#LyX 1.5.1 created this file. For more info see http://www.lyx.org/
\lyxformat 276
\begin_document
\begin_header
\textclass amsart
\language english
\inputencoding auto
\font_roman default
\font_sans default
\font_typewriter default
\font_default_family default
\font_sc false
\font_osf false
\font_sf_scale 100
\font_tt_scale 100
\graphics default
\paperfontsize default
\spacing double
\papersize default
\use_geometry false
\use_amsmath 1
\use_esint 1
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
\leftmargin 1.2in
\topmargin 1.2in
\rightmargin 1.2in
\bottommargin 1.2in
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\author "" 
\author "" 
\end_header

\begin_body

\begin_layout Title
Math Review #2 Supplement (Math Camp 2007)
\end_layout

\begin_layout Author
John Morrow
\end_layout

\begin_layout Date
8/23/07
\end_layout

\begin_layout Standard
What follows is a rough sketch of material for Day 2 and a few more problems.
\end_layout

\begin_layout Section
Day 2
\end_layout

\begin_layout Enumerate
Functions
\end_layout

\begin_deeper
\begin_layout Enumerate
Example: 
\begin_inset Formula $\mathbb{R}^{1}\rightarrow\mathbb{R}^{1}$
\end_inset


\end_layout

\begin_layout Enumerate
Monotone functions.
\end_layout

\begin_layout Enumerate

\series bold
Problem:
\series default
 Construct examples on 
\begin_inset Formula $[0,1]$
\end_inset

 of the following types of functions and find the points in 
\begin_inset Formula $[0,1]$
\end_inset

 where they are maximized and minimized:
\end_layout

\begin_deeper
\begin_layout Enumerate
Strictly increasing.
\end_layout

\begin_layout Enumerate
Strictly decreasing.
\end_layout

\begin_layout Enumerate
Increasing but not strictly increasing.
\end_layout

\begin_layout Enumerate
A function which fails both definitions of increasing and decreasing functions.
\end_layout

\end_deeper
\begin_layout Enumerate
Example: 
\begin_inset Formula $\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$
\end_inset


\end_layout

\begin_layout Enumerate
Example: 
\begin_inset Formula $\mathbb{R}^{N}\rightarrow\mathbb{R}^{1}$
\end_inset


\end_layout

\begin_layout Enumerate
Concave/Convex functions.
\end_layout

\end_deeper
\begin_layout Enumerate
Smoothness
\end_layout

\begin_deeper
\begin_layout Enumerate
Sequences.
\end_layout

\begin_layout Enumerate
Open/closed intervals.
\end_layout

\begin_layout Enumerate
Closed and bounded intervals, convergence of sequences.
\end_layout

\begin_layout Enumerate

\series bold
Problem:
\series default
 Suppose for a given sequence 
\begin_inset Formula $\{a_{n}\}$
\end_inset

 that 
\begin_inset Formula $|a_{n}|\leq M$
\end_inset

 for some constant 
\begin_inset Formula $M$
\end_inset

.
 Show that 
\begin_inset Formula $\{a_{n}\}$
\end_inset

 has a convergent subsequence.
 Does 
\begin_inset Formula $\{a_{n}\}$
\end_inset

 have to converge to a unique number 
\begin_inset Formula $a$
\end_inset

? (Prove or provide a counterexample.)
\end_layout

\begin_layout Enumerate
Continuity.
\end_layout

\begin_layout Enumerate
Continuous functions on closed and bounded sets always take on maximal and
 minimal values.
\end_layout

\begin_layout Enumerate

\series bold
Problem:
\series default
 
\begin_inset Formula $f(x)\equiv1/|x|$
\end_inset

 fails to take on a maximal value.
 How does this fail to happen?
\end_layout

\begin_layout Enumerate
The derivative on 
\begin_inset Formula $\mathbb{R}^{1}$
\end_inset

.
\end_layout

\begin_layout Enumerate

\series bold
Problem: 
\series default
Show that continuity of 
\begin_inset Formula $f(x)$
\end_inset

 at 
\begin_inset Formula $y$
\end_inset

 is a necessary condition for 
\begin_inset Formula $f$
\end_inset

 to be differentiable at 
\begin_inset Formula $y$
\end_inset

.
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\mathcal{C}^{1}$
\end_inset

, 
\begin_inset Formula $\mathcal{C}^{2}$
\end_inset

, 
\begin_inset Formula $\mathcal{C}^{\infty}$
\end_inset

.
\end_layout

\end_deeper
\begin_layout Enumerate
Optimization Primer
\end_layout

\begin_deeper
\begin_layout Enumerate
Local/Global Maxima and Minima.
\end_layout

\begin_layout Enumerate
Uniqueness of maxima/minima for strictly concave/convex functions.
\end_layout

\begin_layout Enumerate

\series bold
Problem:
\series default
 Provide an example of a function which is concave and convex.
 
\end_layout

\begin_layout Enumerate
Necessity of critical points for maxima/minima.
\end_layout

\begin_layout Enumerate

\series bold
Problem:
\series default
 Construction a function 
\begin_inset Formula $f(x)$
\end_inset

 where for some 
\begin_inset Formula $y$
\end_inset

 we have 
\begin_inset Formula $f'(y)=0$
\end_inset

 but 
\begin_inset Formula $f'(y)$
\end_inset

 is not a local max or min.
\end_layout

\begin_layout Enumerate
Critical points sufficient for maxima/minima under concavity/convexity.
\end_layout

\end_body
\end_document