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\noindent
{\Large {\bf Mathematics 1210  \hspace{5ex} PRACTICE EXAM I 
\hspace{6ex}Fall 2004}}\\ 
%\vspace{2ex}

\begin{enumerate}

\item[1.]  Calculate the following limits.  If a
      particular limit does not exist, state this clearly
      and tell why.



\begin{enumerate}
 \item[(a)] $\ds \lim_{x \rightarrow \sqrt{2}} 
                3x^2$
 \hspace{4ex}
% \item[(b)] 
(b) $\ds \lim_{\theta \rightarrow \pi/2} 
                \tan{\theta}$
\hspace{4ex}
% \item[(c)] 
(c) $\ds \lim_{x \rightarrow -1} \frac{x^2-x+2}{x+1}$
\hspace{4ex}
% \item[(d)] 
(d) $\ds \lim_{x \rightarrow 0^+} \sqrt{x}\sin{\left( \frac{1}{x^2}\right)}$

\vspace{1ex}
 \item[(e)] $\ds \lim_{x \rightarrow 0} \frac{\sin{(x^2)}}{x}$
\hspace{4ex}
% \item[(f)] 
(f) $\ds \lim_{x \rightarrow +\infty} 
                \frac{\sin{x}}{x}$
\hspace{4ex}
% \item[(g)] 
(g)  $\ds \lim_{x \rightarrow 2} f(x)$, ~~~ 
where $f (x) = \left \{
\begin{array}{ll}
x^3, & x \leq 2 \\
x, & x > 2
\end{array}
\right.$
\vspace{1ex}
\item[(h)]  $\ds \lim_{x \rightarrow \pi} f(x)$, ~~~ 
where $f (x) = \left \{
\begin{array}{ll}
0, & x \hspace{1.5ex} \mbox{irrational} \\
\sin{\left( \frac{1}{q}\right)}, & x=\frac{p}{q} \hspace{1.5ex} \mbox{rational}
\end{array}
\right.$
\hspace{2ex}
(i) $\ds \lim_{x \rightarrow +\infty} \sqrt[3]{\frac{8x^7+3x^5}{x^7+6x^2}}$


\end{enumerate}



\item[2.] 
\begin{enumerate}
\item[(a)] Let $f(x) = \sqrt{x}$.  Using the {\it
      definition} of the derivative, calculate $f'(x)$. 
      Do the same for $g(x) = 1/x$.     
\item[(b)]  Using your result from (a), find the equation of
      the line tangent to the graph of $f(x) = \sqrt{x}$ at 
      $x = 1$. Do the same for $g(x) = 1/x$. 
\end{enumerate}


\item[3.] Let $f(x) = -x$ when $x\leq0, x\neq-1; \;\;\;2$ when $x=-1; 
         \;\;\; \sqrt{x}$ when $0<x<1; \;\;\;\sqrt[3]{3-x}$ when $x\geq1$.
         Sketch the graph of $f(x)$.
\begin{enumerate}
\item[(a)]  For which points $c$ does $\ds \lim_{x
      \rightarrow c} f(x)$ exist? \hspace{3ex}
%\vfill
(b) For which points is $f$ continuous?
%\vfill
\item[(c)]  For which points is $f$ differentiable?
\end{enumerate}


\item[4.] Let $f(x) = x+2$ when $\ds x\leq0; \;\;\; -\frac{1}{2}x+2$ when 
         $0<x\leq2;
         \;\;\; \sqrt{x-2}\; + \;1$ when $x>2$.
         Sketch the graph of $f(x)$, and then using your result
         sketch the graph of $f'(x)$.


\item[5.] Find the derivative and antiderivative of 
          (a) $f(x)=12x^5+5x^4+x^2+2x+1$,
          (b) $f(x)=(x+1)^3$, (c) $f(x)=(3x^2-2x+1)(x-1)$.
         


%\item[6.]  Calculate the following derivatives 
%(a) $\ds \frac{d}{dx} \left( \frac{x+1}{x-1} \right)$,
%(b) $\ds \frac{d}{dx} (\sqrt{x})^3$,\\
%(c) $\ds \frac{d}{dx} (x^3+3x^2+2x+1)^{-1}$,
%(d) $\ds \frac{d}{dx} \left((x^2+1)\cos{x}\right)$.



\item[6.] Let the position $x(t)$ of a particle at time $t$
          be given by $x(t) = 3t^2-2t+1$. Find the
          instantaneous velocity $v(t)$ of the particle
          for any time $t$. Where is the particle when
          its velocity is zero?

\item[7.] A clever tick falls strategically from the top of a 22 foot tree
         onto the top of the head of a 6 foot tall hiker.
          How long does it take the tick to hit the hiker's head (neglecting
          air friction), and what is the tick's velocity when it hits?



\item[8.] On earth, the acceleration
$a(t)$ due to gravity is essentially constant in time,
with $a(t)=-g$, where $g=32\; f/s^2$. On nearby planet
$\Psi$, scientists have discovered how to vary their
planet's gravitational force with time. If the acceleration
due to gravity on $\Psi$ is $a(t)=-t$, find the 
analog of the earth formula $x(t)=-16t^2 + {v_0}t + x_0$  
for planet $\Psi$. That is, find $x(t)$ for $a(t)=-t$
with initial velocity $v_0$ and position $x_0$.
%(HINT: $a(t)=\frac{dv}{dt}$ and $v(t)=\frac{dx}{dt},$
%or equivalently, $a(t)=\frac{d^2x}{dt^2}$.) 
Using your expression for $x(t)$, find how long it will 
take for a ball thrown upward from the ground on $\Psi$
at $t=0$ with initial velocity 6$\;f/s$ to hit the ground. 




\end{enumerate}

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