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\begin_body

\begin_layout Title
Math Review #1 Supplement (Math Camp 2007)
\end_layout

\begin_layout Author
John Morrow
\end_layout

\begin_layout Date
8/21/07
\end_layout

\begin_layout Standard
What follows is a supplement of the document labeled 
\begin_inset Quotes eld
\end_inset

MATH REVIEW #1 (short version).
\begin_inset Quotes erd
\end_inset

 Sections are provided to correspond roughly with what I intend to cover
 each day of Math Camp 2007.
 These questions are primarily designed to get you thinking, not review
 material.
\end_layout

\begin_layout Section
Day 1
\end_layout

\begin_layout Subsection
Some Examples of Sets
\end_layout

\begin_layout Standard
Some familar sets should be 
\begin_inset Formula $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{R}^{n},\mathbb{C},\mathcal{C}^{0},\mathcal{C}^{n},\mathcal{C}^{\infty}$
\end_inset

 as well as the set of all polynomial functions with real coefficients (say
 
\begin_inset Formula $\mathbb{P}$
\end_inset

) and linear spaces on 
\begin_inset Formula $\mathbb{R}^{n}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula \begin{eqnarray*}
\mathcal{V} & \equiv & \{x\mbox{ is a letter}:\quad x\mbox{ is a vowel}\}\\
\mathcal{G} & \equiv & \{x\mbox{ is a letter}:\quad x\mbox{ appears in the word "Cheese"}\}\\
\mathcal{A} & \equiv & \{x\in\mathbb{R}:\quad x\geq0\mbox{ and }x\leq1\}\end{eqnarray*}

\end_inset


\end_layout

\begin_layout Enumerate
Some questions:
\end_layout

\begin_deeper
\begin_layout Enumerate
How many elements does 
\begin_inset Formula $\mathcal{G}$
\end_inset

 have? What is 
\begin_inset Formula $\mathcal{V}\cap\mathcal{G}$
\end_inset

? 
\end_layout

\begin_layout Enumerate
Relationships between 
\begin_inset Formula $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$
\end_inset

? 
\end_layout

\begin_layout Enumerate
What about 
\begin_inset Formula $\mathbb{Z}^{2}$
\end_inset

 and 
\begin_inset Formula $\mathbb{R}^{2}$
\end_inset

? 
\end_layout

\begin_layout Enumerate
Relationships between 
\begin_inset Formula $\mathcal{C}^{0},\mathcal{C}^{n},\mathcal{C}^{\infty},\mathbb{P}$
\end_inset

?
\end_layout

\begin_layout Enumerate
Another way of writing 
\begin_inset Formula $\mathcal{A}$
\end_inset

?
\end_layout

\begin_layout Enumerate
Is 
\begin_inset Formula $\mathcal{A}$
\end_inset

 convex? What about 
\begin_inset Formula $\mathcal{B}\equiv\{x\in\mathbb{R}:\quad x>1\mbox{ and }x<2\}$
\end_inset

?
\end_layout

\begin_layout Enumerate
What is 
\begin_inset Formula $\mathcal{A}\cap\mathcal{B}$
\end_inset

? How about 
\begin_inset Formula $\mathcal{D}\equiv\{x\in\mathbb{R}:\quad x>2\mbox{ and }x<1\}$
\end_inset

? And 
\begin_inset Formula $\mathcal{D}^{c}$
\end_inset

?
\end_layout

\begin_layout Enumerate
What is 
\begin_inset Formula $\mathcal{E}\equiv(((\mathcal{A}\cap\mathcal{B})^{c}\cap\mathcal{B})\cup\mathcal{A})\cap\mathcal{Q}$
\end_inset

? How about 
\begin_inset Formula $\mathcal{E}\cap\mathbb{Z}$
\end_inset

?
\end_layout

\begin_layout Enumerate
Is the union of two convex sets convex? (Prove or provide a counterexample.)
\end_layout

\begin_layout Enumerate
Is the intersection of two convex sets convex? (Prove or provide a counterexampl
e.)
\end_layout

\end_deeper
\begin_layout Subsection
Functions and Continuity
\end_layout

\begin_layout Enumerate
Some questions:
\end_layout

\begin_deeper
\begin_layout Enumerate
Is 
\begin_inset Formula $f(x)\equiv\pm\sqrt{x}$
\end_inset

 a function? If 
\begin_inset Formula $g$
\end_inset

 is a function is 
\begin_inset Formula $g^{-1}(x)$
\end_inset

 always a function? (If not provide a counterexample.)
\end_layout

\begin_layout Enumerate
If 
\begin_inset Formula $g$
\end_inset

 is a continuous function and 
\begin_inset Formula $g^{-1}(x)$
\end_inset

 is a function, is 
\begin_inset Formula $g^{-1}$
\end_inset

 continuous? (If not provide a counterexample.)
\end_layout

\begin_layout Enumerate
On 
\begin_inset Formula $[0,1]$
\end_inset

 are monotone functions invertible? Is 
\emph on
strict 
\emph default
monotonicity involved? What about continuity?
\end_layout

\begin_layout Enumerate
Suppose 
\begin_inset Formula $f$
\end_inset

 is continuously differentiable, 
\begin_inset Formula $f(1)=10$
\end_inset

 and 
\begin_inset Formula $a_{n}\rightarrow1$
\end_inset

.
 Let 
\begin_inset Formula $b_{n}\equiv f(a_{n})$
\end_inset

.
 What is 
\begin_inset Formula $\lim_{n\rightarrow\infty}b_{n}$
\end_inset

? 
\end_layout

\begin_layout Enumerate
Let 
\begin_inset Formula $f$
\end_inset

 be as in (d) and let 
\begin_inset Formula $g(x)\equiv\frac{1}{x-10}$
\end_inset

.
 Suppose 
\begin_inset Formula $f'(1)>0$
\end_inset

.
 Can you compute 
\begin_inset Formula $\lim_{n\rightarrow\infty}g\circ f(a_{n})$
\end_inset

 (allowing for 
\begin_inset Formula $\pm\infty$
\end_inset

)? What if 
\begin_inset Formula $a_{n}\geq1$
\end_inset

 for all 
\begin_inset Formula $n$
\end_inset

?
\end_layout

\end_deeper
\begin_layout Subsection
Derivatives
\end_layout

\begin_layout Enumerate
Some questions:
\end_layout

\begin_deeper
\begin_layout Enumerate
If 
\begin_inset Formula $g$
\end_inset

 is a 
\begin_inset Formula $\mathcal{C}^{1}$
\end_inset

 function on 
\begin_inset Formula $[0,1]$
\end_inset

 with 
\begin_inset Formula $g'>0$
\end_inset

, is 
\begin_inset Formula $g^{-1}$
\end_inset

 a function? (If not provide a counterexample.) How about if just 
\begin_inset Formula $g'\neq0$
\end_inset

? If 
\begin_inset Formula $g'\equiv0$
\end_inset

? Can you connect this to any 
\begin_inset Quotes eld
\end_inset

famous
\begin_inset Quotes erd
\end_inset

 results?
\end_layout

\begin_layout Enumerate
If 
\begin_inset Formula $f$
\end_inset

 is differentiable at 
\begin_inset Formula $x$
\end_inset

, what are
\begin_inset Formula \begin{alignat*}{7}
(\ln f(x))' &  & \; & \; & (\exp\{f(x)\})' &  & \; & \; & (5^{f(x)})' &  & \; & \; & (f(x)^{f(x)})'\end{alignat*}

\end_inset


\end_layout

\begin_layout Enumerate
Assume 
\begin_inset Formula $f,g,h$
\end_inset

 are differentiable at 
\begin_inset Formula $x$
\end_inset

.
 What is 
\begin_inset Formula $(f\circ g\circ h(x))'$
\end_inset

?
\end_layout

\begin_layout Enumerate
Assume 
\begin_inset Formula $f$
\end_inset

 and 
\begin_inset Formula $f^{-1}$
\end_inset

 are diffentiable functions and use 
\begin_inset Formula $f(f^{-1}(x))'=(x)'=1$
\end_inset

 to derive an expression for 
\begin_inset Formula $f^{-1}(x)'$
\end_inset

.
\end_layout

\begin_layout Enumerate
Suppose 
\begin_inset Formula $f$
\end_inset

 is continuously differentiable on 
\begin_inset Formula $(0,1)$
\end_inset

 and 
\begin_inset Formula $f$
\end_inset

 is strictly increasing.
 Use the definition of the derivative to conclude that 
\begin_inset Formula $f'(x)\geq0$
\end_inset

 for 
\begin_inset Formula $x\in(0,1)$
\end_inset

.
 
\end_layout

\begin_layout Enumerate
A twice continuously differentiable function 
\begin_inset Formula $f$
\end_inset

 is said to satisfy the inada conditions if
\begin_inset Formula \begin{alignat*}{3}
\lim_{x\rightarrow0}f'(x) & =\infty & \quad & \quad & \lim_{x\rightarrow\infty}f'(x) & =0\end{alignat*}

\end_inset

What do the inada conditions have to do with the equation 
\begin_inset Formula $f'(x)=25$
\end_inset

? When considering 
\begin_inset Formula $f'(x)=25$
\end_inset

, would it be useful to know something like 
\begin_inset Formula $f''<0$
\end_inset

?
\end_layout

\begin_layout Enumerate
Construct a function 
\begin_inset Formula $f$
\end_inset

 which satisfies the inada conditions.
 Let 
\begin_inset Formula $g(x)\equiv f'(x)$
\end_inset

 and find 
\begin_inset Formula $g^{-1}(x)$
\end_inset

.
 How does 
\begin_inset Formula $g^{-1}$
\end_inset

 relate to (f)?
\end_layout

\end_body
\end_document