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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Review Sheet -- Midterm Exam 3}
\centerline{April 22, 2014}
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\noindent
{\it General Information}
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As you know from the course syllabus, the third and final 
midterm exam for the course will be given in class on 
Friday, May 2.  This be an individual closed book exam
similar in format to the other midterms we have done this
semester.  {\it You may use a calculator, but you may not use
any other electronic devices during the exam.}  
I will be happy 
to hold a late afternoon or evening review session to help you 
prepare.  Late afternoon or evening times are possible Wednesday, April 30
or Thursday, May 1.  
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\noindent
{\it Topics to be Covered}
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The exam will cover the material we have covered since the last 
exam (problem sets 7,8,9).
This is sections 3.5, 3.6, 4.1, 4.2, 4.3, 5.1, 5.2, 5.3, and 5.4
in the text, but 
as before, not all the topics in those sections were discussed in 
class. You are responsible for only what we did talk about:
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\item{1)}  Properties of continuous functions on closed intervals:
The Intermediate Value and Extreme Value Theorems (IVT, EVT), uniform continuity.  
(Note: This overlaps Exam 2 to some extent, but it is also an important 
component of several other topics from this part of the course.  So this
exam will cover some of these topics as well.)  
\item{2)}  The definition of differentiability and examples.
\item{3)}  Rolle's Theorem, the Mean Value Theorem (MVT) and its consequences.
\item{4)}  The definition of integrability, criteria for integrability, computations
of definite integrals from the definition.
\item{5)}  The Fundamental Theorem of Calculus (FTC).
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\noindent
{\it What to Expect}
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The exam will have four or five questions, each possibly with several
parts.  Some questions will ask for a precise statement of a 
definition or a theorem we have discussed.  Be prepared to give
careful statements of the definitions noted above and know 
how to use them (for instance how to show that a function limit
exists using the $\varepsilon,\delta$ definition).
\noindent
Also know and be able to give these proofs:
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\item{1)}  The Mean Value Theorem (including the special case known as
Rolle's Theorem -- deduce the general result from that).  (You may use
other results needed in the proof, such as the Extreme Value Theorem, 
without proving them here).
\item{2)}  A monotone increasing function on an interval $[a,b]$
is integrable.
\item{3)}  A continuous function on $[a,b]$ is integrable.  
\item{4)}  Part 1 of the FTC:  If $f$ is continuous on $[a,b]$, and
$F(x) = \int_a^x f$, then $F'(x) = f(x)$.  
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\noindent
The other questions will be similar to questions from the problem sets. 
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\noindent
{\it Review Problems}
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\noindent
A)  For each of the following, give an example or a short proof
that no such examples exist:
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\item{1)}  A function $g : {\bf R} \to {\bf R}$ that is continuous at 
no $x \in {\bf R}$.
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\item{2)}  A function $f : [0,1] \to {\bf R}$ such 
that $3 = {\rm lub}\{f(x) : x\in [0,1]\}$, but there is no $x\in [0,1]$
with $f(x) = 3$.  
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\item{3)}  A continuous function $f : [0,1] \to {\bf R}$ such 
that $3 = {\rm lub}\{ f(x) : x\in [0,1] \}$, but there is no $x\in [0,1]$
with $f(x) = 3$. 
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\item{4)}  A continuous function $f : [0,1] \to [0,1]$ with
$f(0) = 0$, $f(1) = 1$, but such that there is no $x\in [0,1]$
with $f(x) = 1/3$.
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\item{5)}  A function $f: [a,b] \to {\bf R}$ that is
differentiable on $[a,b]$, satisfies $f(a) = f((a+b)/2) = f(b)$,
but has only one critical point in $(a,b)$.  (Recall, a 
critical point is a $c$ where $f'(c) = 0$ or $f'(c)$ does
not exist.)
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\item{7)}  A differentiable function $f : {\bf R} \to {\bf R}$ such that 
$f'(x) = e^{-x^2}$ and $f(0) = 0$.  
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\noindent
B)  Let 
$$f(x) = \cases{1 & if $x < 0$\cr
                1 + x + x^2 & if $x \ge 0$\cr}$$
Potentially useful information:  $\sum_{i=1}^n i = {n(n+1)\over 2}$,
and $\sum_{i=1}^n i^2 = {n(n+1)(2n+1)\over 6}$.
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\item{1)} Using the definition of $f'(0)$, determine whether
$f(x)$ is {\it differentiable} at $x = 0$.
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\item{2)}  Show directly from the definition that $f(x)$ is integrable on $[-1,1]$.
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\item{3)} State a general theorem that implies $f(x)$ is integrable on $[-1,1]$.
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\item{4)}  Using the Fundamental Theorem of 
Calculus and any other facts about integrals you need, 
find the value of $\int_{-1}^1 f(x)\ dx$.
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\noindent
C)  Show that if $g : (a,b) \to {\bf R}$ is differentiable
at some $c \in (a,b)$ with $g'(c) \ne 0$, then
there is a $\delta > 0$ such that $g(x) \ne g(c)$ for 
all $x$ with $0 < |x - c| < \delta$.  (Hint:  One method is to argue
by contradiction, and ``think sequentially.'') 
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\noindent
D) 
\item{1)} Assume that $g$ is differentiable on $[a,b]$ and satisfies
$g'(a) < 0$, $g'(b) > 0$.  Show that there exists some $x_1\in (a,b)$
where $g(x_1) < g(a)$ and also some $x_2 \in (a,b)$ such that $g(x_2) < g(b)$.
\item{2)}  Deduce from part 1 that there is some $c \in (x_1,x_2)$
where $g'(c) = 0$.  (Hint: ``think Rolle.'')
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\noindent
E)  Let $f$ be differentiable on $(a,b)$
and continuous on $[a,b]$.  Assume also that 
and $|f'(x)| < 1$ for all $x \in (a,b)$.  Show 
that there exists $0 \le c \le 1$ such that 
$$|f(x) - f(y)| \le c |x - y|$$
for all $x,y \in [a,b]$.
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\noindent
F)  
\item{1)}  Let $f(x) = 2x^2 + 3x + 3$.  Show directly 
(i.e. using the definition via upper and lower sums) that
$f$ is integrable on $[0,1]$, and determine the value
of $\int_0^1 2x^2 + 3x + 3\ dx$.
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\item{2)}  Let 
$$g(x) = \cases{x & if $0\le x \le 1$\cr
                              3 - x & if $1 \le x \le 2$\cr}$$
Is $g$ integrable on $[0,2]$?  Why or why not?  If so,
determine the value $\int_0^2 g(x)\ dx$.  
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\noindent
G)  Let $f : {\bf R} \to {\bf R}$ be defined by $f(x) = 2x^2 + 3x + 3$.
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\item{1)}  Show that $f$ is uniformly continuous on $[0,1]$ directly from
the definition.
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\item{2)}  Is there a general theorem that gives the same result as part 1?
Explain.
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\item{3)}  Is $f$ uniformly continuous on ${\bf R}$?  Prove your assertion.  
\bye
