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>

  <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>


  <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>

  <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>

  <p>
    Because a definite integral can be computed using an antiderivative,
    we have the following definition.
  </p>

  <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>