108 lines
2.8 KiB
TeX
108 lines
2.8 KiB
TeX
% Copyright (C) 2023 <Mark (mark@betalupi.com)>
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% This program is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% You may have received a copy of the GNU General Public License
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% along with this program. If not, see <https://www.gnu.org/licenses/>.
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% If you edit this, please give credit!
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% Quality handouts take time to make.
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\section{Dual Numbers}
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\definition{}
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In the problems below, $\varepsilon$ an infinitesimal so that $\varepsilon^2 = 0$. \par
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Note that $\varepsilon \neq 0$.
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\definition{}
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The set of \textit{dual numbers} is a nonarchimedian extension of $\mathbb{R}$ \par
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that consists of elements that look like $a + b\varepsilon$, where $a, b \in \mathbb{R}$.
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\problem{}
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Compute $(a + b\varepsilon) \times (c + d\varepsilon)$.
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\vfill
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\definition{}
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Let $f(x)$ be an algebraic function $\mathbb{R} \to \mathbb{R}$. \par
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(that is, a function we can write using the operators $+-\times\div$, powers, and roots) \par
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\note[Note]{
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Why this condition? These are the only operations we have in an ordered field! \\
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Powers, roots, and division aren't directly available, but are fairly easy to define.
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}
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\vspace{2mm}
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the \textit{derivative} of such an $f$ is a function $f'$ that satisfies the following:
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\begin{equation*}
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f(x + \varepsilon) = f(x) + f'(x)\varepsilon
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\end{equation*}
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If $f(x + \varepsilon)$ is not defined, we will say that $f$ is not \textit{differentiable} at $x$.
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\problem{}
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What is the derivative of $f(x) = x^2$?
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\vfill
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\problem{}
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What is the derivative of $f(x) = x^n$?
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\vfill
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\problem{}
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Say the derivatives of $f$ and $g$ are known. \par
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Find the derivatives of $h(x) = f(x) + g(x)$ and $k(x) = f(x) \times g(x)$.
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\vfill
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\pagebreak
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\problem{}
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When can you divide dual numbers? \par
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That is, for what numbers $(a + b\varepsilon)$ is there a $(x + y\varepsilon)$ such that $(a +b\varepsilon)(x+y\varepsilon) = 1$?
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\vfill
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\problem{}
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Find an explicit formula for the inverse of a dual number, \par
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and use it to find the derivative of $f(x) = \frac{1}{x}$.
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\vfill
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\problem{}
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Which dual numbers have a square root? \par
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That is, for which dual numbers $(a + b\varepsilon)$ is there a dual number
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$(x + y\varepsilon)$ such that $(x + y\varepsilon)^2 = a + b\varepsilon$?
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\vfill
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\problem{}
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Find an explicit formula for the square root and use it to find the derivative of $f(x) = \sqrt{x}$
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\vfill
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\problem{}
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Find the derivative of the following functions:
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\begin{itemize}
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\item $f(x) = \frac{x}{1 + x^2}$
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\item $g(x) = \sqrt{1 - x^2}$
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\end{itemize}
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\vfill
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\problem{}
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Say the derivatives of $f$ and $g$ are known. \par
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What is the derivative of $f(g(x))$?
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\vfill
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\pagebreak |