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Advanced/Nonstandard Analysis/parts/dual.tex

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+
+\section{Dual Numbers}
+
+\definition{}
+In the problems below, let $\varepsilon$ a positive infinitesimal so that $\varepsilon^2 = 0$. \par
+\note{Note that $\varepsilon \neq 0$.}
+
+\definition{}
+The set of \textit{dual numbers} consists of elements of the form $a + b\varepsilon$, where $a, b \in \mathbb{R}$.
+
+\problem{}
+Compute $(a + b\varepsilon) \times (c + d\varepsilon)$.
+
+\vfill
+
+
+
+\definition{}
+Let $f(x)$ be an algebraic function $\mathbb{R} \to \mathbb{R}$. \par
+(that is, a function we can write using the operators $+-\times\div$ and integer powers) \par
+
+\vspace{2mm}
+
+the \textit{derivative} of such an $f$ is a function $f'$ that satisfies the following:
+\begin{equation*}
+	f(x + \varepsilon) = f(x) + f'(x)\varepsilon
+\end{equation*}
+If $f(x + \varepsilon)$ is not defined, we will say that $f$ is not \textit{differentiable} at $x$.
+
+\problem{}
+What is the derivative of $f(x) = x^2$?
+
+\vfill
+
+\problem{}
+What is the derivative of $f(x) = x^n$?
+
+\vfill
+
+\problem{}
+Assume that the derivatives of $f$ and $g$ are known. \par
+Find the derivatives of $h(x) = f(x) + g(x)$ and $k(x) = f(x) \times g(x)$.
+
+\vfill
+\pagebreak
+
+
+
+
+\problem{}
+When can you divide dual numbers? \par
+That is, for what numbers $(a + b\varepsilon)$ is there a $(x + y\varepsilon)$ such that $(a +b\varepsilon)(x+y\varepsilon) = 1$?
+
+\vfill
+
+\problem{}
+Find an explicit formula for the inverse of a dual number $(a + b\varepsilon)$, assuming one exists. \par
+Then, use this find the derivative of $f(x) = \frac{1}{x}$.
+
+\vfill
+
+\problem{}
+Which dual numbers have a square root? \par
+That is, for which dual numbers $(a + b\varepsilon)$ is there a dual number
+$(x + y\varepsilon)$ such that $(x + y\varepsilon)^2 = a + b\varepsilon$?
+
+\vfill
+
+\problem{}
+Find an explicit formula for the square root and use it to find the derivative of $f(x) = \sqrt{x}$
+
+\vfill
+
+\problem{}
+Find the derivative of the following functions:
+\begin{itemize}
+	\item $f(x) = \frac{x}{1 + x^2}$
+	\item $g(x) = \sqrt{1 - x^2}$
+\end{itemize}
+
+\vfill
+
+\problem{}
+Assume that the derivatives of $f$ and $g$ are known. \par
+What is the derivative of $f(g(x))$?
+
+\vfill
+\pagebreak