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\centerline {\largerbold Problems, Week 1}

\bigskip

%principally a review from the final exam. the first two represent
%problems that i thought students did quite poorly. the third is
%essentially the Fund. Theorem problem from the final, and the fourth
%is a problem from the text. i will try to do select problems from the
%text more regularly this semester. several people indicated that they
%thought this would be a good idea. 

1.~Suppose that $Q$ is the function 

$$Q(x)={\biggl( \arctan \Bigl( \ln \bigl( \sqrt{x} -1 \bigr) \Bigr)
\biggr)^{\! 3} }\; .$$

a) What are the domain and range of $Q$?  Answers should {\it not} be
numerical approximations, but should be written if needed in terms of
constants studied in calculus such as $\pi$ and $e$.

\medskip

b) Suppose $y=Q(x)$. Write a formula for $x$ in terms of $y$. 

\bigskip

2.~A radioactive substance $A$ decays at a rate proportional to
the amount of the substance present.  

\smallskip

a) Suppose that an initial amount of 10 micrograms decays after 8
hours to 7 micrograms. Determine a formula for $A(t)$, the amount of
substance $A$ present at time $t$.

\medskip

b) In the presence of a certain gamma ray flux, the radioactive decay
of the substance is increased. In fact, when an initial amount of 10
micrograms of $A$ is subject to this radiation, after 8 hours only 2
micrograms of $A$ remain. Determine a formula for $B(t)$, the amount
of substance $A$ present at time $t$ when the radiation mentioned is
present. 

\medskip

c) Suppose we are presented with 10 micrograms of substance $A$ and
wish to have 5 micrograms after 8 hours. We are allowed to ``turn on''
the gamma radiation at some time during the 8 hours (but it must
stay on after it is turned on!). At what time should the radiation be
introduced in order to obtain 5 micrograms of $A$ after 8 hours?

\bigskip

3.~Suppose the graph of $f$ is given by:
\bigskip
\centerline{\epsfbox{pic1-1.eps}} 
\centerline{The graph of $y=f(t)$}

\bigskip

Let $\d{g(x)=\int_0^x f(t)\; dt}$ for $0 \le x \le 4$.

\bigskip

a) Find $g(0)$, $g(2)$, and $g(4)$.

\medskip

b) Find all values of $x$ for which $g(x)$ is increasing, and explain
your answer. Find all values of $x$ for which $g(x)$ is decreasing,
and explain your answer.

\medskip
 
c) What is the maximum value of $g(x)$? 

\bigskip

%p.386 of Stewart, problem # 56, in section 6.1

4.~a) Find the number $a$ such that the line $x=a$ bisects the area
under the curve $y= 1/x^2$, $1 \le x \le 4$.

\medskip

b) Find the number $b$ such that the line $y=b$ bisects the area in
part a). 

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