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\section{Dual Numbers}

\definition{}
In the problems below, $\varepsilon$ an infinitesimal so that $\varepsilon^2 = 0$. \par
Note that $\varepsilon \neq 0$.

\definition{}
The set of \textit{dual numbers} is a nonarchimedian extension of $\mathbb{R}$ \par
that consists of elements that look like $a + b\varepsilon$, 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 $+-\times\div$, powers, and roots) \par
\note[Note]{
	Why this condition? These are the only operations we have in an ordered field! \\
	Powers, roots, and division aren't directly available, but are fairly easy to define.
}

\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{}
Say 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, \par
and use it to 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{}
Say the derivatives of $f$ and $g$ are known. \par
What is the derivative of $f(g(x))$?

\vfill
\pagebreak