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\leftline{MATH 131 -- Calculus for Physical and Life Sciences}
\leftline{Practice Questions for Exam 1}
\leftline{September 17, 2007}
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\noindent
{\it Disclaimers:}  The actual exam questions may be formatted differently and 
may ask things in different ways, explore different aspects of the topics
we have discussed, etc.  The following
questions {\it only} indicate the range of topics that will be covered and the 
approximate mix of topics and degree of difficulty of the questions on the real exam.  
There are probably a few more parts here than will appear on the real exam.
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\noindent
I.  Let $f(x) = \sqrt{1 - x}$ and $g(x) = e^{-x}$.  
\begin{enumerate}
\item[A)]  What are the functions $(f\circ g)(x) = f(g(x))$ and 
$(g\circ f)(x) = g(f(x))$?
\item[B)]  What is the domain of $f \circ g$?  
\item[C)]  Is $f\circ g$ a one-to-one function?  If so, find its inverse
function $(f\circ g)^{-1}$.  If not, say why not. 
\end{enumerate}
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\noindent
II.  Consider the function $K(t)$, which gives 
the height (in feet) of a kite above the ground at time $t$ (in 
seconds):
\[
\begin{array}{ccccccc}
t & \vline & 2.0 & 2.5 & 3.0 & 3.5 & 4.0\\ \hline
K(t) & \vline & 47 & 54 & 61 & 68 & 75
\end{array}
\]
\begin{enumerate}
\item[A)]  Explain why it is reasonable to suspect that $K(t)$ is a linear function,
and give a possible formula for $K(t)$.
\item[B)]  If $K(t)$ is given by your formula from part A, $K$ a one-to-one
function.  Give a formula for $K^{-1}$.  
\item[C)]  Using your formula from part A, what is the value $K(5)$ and what
does this number represent?
\item[D)]  What does the number $K^{-1}(70)$ represent.  
\end{enumerate}
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\noindent
III.  Charcoal is created by burning 
wood.  The amount of radioactive $C^{14}$ left in 
in the charcoal decays over time.  One half of the original amount is left
after $5730$ years.  
\begin{enumerate}
\item[A)]  If 300 grams are present originally, give a formula
for the amount left after $t$ years.  
\item[B)]  How much is left after $17190$ years?  
\item[C)]  How much is left after $30000$ years?  Express your
answer in {\it exact form}, not as a decimal approximation.
\end{enumerate}
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\noindent
IV.  Short answer:
\begin{enumerate}
\item[A)]  Find all solutions of the equation $|\sin(2x)| = 1$ for $0 \le x \le 2\pi$
(Note:  $|x|$ is the absolute value of $x$:  $|x| = x$ if $x \ge 0$ and $|x| = -x$
if $x < 0$.)
\item[B)]  Express as a single logarithm value:  $-\frac{1}{2} \ln(x) + 3\ln(x - 1)$.
\item[C)]  True or False:  For all real $x$, $\tan^{-1}(\tan(x)) = x$.  Explain.
\item[D)]  Simplify:  $\tan(\sin^{-1}(3x))$.  (Hint:  Set up a right triangle containing
an angle $\theta = \sin^{-1}(3x)$.)
\end{enumerate}
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\noindent
V.  Use graph transformations to sketch the following graphs.  Indicate
clearly where your graphs cross the $x$- and $y$-axes:
\begin{enumerate}
\item[A)] $y = 5\sin(x - \pi/2) + 3$  for $0 \le x \le 2\pi$
\item[B)] $y = 5 - 2^x$
\item[C)] $y = 5 - 2^{-x}$
\end{enumerate}
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\noindent
VI.  The graph $y=f(x)$ and four graphs obtained by transforming it
    are shown. Match the given formulas with the corresponding graph.
    Note that there is an extra graph.

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(a) $y=f(\frac{1}{2}x):\qquad$   
(b) $y=\frac{1}{2}f(x):\qquad$ 
(c) $y=\frac{1}{2}-f(x): \qquad$

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