded by the graphs of $y=1/x, x=1,x=3$ and $y=0.$  Find 
the volume of the solid generated by rotating $R$ about the $x$-axis.

 
\correct $2\pi/3$

\wrong $3\pi/2$

\wrong $2\pi$

\wrong $3\pi$

\wrong $2\pi/5$



\appendix
%That's all folks!

\end
