\def\wsn{28}
\input worksheet.tex

\item{1.} Find all functions which are their own derivative.  Graph 4
diffferent ones on the same axes. 

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\item{2.}  In lecture, Dr. Davis noted that the expression
$\ln(x+1)\over\ln x$ cannot be simplified.
Nevertheless, solve
$$
{\ln(x+1)\over\ln x}=2.
$$


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\item{3.}  Let $f(x)=\ln x$ and $g(x)=\ln|x|$.

\medskip
\itemitem{a)}  Graph $f$ and $g$ on separate axes.

\medskip
\itemitem{b)}  Given that $f'(x)=\displaystyle{1\over x}$, prove that
$g'(x)=\displaystyle{1\over x}$. 

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\item{4.}  Find the derivatives of each of the following.  Simplify
your answer as much as possible.
$$
{\eqalign{\pt a & y=\sin(e^{x^2+x-1})\cr
\pt c & y=a^xx^a,\quad a>0\qquad a,\hbox{ constant}\cr}}\hskip1.5in
{\eqalign{\pt b & y=\sqrt{1+e^x}\cr
\pt d & y=2^{(3^x)}.\cr}}\hskip1.4in
$$

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\item{5.}  Find all vertical and horizontal asymptotes of the following
functions: 
$$
\pt a f(x)=\ln(x+2)-\ln(x+1)\qquad
\pt b g(x)=\displaystyle{\ln x+5\over\ln x^2-5}\qquad
\pt c h(x)=e^{-1/x^2}
$$ 

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\item{6.}  Given that you know $\displaystyle{{d\over dx}[e^x]=e^x}$,
prove the folowing:
$$
\pt a {d\over dx}[a^x]=a^x\ln a\qquad
\pt b {d\over dx}[a^u]=a^u\ln a{du\over dx}\qquad
\pt c {d\over dx}[\log_ax]={1\over x\ln a}\qquad
\pt d {d\over dx}[\log_au]={1\over u\ln a}{du\over dx}
$$
Hint for part c - Set $u=\log_ax$ and use part b.

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\item{7.} In a certain village, there are 50 couples. As it turns out,
everyone in the village is having an affair. This despite a
particularly gruesome custom which requires a wife, upon discovering
that her husband is having an affair, to kill him the following
morning. Even more oddly, the women in the town talk quite freely
about their activities! In fact they are all perfectly aware that any
woman who is having an affair will tell EVERY other woman in the
village except, of course, for the wife of the man with whom they are
having the affair. Yet life there goes on quite peacefully since no
woman can know for sure that her husband is actually having an
affair. One day, a well revered wise man visits the village and
announces that someone is having an affair. What happens after this?



\bye