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mark 2024-01-31 11:31:42 -08:00
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14
Advanced/Nonstandard Analysis/parts/1 extensions.tex
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@ -116,7 +116,7 @@ Then, show that a negative $\delta$ is infinitesimal if and only if it is bigger
Prove the following statements: \par Prove the following statements: \par
\begin{itemize} \begin{itemize}
\item If $\delta$ and $\varepsilon$ are infinitesimal, then $\delta + \varepsilon$ is infinitesimal. \item If $\delta$ and $\varepsilon$ are infinitesimal, then $\delta + \varepsilon$ is infinitesimal.
\item If $\delta$ is infinitesimal and $x$ is limited, then $a\delta$ is infinitesimal. \item If $\delta$ is infinitesimal and $x$ is limited, then $x\delta$ is infinitesimal.
\item If $x$ and $y$ are limited, $xy$ and $x+y$ are too. \item If $x$ and $y$ are limited, $xy$ and $x+y$ are too.
\item A nonzero $\delta$ is infinitesimal iff $\delta^{-1}$ is unlimited. \item A nonzero $\delta$ is infinitesimal iff $\delta^{-1}$ is unlimited.
\end{itemize} \end{itemize}
@ -126,9 +126,9 @@ Prove the following statements: \par
\problem{} \problem{}
Let $\delta$ be a positive infinitesimal. Which is greater? Let $\delta$ be a positive infinitesimal. Which is greater?
\begin{itemize} \begin{itemize}
\item $\delta$ or $\delta^2$? \item $\delta$ or $\delta^2$
\item $(1 - \delta)$ or $(1 + \delta^2)^{-1!}$? \item $(1 - \delta)$ or $(1 + \delta^2)^{-1!}$
\item $\frac{1 + \delta}{1 + \delta^2}$ or $\frac{2 + \delta^2}{2 + \delta^3}$? \par \item $\frac{1 + \delta}{1 + \delta^2}$ or $\frac{2 + \delta^2}{2 + \delta^3}$ \par
\note[Note]{we define $\frac{1}{x}$ as $x^{-1}$, and thus $\frac{a}{b} = a \times b^{-1}$} \note[Note]{we define $\frac{1}{x}$ as $x^{-1}$, and thus $\frac{a}{b} = a \times b^{-1}$}
\end{itemize} \end{itemize}
@ -138,13 +138,13 @@ Let $\delta$ be a positive infinitesimal. Which is greater?
\definition{} \definition{}
We say two elements of an ordered field are \textit{infinitely close} if $x - y$ is infinitesimal. \par We say two elements of an ordered field are \textit{infinitely close} if $x - y$ is infinitesimal. \par
We say that $x_0 \in \mathbb{R}$ is a \textit{standard part} of $x$ if it is infinitely close to $x$. \par We say that $x_0 \in \mathbb{R}$ is the \textit{standard part} of $x$ if it is infinitely close to $x$. \par
\problem{} \problem{}
We will denote the standard part of $x$ as $\text{st}(x)$. \par We will denote the standard part of $x$ as $\text{st}(x)$. \par
Show that $\text{st}(x)$ is well-defined for limited $x$. \par Show that $\text{st}(x)$ is well-defined for limited $x$. \par
(In other words, Show that $x_0$ exists and is unique for limited $x$). \par (In other words, show that $x_0$ exists and is unique for limited $x$). \par
\hint{To prove existance, consider $\text{sup}(\{a \in \mathbb{R} ~|~ a < x\}$)} \hint{To prove existence, consider $\text{sup}(\{a \in \mathbb{R} ~|~ a < x\}$)}
\vfill \vfill

4
Advanced/Nonstandard Analysis/parts/2 dual.tex
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@ -58,7 +58,7 @@ What is the derivative of $f(x) = x^n$?
\vfill \vfill
\problem{} \problem{}
Say the derivatives of $f$ and $g$ are known. \par Assume that the derivatives of $f$ and $g$ are known. \par
Find the derivatives of $h(x) = f(x) + g(x)$ and $k(x) = f(x) \times g(x)$. Find the derivatives of $h(x) = f(x) + g(x)$ and $k(x) = f(x) \times g(x)$.
\vfill \vfill
@ -101,7 +101,7 @@ Find the derivative of the following functions:
\vfill \vfill
\problem{} \problem{}
Say the derivatives of $f$ and $g$ are known. \par Assume that the derivatives of $f$ and $g$ are known. \par
What is the derivative of $f(g(x))$? What is the derivative of $f(g(x))$?
\vfill \vfill