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

\preface
\magnification=1200

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\def\tank{\scaledpicture .5in by 0in (tank scaled 700)}
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\title
\noindent
{\bf Math 120: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\hfill
Tutorial Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl May 9, 1995}\hfill
Tutorial Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Calculators are {\it not} allowed on this exam.  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  29 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 Express the repeating decimal $\displaystyle 1.259595959...$ as a
fraction.
\correct $\displaystyle {{1247} \over {990}}$
\wrong $\displaystyle {{11} \over {9}}$
\wrong $\displaystyle {{1258} \over {999}}$
\wrong $\displaystyle {{113} \over {90}}$
\wrong $\displaystyle {{124} \over {99}}$

\note %2
\problem Find the sum of the following geometric series:
$$
\sum_{k=0}^{\infty} {{2^{k+2}} \over {3^{k+1}}}
$$
\correct 4
\wrong 1
\wrong 2
\wrong 3
\wrong 5

\note %3
\problem Joe has six cards: four of them are aces and two are not.  If Diane
takes away two cards, chosen at random, what is the probability that both are aces?
\correct $\displaystyle {2 \over 5}$
\wrong $\displaystyle {3 \over 5}$
\wrong $\displaystyle {2 \over 3}$
\wrong $\displaystyle {1 \over 3}$
\wrong $\displaystyle {1 \over 2}$

\note %4 
\problem Let $f(x) = \ln (1-x^2 )$.  Find the second degree Taylor polynomial
$P_2 (x)$ of $f(x)$.
\correct $-x^2$
\wrong $1 - x^2$
\wrong $1 + x^2$
\wrong $x^2$
\wrong $1+x+x^2$

\note %5
\problem Let $\displaystyle f(x) = e^{3x}$.  Find the $n$th Taylor polynomial for
$f(x)$.  (Hint: You can use the fact that the $n$th Taylor polynomial for $e^x$ is 
$
\displaystyle 1 + x + {{x^2} \over {2!}} + {{x^3} \over {3!}} + \cdots + {{x^n}
\over {n!}}
$.)
\correct $\displaystyle \sum_{k=0}^n 3^k \cdot {{x^k} \over {k!}}$
\wrong $\displaystyle 3 \cdot \sum_{k=0}^n {{x^k} \over {k!}}$
\wrong $\displaystyle \sum_{k=0}^n {{x^k} \over {(3k)!}}$
\wrong $\displaystyle \sum_{k=0}^n {{x^{3k}} \over {(3k)!}}$
\wrong $\displaystyle \sum_{k=0}^n {{x^{3k}} \over {k!}}$

\note %6
\problem For the following problem you may use any of the following facts (not
all will necessarily be used):
$$
\int_{- \pi / 2}^{\pi /2} \cos^2 x \, dx = \pi /2 \ \ \ \ 
\int_{- \pi / 2}^{\pi /2} x \cos x \, dx = 0 
$$
$$\int_{- \pi / 2}^{\pi /2} \cos x \, dx = 2 \ \ \ \ \
\int_{- \pi / 2}^{\pi /2} \sin x\cos x \, dx = 0 
$$
Find the centroid (= center of mass) of the region bounded by the curve $y = \cos
x$ and the $x$-axis, from $x = -\pi /2$ to $x = \pi /2$.
\correct $\displaystyle (0, {{\pi} \over 8})$
\wrong $\displaystyle (0,{{\pi} \over 4})$
\wrong $\displaystyle (0,0)$
\wrong $\displaystyle (0,{{\pi} \over 2})$
\wrong $\displaystyle (0,2)$

\note %7
\problem Which of the following integrals represents the length of the arc of the
curve $y^2 = x^3$ from $(0,0)$ to $(2, \sqrt{8})$?
\correct $\displaystyle \int_0^2 \sqrt{1 + {9 \over 4} x} \, \, dx$
\wrong $\displaystyle \int_0^2 \sqrt{1 + {9 \over {4x}} } \, \, dx$
\wrong $\displaystyle \int_0^2 \sqrt{1 + {3 \over 2}x^{1/2} } \, \, dx$
\wrong $\displaystyle \int_0^{\sqrt{8}} \sqrt{1 + {9 \over 4}x } \, \, dx$
\wrong $\displaystyle \int_0^{\sqrt{8}} \sqrt{1 + {3 \over 2}x^{1/2} } \, \, dx$

\note %8
\problem The error bound for Simpson's rule is
$$
|\hbox{error}| \leq {{M(b-a)^5} \over {180n^4}}
$$ 
where $\displaystyle M = \max \left |f^{(4)} (x) \right |$, $a \leq x \leq b$.
If you apply Simpson's
rule with $n=6$ to the integral
$$ \int_1^3 {1\over x}\,dx$$ you get $\displaystyle {{2077} \over
{1890}}$ as an answer.  On
the other hand, we know that $\displaystyle \int_1^3 {1\over x}\,dx = \ln 3 - \ln
1$.   It follows that
$$
\left | \ln 3 - {{2077} \over {1890}} \right |
$$
is at most which of the following? (Note that the fourth derivative of $\displaystyle {1
\over x}$ is
$\displaystyle  {{24}
\over {x^5}}$.)
\correct $\displaystyle {{24 \cdot 2^5} \over {180 \cdot 6^4}}$
\wrong $\displaystyle {{24 \cdot 2^5} \over {3^5 \cdot 180   \cdot 6^4}}$
\wrong $\displaystyle 1 + {{24 \cdot 2^5} \over {3^5 \cdot 180   \cdot 6^4}}$
\wrong $\displaystyle {{24 \cdot 3^5} \over {2^5 \cdot 180   \cdot 6^4}}$
\wrong $\displaystyle {{8 \cdot 2^5} \over {3^5 \cdot 180   \cdot 6^4}}$

\note %9
\problem Find the following antiderivative:
$$ \int {{dx} \over {(x-1)(x+2)}}$$
\correct $\displaystyle {1 \over 3} \ln \left | {{x-1} \over {x+2}} \right | + C$
\wrong $\displaystyle {1 \over 3} \ln \left | (x-1)(x+2) \right |+ C$
\wrong $\displaystyle\ln \left | {{(x-1)^2} \over {x+2}} \right |+ C$
\wrong $\displaystyle {1 \over 3} \left [ (x-1)^{-2} + (x+2)^{-2} \right ]+ C$
\wrong $\displaystyle  \ln \left | (x-1)(x+2) \right |+ C$

\note %10
\problem In order to evaluate the following definite integral:
$$ \int_2^{2\sqrt{3}} {{dx} \over {4+x^2}}$$
what substitution should you make?
\correct $x = 2 \tan u$
\wrong $u = 2 \tan x$
\wrong $u = 4 + x^2$
\wrong $\displaystyle x = {1 \over 2} \tan u$
\wrong $\displaystyle u = {1 \over 2} \tan x$

\note %11
\problem Find the following definite integral:
$$
\int_0^{\pi /2} x \sin x \, dx
$$
\correct 1
\wrong 0
\wrong $-1$
\wrong $\displaystyle -{{\pi} \over 2} + 1$
\wrong $\displaystyle -{{\pi} \over 2}$

\note %12
\problem Find the following antiderivative:
$$
\int x^2 \ln x \, dx
$$
(Hint: at the end you'll have to pull out a common factor.)
\correct $\displaystyle {1 \over 3} x^3 \left [ \ln x - {1 \over 3} \right ]+ C$
\wrong $\displaystyle {1 \over 3} x^3 \left [ \ln x + {1 \over 3} \right ]+ C$
\wrong $\displaystyle x[1 + 2 \ln x ]+ C$
\wrong $\displaystyle {1 \over 3} x^3 \left [ x\ln x - x \right ]+ C$
\wrong $\displaystyle {1 \over 3} x^3 \left [ x\ln x + x \right ]+ C$

\note %13
\problem Mary plans to invest some money in a savings account earning 10\%
interest, compounded continuously.  Her goal is to have a total of \$10,000 in the
account after 10 years.  How much money (in dollars) should she invest now to
achieve that goal?
\correct $\displaystyle {{10,000} \over e}$
\wrong $\displaystyle \$ 1,000$
\wrong $\displaystyle 10,000 e$
\wrong $\displaystyle {{10,000} \over {e^{10}}}$
\wrong $\displaystyle 10,000 e^{10}$

\note %14
\problem A certain radioactive substance has a half-life of 4 years.  If a sample
of this substance has an initial mass of 50 kilograms, how much remains after 6
years (in kilograms)?
\correct $\displaystyle 50 \cdot 2^{-3/2}$
\wrong $\displaystyle 50 \cdot 4^{-6}$
\wrong $\displaystyle 50 \cdot 2^{-2/3}$
\wrong $\displaystyle 50 \cdot 4^{-3/2}$
\wrong $\displaystyle {2 \over 3} \cdot 50$

\note %15 
\problem Find the derivative of $\displaystyle f(x) = 2^x$
\correct $\displaystyle 2^x \cdot \ln 2$
\wrong $\displaystyle 2^x $
\wrong $\displaystyle 2 \cdot 2^x $
\wrong $\displaystyle e^x \cdot \ln 2$
\wrong $\displaystyle x \cdot 2^{x-1}$ 

\note %16
\problem Find the domain of the function $\displaystyle f(x) = e^{x^2} \ln (2-x)$.
\correct $-\infty < x < 2$
\wrong $-\infty < x \leq 2$
\wrong $2 < x < \infty$
\wrong  $2\leq x < \infty$
\wrong $-\infty < x < 2$ except $x=0$

\note %17
\problem Solve for $x$: $\ln (\ln x) = 1$
\correct $\displaystyle x = e^e$
\wrong There is no such $x$
\wrong $x = e$
\wrong $x = 0$ 
\wrong $x = 1$

\note %18 
\problem Find the derivative of $\displaystyle f(x) = \sqrt{e^{(x^2)}}$.  (Hint:
$\displaystyle \sqrt{a} = a^{1/2}$ and 
$\displaystyle (a^b )^c = a^{bc}$)
\correct $ \displaystyle x \sqrt{e^{(x^2)}}$
\wrong $e^x$
\wrong $\displaystyle {1 \over {2 \sqrt{ e^{(x^2)}}}} $
\wrong $\displaystyle {x \over { \sqrt{ e^{(x^2)}}}} $
\wrong $\displaystyle {{2x} \over { \sqrt{ e^{(x^2)}}}} $

\note %19
\problem Evaluate the following integral: $\displaystyle \int_0^1 e^{3x} dx$
\correct $\displaystyle {1 \over 3} \left (e^3 - 1 \right )$
\wrong $\displaystyle e^3 - 1 $
\wrong $\displaystyle {1 \over 3} e^3 $
\wrong $\displaystyle {{e^3} \over 3}  -1$
\wrong $\displaystyle {1 \over 3} \left (e - 1 \right )$

\note %20
\problem Simplify: $\displaystyle {{z^{-1/2} z^2} \over {z^{1/2}}}$
\correct $z$
\wrong $z^2$
\wrong $z^{7/2}$
\wrong $z^{-3/2}$
\wrong $z^{1/2}$

\note %21
\problem Find the average value of the function $f(x) = x^2$ on the interval
$[0,2]$
\correct $\displaystyle {4 \over 3}$
\wrong $\displaystyle {8 \over 3}$
\wrong 1
\wrong 2
\wrong $\displaystyle {3 \over 2}$

\note %22
\problem A tank, shown below, has water in it up to a height of 3 m.  Which of
the following integrals represents the work required to pump all the water out of
the tank?  Note that the water is not included in the picture.

\vskip 1.4in

\hskip 1in \tank

\bigskip

\correct $\displaystyle \int_0^3 9800 \cdot 8y (5-y)\, dy$
\wrong $\displaystyle \int_0^3 9800 \cdot {{4y} \over 5} (5-y)\, dy$
\wrong $\displaystyle \int_0^5 9800 \cdot {{4y} \over 5} (3-y)\, dy$
\wrong $\displaystyle \int_0^5 9800 \cdot 8y (3-y)\, dy$
\wrong $\displaystyle \int_0^3 9800 \cdot 4y (5-y)\, dy$

\note %23
\problem Which of the following integrals represents the volume of the solid
obtained by rotating about the $x$-axis the region bounded by the curves $y =
e^x$, $x = 2$, $x = 5$ and the $x$-axis?
\correct $\displaystyle \int_2^5 \pi e^{2x} dx$
\wrong $\displaystyle \int_2^5 \pi e^{(x^2)} dx$
\wrong $\displaystyle \int_2^5 \pi e^{x} dx$
\wrong $\displaystyle \int_{e^2}^{e^5} \pi e^{2x} dx$
\wrong $\displaystyle \int_{e^2}^{e^5} \pi e^{(x^2)} dx$

\note %24
\problem Which of the following integrals represents the volume of the solid
obtained by rotating {\it about the line} $x = 3$ the region bounded above by $y =
x^2$, below by the $x$-axis, to the left by the $y$-axis and to the right by the
line $x=3$ (see picture)?

\vskip 1.8in

\hskip 1.8in \region

\correct $\displaystyle \int_0^9 \pi (3 - \sqrt{y} )^2 dy$
\wrong $\displaystyle \int_0^9 \pi \left ( \sqrt{y} \right )^2 dy$
\wrong $\displaystyle \int_0^3 \pi \left ( x^2 \right )^2 dx$
\wrong $\displaystyle \int_0^3 \pi (3-x^2 )^2 dx$
\wrong $\displaystyle \int_0^9 \pi \left ( 9 - (\sqrt{y})^2 \right ) dy$

\note %25 
\problem Evaluate the following definite integral: $\displaystyle \int_0^1 2x(x^2
-1)^8 dx$
\correct $\displaystyle {1 \over 9}$
\wrong $\displaystyle -{1 \over 9}$
\wrong $\displaystyle 0$
\wrong $\displaystyle -{1 \over 8}$
\wrong $\displaystyle 1$

\note %26 
\problem Let $\displaystyle f(x) = \int_1^{x^2} \sin t \, dt$.  What is $f'(x)$?
\correct $2x \sin (x^2 )$
\wrong $2x \cos (x^2 )$
\wrong $\cos (x^2 )$
\wrong $\sin (x^2 )$
\wrong $\sin (2x)$

\note %27
\problem Over the course of his basketball career, Julius succeeded in making a
free throw in 75\% ($= 3/4$) of his attempts.  In a randomly selected game, he
attempted to make a free throw 12 times.  What is the probability that he
succeeded in 8 of the 12 attempts?  (You don't need to know anything about
basketball to answer this problem.)

\correct $\displaystyle {{12} \choose 8} \left ( {3 \over 4} \right )^8 \left (
{1 \over 4} \right )^4$
\wrong $\displaystyle {2 \over 3}$
\wrong $\displaystyle {{8 \choose 8} \over {{12} \choose 8}}$
\wrong $\displaystyle \left ( {3 \over 4} \right ) \cdot {{8 \choose 8} \over {{12}
\choose 8}}$
\wrong $\displaystyle {3 \over 4} \cdot \left ( {2 \over 3} \right )^8 \left (
{1 \over 3} \right )^4$

\note %28
\problem Find the following antiderivative: $\displaystyle \int x \sin (x^2 ) dx$
\correct $\displaystyle - {1 \over 2} \cos (x^2 ) + C$
\wrong $\displaystyle - {1 \over 2} x \cdot \cos (x^2 ) + C$
\wrong $\displaystyle  {1 \over 2} \cos (x^2 ) + C$
\wrong $\displaystyle - 2 \cos (x^2 ) + C$
\wrong $\displaystyle 2 \cos (x^2 ) + C$

\note %29
\problem A metal chain weighs 2 lbs per ft and is hanging to a length of 20
ft off the top of a tall building.  How much work is done in raising the bottom
of the chain 15 ft?
\correct $\displaystyle \int_0^{15} 2(20-y)dy$
\wrong $\displaystyle \int_5^{20} 2(20-y)dy$
\wrong $\displaystyle \int_0^{20} 2(15-y)dy$
\wrong $\displaystyle \int_0^{15} 2ydy$
\wrong $\displaystyle \int_5^{20} 2(15-y)dy$

\appendix
%That's all folks!

\end
