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\hspace{2.5in}Exam. II, 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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2.

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3.

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4.

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5.

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6.

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7.

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8.

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9.

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10.

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11.

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12.


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1.  Sketch the graph of $f(x)=3^{-x}$.  Your graph should make clear
whether the function is increasing or decreasing, the value at $x=0$,
and and what happens as $x\rightarrow\infty$ and as
$x\rightarrow\ - \infty$.
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2.  If $\ln\ln x = 1$, what is $x$?

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(a)  $e$\hspace{.3in}
(b)  $e^{e}$\hspace{.3in}
(c)  $1$\hspace{.3in}
(d)  $\frac{1}{e}$\hspace{.3in}
(e)  $\frac{1}{e^{e}}$

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3.  What is $\frac{d}{dx}\,x^{x}$ ?

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(a)  $x^{x}(1+\ln x)$\hspace{.3in}
(b)  $x^{x-1}$\hspace{.3in}
(c)  $x^{x}$\hspace{.3in}
(d)  $1+\ln x$\hspace{.3in}
(e)  $\frac{1}{x^{x}}$

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4.  If $y=\frac{(x+1)^{2}x^{3}}{x+3}$, what is $y'$?

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(a)  $\frac{(x+1)^{2}x^{3}}{x+3}\,(6+2x - 3x^{2})$\hspace{.3in}
(b)  $\frac{(x+1)^{2}x^{3}}{x+3}\,(\frac{2}{x+1}+\frac{3}{x} -
\frac{1}{x+3})$

(c)  $6\,(x+1)\,x^{2}$\hspace{.3in}
(d)  $2\,\ln (x+1)+3\,\ln x - \ln(x+3)$

(e)  this derivative is too gruesome to think about

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5.  If $f'(x)=3\,f(x)$, and $f(0)=2$, then what is $f(2)$?

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(a)  $12e$\hspace{.3in}
(b)  $4e^{3}$\hspace{.3in}
(c)  $3e^{4}$\hspace{.3in}
(d)  $6e^{2}$\hspace{.3in}
(e)  $2e^{6}$

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6.  If \$$1000$ is invested at $8$\% interest, compounded quarterly (i.e.,
four times per year), what is the value of the investment after one
year?

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(a)  $1080\,e^{4}$	\hspace{.8in}
(b)  $1000\,e^{.32}$\hspace{.8in}
(c)  $1000\,e^{.08}$

(d)  $1000(1.02)^{4}$\hspace{.8in}
(e)  $1000(1.08)$

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7.  What is $\lim_{n\rightarrow\infty} (1+\frac{.08}{n})^{n}$?

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(a)  $0$\hspace{.3in}
(b)  $(.08)^{e}$\hspace{.3in}
(c)  $e^{.08}$\hspace{.3in}
(d)  $\frac{1}{.08 e}$\hspace{.3in}
(e)  $\infty$

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8. The population of geese at ND was 50 in June of 1995, and 200 in June
of 1999.  Using an exponential model, predict the population of geese for
June of 2000.

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(a)  $200\,e$\hspace{.3in}
(b)  $50e^{5}$\hspace{.3in}
(c)  $200\sqrt{2}$	\hspace{.3in}
(d)  $250$\hspace{.3in}
(e)  $250\sqrt{2}$

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9.  Find $\lim_{x\rightarrow 1} \frac{\ln x}{x^{2}-1}$.

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(a)  $\frac{1}{4}$\hspace{.3in}
(b)  $\frac{1}{2}$\hspace{.3in}
(c)  $\frac{3}{4}$\hspace{.3in}
(d)  $1$\hspace{.3in}
(e)  $\infty$

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10.  What is $\int^{1}_{0} x e^{2x} dx$?

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(a)  $\frac{1}{2}\,(e-1)$\hspace{.5in}
(b)  $\frac{1}{4}\,(e^{2}-2)$\hspace{.5in}
(c)  $\frac{1}{2}(e^{2}+e)$

(d)  $\frac{1}{2}(e^{2}-e)$\hspace{.5in}
(e)  $\frac{1}{4}\,(e^{2}+1)$

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11.  What is $\int \frac{2}{(x-1)(x+2)} dx$?

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(a)  $\frac{1}{2}\tan^{-1} x + c$\hspace{.2in}
(b)  $3\ln|x+2| - 2\ln |x-1| + c$

(c)  $\frac{2}{3}(\ln |x-1| - \ln |x+2|) + c$\hspace{.2in}
(d)  $\frac{1}{2}\ln |x-1|+\frac{1}{3}\tan^{-1} x + c$

(e)  the integral is indeterminate

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12.  What is $\int cos^{3}x dx$ ?

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(a)  $\sin x - \frac{1}{3}\sin^{3} x+c$\hspace{.3in}
(b)  $\frac{1}{4}\cos^{4}(x)+c$\hspace{.3in}
(c)  $\sin^{3}(x)+c$

(d)  $ - 3\cos^{2} x \sin x+c$\hspace{.3in}
(e)  $\ln (\cos x - \sin x)+c$

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