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{\bf date:} the $20^{\rm th}$ of November, 1998 

{\bf place:} room 221 Hayes

{\bf time:} 8:30-9:20

{\bf 111 - Exam III}

This exam contains 10 problems worth 10 points each. You may use only a
blank sheet of paper, a pencil, a rubber gum, a ruler and a small
calculator. You can use your calculator only to add, substract,
multiply or divide two numbers. This exam is taken under the honor
code. \\

{\bf Name:}
\\

{\bf Recommendation}

{\it Never give a "solitary" answer without justifying it by previous calculations or reasoning. } \newpage
{\bf Problems}
\\

{\bf 1.} Find a formula for $\frac{d}{dx}f(g(x))$, where $f(x)$ is a
function such that $f'(x)=x\ \root\of{1-x^2}$ and
$g(x)=x^{\frac{3}{2}}$.

%{\bf 1. bis.} Find $\frac{dy}{dx}$ where $y$ is a function of $u$ such
that $$\frac{dy}{du}=\frac{u}{u^2+1}$$ and $u=\frac{2}{x}$.


{\bf 2.} Calculate $4^{.2}\cdot 4^{.3}$. 

%{\bf 3.} Simplify $$\frac{e^{\frac{5x}{2}}-e^{3x}}{\root\of{e^x}}.$$ 

{\bf 3.} Simplify the expression $$e^{\ln(x^2)}.$$ 

{\bf 4.} Solve the equation $e^{-5x}\cdot e^4=e$ for $x$. 

%{\bf 4. bis.} Solve the equation $\ln(\ln(3t))=0$ for $t$. 

%{\bf 5.} Differentiate the function $x^e$. 

{\bf 5.} Differentiate the function $\ln(x^2+e^x)$. 

{\bf 6.} Determine all solutions of the differential equation
$y'=\frac{1}{3}y$.


{\bf 7.} The herring gull population in North America has been doubling
every 13 years since 1900. Give a differential equation satisfied by
$P(t)$, the population $t$ years after 1900 ({\it hint: $\ln2=.69$}).

{\bf 8.} One thousand dollars is deposited in a savings account at 10
\% interest compounded continuously. How many years are required for
the balance in the account to reach \$ 3000 ({\it hint: $\ln3=1.1$})?

{\bf 9.} A piece of charcoal found at Stonehenge contained 63 \% of the
level of $^{14}C$ found in living trees. Approximately how old is the
charcoal ({\it hint: the decay constant $=.00012$; $\ln0.63=-.46$})?

{\bf 10.} A stock portfolio increased in value from \$ 100,000 to \$
117,000 in 2 years. What rate of interest, compounded continuously, did
this investment earn ({\it hint: $\ln1.17=0.16$})?\\

{\bf Good luck!}

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