\def\wsn{30}
\input worksheet.tex

\item{1.}  Let $f(x)=x(1-x)$.  Break $[0,1]$ into four equal intervals.

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\itemitem{a)}  Use these to find an upper sum for $\int_0^1x(1-x)\,dx$.

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\itemitem{b)}  Use these to find a lower sum for  $\int_0^1x(1-x)\,dx$.

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\itemitem{c)}  Find the average of the previous two answers.

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\itemitem{d)}  WITHOUT computing  $\int_0^1x(1-x)\,dx$ show that it is
within $1/16$ of the average you computed in part c).

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\itemitem{}  Hint: plot the upper sum, lower sum, and average on a
number line.  What do you know about where the actual integral lies?
Who won the Nobel Peace Prize this year?

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\itemitem{e)}  Compute  $\int_0^1x(1-x)\,dx$.


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\item{2.}  Let $f(x)=x^2$.

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\itemitem{a)}  Into how many equal intervals must you subdivide
$[0,1]$ for the average of the upper and lower sums to accurately
estimate $\int_0^1x^2\,dx$ to two decimal places?

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\itemitem{b)}  Compute the upper and lower sums for this partition and
find the average.

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\itemitem{c)}  Compute $\int_0^1x^2\,dx$.  To how many decimal places
was your estimate in part b) correct?

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\itemitem{3.\hskip12pt a)}  Suppose that $f'(x)=f(x)g'(x)$ for some
$g$.  Show that $f(x)=Ke^{g(x)}$ for some $K$.

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\itemitem{b)}  Newton's Law of Cooling states that the rate of
change of the temperature $T$ of an object is proportional to the
difference between $T$ and the temperature $\tau$ of the surrounding
medium.  A cup of coffee at $200^\circ$ in a room of temperature
$70^\circ$ is stirred continually and reaches $100^\circ$ after ten
minutes.  At what time was it at $120^\circ$?

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\item{4.}  Your well-intentioned roommates wake you up by bringing you
a fresh cup of coffee.  What they didn't know is that you ALWAYS take
a shower before drinking your coffee.  Also, you drink your coffee
with milk in it.  If you want your coffee to be as hot as possible
when you get out of the shower, should you (a) go to the refrigerator
and put milk in it now or (b) wait until you get out of the shower to
do this?

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\item{5.}  Two identical ice trays are filled, one with two cups of
water at room temperature and the other with two cups of boiling
water.  Both are placed in the freezer.  Which tray has solid ice
first?

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\item{6.}  The half-life of a radioactive material is the number of
years required for $1/2$ of the atoms in a sample to decay.  

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\itemitem{a)}  Suppose that after one year, only 36.79\% of an initial
amount of radioactive material remains.  Find the half-life.

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\itemitem{b)}  From the information given, can you compute the initial
amount present?

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\item{7.}  A year ago, there were 4 grams of a radioactive substance.
Now there are 3 grams.  How much was there 10 years ago?

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\item{8.}  {\bf Carbon-14 Dating.}  The half-lives of radioactive
elements can sometimes be used to date events from the Earth's past.
The ages of rocks more than 2 billion years old have been measured by
the extent of the radioactive decay of uranium (half-life 4.5 billion
years!).  In a living organism, the ratio of radioactive carbon stays
fairly constant during the lifetime of the organism, being
approximately equal to the ratio in the organism's surroundings at the
time.  After the organism's death, however, no new carbon is ingested,
and the proportion of carbon-14 in the organism's remains decreases as
the carbon-14 decays.  Since the half life of carbon 14 is known to be
about 5700 years, it is possible to estimate the age of organic
remains by comparing the proportion of carbon-14 they contain with the
proportion assumed to have been in the organism's environment at the
time it lived.  Archeologists have dated shells (which contain
CaCO$_3$), seeds, and wooden artifacts this way.

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\itemitem{a)} Find $k$ in the equation ${dy\over dt}=ky$ for
carbon-14.

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\itemitem{b)}  What is the age of a sample of charcoal in which 90\%
of the carbon-14 has decayed?

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\itemitem{c)}  The charcoal from a tree killed in the volcanic
eruption that formed Crater Lake in Oregon contained 39.2\% of the
carbon-14 found in living matter.  About how old is Crater Lake?

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