\documentclass{exam}
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\begin{document}

\noindent
\textbf{Math 105 Exam II.}
\hspace{1.6in}
\hbox to 3.3in{Name:\enspace\hrulefill}

\vspace{0.2in}

\noindent
March 2, 1999
\hspace{2.1in}
\hbox to 3.3in{Instructor's name:\enspace\hrulefill}

\vspace{0.2in}

\noindent
\hspace{3.15in}
\hbox to 3.3in{Time of MWF class:\enspace\hrulefill}

\vspace{0.2in}

Show all work and put your answers in the spaces provided.  You will
receive no credit if the answer is not in the space provided and no
partial credit for a wrong answer if you do not show your work.
Answers may be approximated by decimals if desired, but full points
will also be awarded for correct answers left in exact form in
terms of fractions, logarithms, or exponentials.

\vspace{0.2in}

\noindent 
Sections of the course: 
\begin{tabbing}
Section 01 \hspace{0.2in}  \= Brown  \hspace{0.2in} \= MWF
10:40-11:30 am \\
Section 02                 \> Palmieri \> MWF 1:55-2:45 pm
\end{tabbing}

\newpage

\begin{questions}



\question[8] Solve the following for $x$.

\begin{parts}

\part $e^{3x-2} = 1$

\vspace{1.5in}

\answer 

\part $\ln (5 e^{-2x}) = 0$ 

\vspace{1.5in}

\answer

\end{parts}



\question[6] Evaluate the following; your answer should be an integer
or fraction.

\begin{parts}

\part $\log_{7}(\frac{1}{49})$

\vspace{1in}

\answer

\part $e^{\ln (2^{3})}$

\vspace{1in}

\answer

\end{parts} 

%%%%%%%%%%%%%%%%

\newpage

\question[5] What is the half-life of a substance with a decay
constant of $0.635$?

\vspace{1.25in}

\answer 

%%%%%%%%%%%%%%%%

\question[7] A substance decays exponentially, so that the amount
after $t$ days is given by the formula $y=A e^{-kt}$.  If exactly
$\frac{1}{3}$ of the initial amount remains after four days, what is
the decay constant $k$ of the substance?

\vspace{1.5in}

\answer 

%%%%%%%%%%%%%%%%

\question You owe a loan shark $\$800$; he charges interest at a
rate of $35\%$.  What is the formula for the amount
of money owed after $t$ years in these cases?

\begin{parts}

\part[4] If the interest is compounded weekly (52 times a year)?

\vspace{1in}

\answer 

\part[4] If the interest is compounded continuously?  

\vspace{1in}

\answer 

\end{parts}

%%%%%%%%%%%%%%%%

\newpage

\question[6] You deposit $\$2000$ into an account paying $7\%$,
compounded continuously.  How long will it take for the account to
grow to $\$3000$?

\vspace{2in}

\answer 

%%%%%%%%%%%%%%%%

\question[8] What is the definition of the derivative of the function
$f(x)$? (We are looking for the formula involving a limit and $h$ and
\dots)

\vspace{1.5in}

\answer 

%%%%%%%%%%%%%%%%

\question[7] Compute the derivative $f'(x)$ of the function $f(x) =
3x^{2} - \frac{2}{x} + \sqrt[3]{x}$ using any rule we have had 
so far.

\vspace{2in}

\answer 

%%%%%%%%%%%%%%%%

\newpage

\question Consider the function $f(x)=7x^{2}$.

\begin{parts}

\part[5] What is the slope of the secant line to this curve
between the points $(1,7)$ and $(1+h,7(1+h)^{2})$?  Simplify as much as
possible.

\vspace{1.7in}

\answer 

\part[3] What is the slope of the tangent line to this curve at the point
$(1,7)$?

\vspace{1.2in}

\answer 

\end{parts}

%%%%%%%%%%%%%%%%

\question[8] For what \emph{two} values of $x$ does the curve
$y=\frac{1}{3}x^{3}+\frac{1}{2}x^{2} -20x + 3$ have slope zero?

\vspace{2.5in}

\answer 

%%%%%%%%%%%%%%%%

\newpage

\question[6] Let $y=x^{1/2} - 3$.  What is the equation of the
line tangent to this curve at the point $(1,-2)$?

\vspace{2in}

\answer 

%%%%%%%%%%%%%%%%

\question[7] Let $f(x) = x^{4}-6x^{2}+7x+1$.  Compute $f''(x)$ using
any rule we have had so far.

\vspace{2in}

\answer 

%%%%%%%%%%%%%%%%

\question[4] You throw a ball into the air at a rate of 80 ft./sec.,
so that its position (height) after $t$ seconds is given by the
formula $s(t) = -16t^{2} + 80t$.  What is the velocity of the ball
after 2 seconds?

\vspace{1.5in}

\answer 

%%%%%%%%%%%%%%%%

\newpage

\question A particle moves along the $x$-axis so that its position
after $t$ seconds is $s(t) = 3t^{3} + 2t + 1$.

\begin{parts}

\part[4] What is the average velocity of the particle between times
$t=0$ and $t=1$?

\vspace{2in}

\answer 

\part[4] What is the instantaneous velocity of the particle when
$t=1$?

\vspace{2in}

\answer 

\part[4] What is the acceleration of the particle when $t=1$?

\vspace{2in}

\answer 

\end{parts}

%%%%%%%%%%%%%%%%

\end{questions}

\end{document}

