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\hspace{2.5in}Exam. III, Math. 120, Fall, 1999

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Your name:

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\textbf{Instructions}:  The exam. consists of 12 problems.  There is
space to work beside each problem.  Most of the problems offer
multiple-choice answers.  You should use the answers to check yourself.
It is important to show your work, and make your steps clear and
readable.   Partial credit may be given for solutions that are set up
correctly, even though the final answer is incorrect, and full credit may
not be given for a correct answer that is not supported by correct work.

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Please remember your obligations under the honor code.

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The spaces below are \emph{not} for answers---but for recording scores.

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1.

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3.

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11.

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12.


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1.  An appropriate trigonometric substitution converts the integral\\
$\int_{2}^{2\sqrt{3}}\frac{x^{2}}{\sqrt{4+x^{2}}} dx$ into which of  the
following?\\

(a)  $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}4\tan^{2}\theta\sec\theta
d\theta$\hspace{.5in}
(b)  $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}2\tan^{2}\theta\sec\theta
d\theta$\\

(c)  $\int_{2}^{2\sqrt{3}}4\tan^{3}\theta d\theta$\hspace{.2in}
(d)  $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}4\sec^{3}\theta d\theta$
\hspace{.2in}
(e)  $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}tan^{3}\theta d\theta$
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2.  What is $\int_{0}^{1}\frac{1}{\sqrt{4 - x^{2}}} dx$?\\

(a)  $0$ \hspace{.2in}
(b)  $\frac{1}{2}$ \hspace{.2in}
(c)  $\frac{1}{4}$ \hspace{.2in}
(d)  $\frac{\pi}{12}$ \hspace{.2in}
(e)  $\frac{\pi}{6}$
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3.  Which of the following integrals represents the length of the curve
$y=x^{3}$, for $0\leq x\leq 1$?\\

(a)  $\int_{0}^{1}(1+9x^{4}) dx$\hspace{.2in}
(b)  $\int_{0}^{1}(1+3x^{2}) dx$\hspace{.2in}
(c)  $\int_{0}^{1}\sqrt{1+x^{3}} dx$\\

(d)  $\int_{0}^{1}\sqrt{1+9x^{4}} dx$\hspace{.8in}
(e)  $\int_{0}^{1} \sqrt{1+x^{6}} dx$
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4.  For the region bounded by $y = x^{2}$, $y = 0$, $x = 0$, $x = 2$, what
is the centroid?

(a)  $(\frac{1}{2},1)$\hspace{.2in}
(b)  $(\frac{3}{2},\frac{6}{5})$\hspace{.2in}
(c)  $(1,2)$\hspace{.2in}
(d)  $(\frac{5}{6},\frac{3}{2})$\hspace{.2in}
(e)  $(\frac{3}{5},1)$
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5.  For the region bounded by $y = \sin x$ and $y = -\sin x$,
for $0\leq x\leq\pi$, what is the centroid ?  [You may avoid some
tedious calculation by appealing to symmetry.]\\

(a)  $(0,\pi)$\hspace{.2in}
(b)  $(0,\frac{\pi}{2})$\hspace{.2in}
(c)  $(\frac{\pi}{2},0)$\hspace{.2in}
(d)  $(\pi,0)$\hspace{.2in}
(e)  $(\frac{\pi}{2},\frac{2}{\pi})$
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6.  For $f(x) = \frac{1}{(2+x)}$ , what is the $4^{th}$
Taylor polynomial $P_{4}(x)$?\\

(a)  $\frac{1}{2}-\frac{x}{4}+\frac{x^{2}}{8}
-\frac{x^{3}}{16}+\frac{x^{4}}{32}$\hspace{.2in}
(b)  $1+\frac{x}{2}+\frac{x^{2}}{4}+\frac{x^{3}}{8}+\frac{x^{4}}{16}$\\

(c)  $\frac{1}{2}+\frac{x}{4}+\frac{x^{2}}{6}+\frac{x^{3}}{12}+
\frac{x^{4}}{48}$\hspace{.2in}
(d)  $1-\frac{x}{2}+\frac{x^{2}}{4}-\frac{x^{3}}{8}+\frac{x^{4}}{16}$\\

(e)  the function has no $4^{th}$ Taylor polynomial


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7.  If $f(x) = 3x^{17}-2x^{16}+\ldots+4x^{5}+\ldots+x+5$, what is
$f^{(5)}(0)$?\\

(a)  $475$\hspace{.2in}
(b)  $480$\hspace{.2in}
(c)  $485$\hspace{.2in}
(d)  $490$\hspace{.2in}
(e)  $495$
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8.  How many different words can be made out of $3$ $S$'s and $5$
$F$'s?\\

(a)  $52$\hspace{.2in}  (b)  $54$\hspace{.2in}  (c)  $56$\hspace{.2in}
(d)  $58$\hspace{.2in}  (e)  $60$
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9.  A box contains $12$ flashbulbs, of which $4$ are defective.  If you
choose $4$ of the $12$, at random, what is the probability that all will
be good?\\

(a)  $\frac{10}{99}$\hspace{.2in}
(b)  $\frac{12}{99}$\hspace{.2in}
(c)  $\frac{14}{99}$\hspace{.2in}
(d)  $\frac{16}{99}$\hspace{.2in}
(e)  $\frac{2}{11}$
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10.  What is the sum of the infinite series
$\sum_{k=0}^{\infty}(-\frac{1}{3})^{k}$?\\

(a)  $\frac{1}{3}$\hspace{.2in}
(b)  $\frac{3}{8}$\hspace{.2in}
(c)  $\frac{2}{3}$\hspace{.2in}
(d)  $\frac{3}{4}$\hspace{.2in}
(e)  $1$
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11.  What is $1+\frac{1}{2}+\frac{1}{2^{2}}+\ldots+\frac{1}{2^{8}}$?\\

(a)  $2-\frac{1}{2^{8}}$\hspace{.4in}
(b) $2$\hspace{.4in}
(c)  $2+\frac{1}{2^{9}}$\hspace{.4in}
(d)  $1$\\

(e)  there is no sum, since $|2|\geq 1$
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12.  A patient begins on Monday taking 1 unit of a drug every day.  For
each unit given, the amount remaining in the patient's system $t$ days
later is $(.2)^{t}$.  How many units of the drug are in the patient's
system just after Wednesday's dose?\\

(a)  $1$\hspace{.2in}
(b)  $1.2$\hspace{.2in}
(c)  $1.22$\hspace{.2in}
(d)  $1.24$\hspace{.2in}
(e)  $1.4$

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