Derivatives and Integrals An Annotated Discourse

    <section xml:id="section-fundamental-theorem" label="section-fundamental-theorem">
      <title>The Fundamental Theorem</title>
      <p>
        There is a remarkable theorem:<fn xml:id="footnote-fermat">
        And fortunately we do not need to try to write it in the margin!
        </fn>
      </p>
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Mathematics:  Wrap inline mathematics in the "m"
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and/or AMSmath here.  You may also use any macro
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If the "m" tag bothers you, you may use \(...\).
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      <theorem xml:id="theorem-FTC">
        <title>The Fundamental Theorem of Calculus</title>
        <idx><h>Fundamental Theorem of Calculus</h></idx>
        <statement>
          <p>
            If <m>f(x)</m> is continuous,
            and the derivative of <m>F(x)</m> is <m>f(x)</m>, then
            <me>
              \definiteintegral{a}{b}{f(x)}{x}=F(b)-F(a)
            </me>
                <idx><h>test: buried in theorem/statement/p</h></idx>
          </p>
        </statement>
        <proof>
          <p>
            Left to the reader.
          </p>
        </proof>
      </theorem>
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Note above:  The "xml:id" attribute is like a LaTeX label.
It is a target for cross-reference, an alias for the theorem
so we can reference it later.  The "theorem" part of the alias
is not necessary, but is a good habit.
-->
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Next paragraph has two citations.  These reference
aliases given by "xml:id"'s in the bibliography.
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      <p>
        You will find almost nothing about all this in the article <xref ref="biblio-lay-article"/>,
        nor in the book <xref ref="biblio-judson-AATA"/>, since they belong in some other article,
        but we can cite them out-of-order for practice anyway.
      </p>
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There are various tools for authors
and we can process the source in order to
make these visible, or not
-->
      <p>
        When we are writing we do not always know what we want to cite,
        or just where subsequent material will end up.
        For example,
        we might want a citation to <xref provisional="some textbook about the FTC"/> or we might want to reference a later<nbsp/><xref provisional="chapter about DiffEq's, and an_underscore"/>.
      </p>
      <p>
        We can also embed
        <q>todo</q>
        s in the source by making an <init>XML</init> comment that begins with the four characters <c>todo</c>,
        and selectively display them,
        so you may not see the one here in the output you are looking at now.
        Or maybe you do see it?
      </p>
<!-- todo: Talk more about author's tools, such as todo's -->
      <p>
        Because a definite integral can be computed using an antiderivative,
        we have the following definition.
      </p>
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Notation may be associated with a definition,
or it can be placed standalone.  If associated
with a definition, future work will reflect that
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      <definition xml:id="definition-indefinite-integral">
        <idx>indefinite integral</idx>
        <idx><h>integral</h><h>indefinite integral</h></idx>
        <notation>
          <usage><m>\int\,f(x)\,dx</m></usage>
          <description>indefinite integral of <m>f(x)</m></description>
        </notation>
        <statement>
          <p>
            Suppose that <m>\frac{d}{dx}F(x)=f(x)</m>.
            Then the <term>indefinite integral</term>
            of <m>f(x)</m> is <m>F(x)</m> and is written as
            <me>
              \int\,f(x)\,dx=F(x)
            </me>.
          </p>
        </statement>
      </definition>
    </section>