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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Midterm Exam 3}
\centerline{December 3, 2004}
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{\it Directions}
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Do all work in the blue exam booklet.  There are 100 possible 
regular points and 10 possible Extra Credit points (see question IV below).
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I.  
\item{A)}  (10) Give the $\epsilon$ - $\delta$ definition for 
the statement $\lim_{x\to c} f(x) = L$.
\item{B)}  (10)  Show using the definition that 
$$\lim_{x\to 2} x^2 - 4x + 2 = -2.$$
\item{C)}  (5)  Using any applicable method, determine the limit 
$$L = \lim_{x\to 0} {x^2\over \sin(2x^2)}.$$

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II.
\item{A)}  (10) State the Intermediate Value Theorem.  
\item{B)}  (15) Let $f(x) = a_1 + a_2\sin(x)$, where $a_1,a_2$ are constants
with $0 < a_1 < a_2$.  Show that there exists some $c \in [-\pi/2,\pi/2]$
such that $f(c) = 0$.
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\noindent  
III.  (20)  Show that every monotone increasing function on a 
closed interval $[a,b]$ is integrable.
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IV.  True-False.  Say whether each of the following statements is true or false,
and give a complete, correct reason for your assertion. 
Do any three parts.  If you submit
solutions for all four, then I will consider the other one for
Extra Credit.  
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\noindent
All parts refer to the function $f$ defined by:
$$f(x) = \cases{ -x - 2 & if $x < 0$\cr
                 2 - x - x^3 & if $x \ge 0$\cr}$$

\item{A)} (10) $f$ is {\it differentiable} at $x = 0$.
\item{B)} (10) $f$ {\it integrable} on $[-2,2]$.
\item{C)} (10) The {\it conclusion} of the Mean Value Theorem is
true for $f$ on the interval $[-2,2]$. 
\item{D)} (10) The {\it conclusion} of the Extreme Value 
Theorem is true for $f$ on the interval $[-2,1]$.
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(Note:  The {\it conclusion} of 
theorem phrased as ``if $p$, then $q$'' is the statement $q$; the 
{\it hypotheses} are the statement(s) $p$.)
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