Derivatives and Integrals: An Annotated Discourse

<section label="section-facts-figures">
  <title>Some Facts and Figures</title>
  <idx>footnotes</idx>

  <p>
    Because of the Fundamental Theorem<fn>
    First test footnote
    </fn>,
    for every derivative we know,
    there is an antiderivative we might find useful.
    Because of the Fundamental Theorem of Calculus<fn>
    Second test footnote
    </fn>,
    we recycle the
    <q><m>\int</m></q>
    symbol as notation for an antiderivative.
    <ul cols="2">
      <li>
        <p>
          Derivatives
          <ol marker="(a)">
            <li><m>\frac{d}{dx}x^n = nx^{n-1}</m></li>
            <li><m>\frac{d}{dx}e^x = e^x</m></li>
            <li><m>\frac{d}{dx}\cos(x) = -\sin(x)</m></li>
          </ol>
        </p>
      </li>
      <li>
        <p>
          Antiderivatives
          <ol marker="i)">
            <li>
              <m>\indefiniteintegral{x^n}{x} = \displaystyle\frac{x^{n-1}}{n+1}\text{ if }n\neq -1</m>
            </li>
            <li><m>\indefiniteintegral{e^x}{x} = e^x</m></li>
            <li><m>\indefiniteintegral{\sin(x)}{x} = -\cos(x)</m></li>
          </ol>
        </p>
      </li>
    </ul>
  </p>
  <remark>
    <p>
      You can<fn>
      Third test footnote
      </fn> gain a greater understanding of derivatives by studying the graphs of functions with their derivatives.
      Can<fn>
      Fourth test footnote
      </fn> you discern the derivative<ndash />antiderivative<fn>
      Fifth test footnote
      </fn> relationship in Figure<nbsp /><xref ref="figure-function-derivative" />?
    </p>
  </remark>

  <figure xml:id="figure-function-derivative">
    <caption>A function and its derivative</caption>
    <image width="50%" source="images/cubic-function.png">
      <shortdescription>
      a third degree polynomial with a local max and a local min;
      its derivative is plotted on the same axes
      </shortdescription>
    </image>
  </figure>
  <p>
    Lists<fn>
    Sixth test footnote
    </fn> can have multiple columns.
    With <init>HTML</init> items displayed in row-major order
    (horizontally first)
    and<fn>
    Seventh test footnote
    </fn> with <latex /> items are displayed in column-major order
    (vertically first).
    When one order, or the other,
    becomes workable in both variants,
    maybe we will be consistent in presentation.
    (Note that with just one row, it makes no difference.)
    We used it above for the two items<mdash />derivatives and integrals<mdash />where each item was a list of its own.
    Here are two more examples, one with short snippets and lots of columns,
    the other with lots of text in paragraphs.<idx xml:id="index-start-multicolumn" finish="index-finish-multicolumn"><h>list</h><h>multicolumn</h></idx>
    <ol cols="5">
      <li>
        <p>
          Red
        </p>
      </li>
      <li>
        <p>
          Blue
        </p>
      </li>
      <li>
        <p>
          Green
        </p>
      </li>
      <li>
        <p>
          Purple
        </p>
      </li>
      <li>
        <p>
          Yellow
        </p>
      </li>
      <li>
        <p>
          Black
        </p>
      </li>
      <li>
        <p>
          Orange
        </p>
      </li>
      <li>
        <p>
          Pink
        </p>
      </li>
      <li>
        <p>
          Salmon
            <idx><h>strange colors</h></idx>
        </p>
      </li>
      <li>
        <p>
          Aqua
        </p>
      </li>
      <li>
        <p>
          Cyan
        </p>
      </li>
      <li>
        <p>
          Puce
            <idx><h>strange colors</h></idx>
        </p>
      </li>
    </ol>
    <ul cols="2">
      <li>
        <p>
          Lorem ipsum dolor sit amet, consectetur adipiscing elit.
          Proin lorem diam, convallis in nulla sed, accumsan fermentum urna.
          Pellentesque aliquet leo elit,
          ut consequat nunc dapibus ac.
          Sed lobortis leo tincidunt, vulputate nunc at, ultricies leo.
          Vivamus purus diam,
          tristique laoreet purus eget, mollis gravida sapien.
          Nunc vulputate nisl ac mauris hendrerit cursus.
          Sed vel molestie velit.
          Suspendisse sem sem, elementum at vehicula id, volutpat ac mi.
          Nullam ullamcorper fringilla purus in accumsan.
          Mauris at nunc accumsan orci dictum vulputate id id augue.
          Suspendisse at dignissim elit, non euismod nunc.
          Aliquam faucibus magna ac molestie semper.
          Aliquam hendrerit sem sit amet metus congue tempor.
          Donec laoreet laoreet metus, id interdum purus mattis vulputate.
          Proin condimentum vitae erat varius mollis.
          Donec venenatis libero sed turpis pretium tempor.
        </p>
        <p>
          Praesent rutrum scelerisque felis sit amet adipiscing.
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          eget elementum neque viverra.
          Integer sagittis dictum turpis vel aliquet.
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          nibh dui vulputate leo, nec tincidunt augue ipsum sed odio.
          Nunc non erat sollicitudin, iaculis eros consequat, dapibus eros.
        </p>
      </li>
      <li>
        <p>
          Donec vestibulum auctor nisl.
          Nullam placerat interdum dui.
          Quisque lobortis scelerisque augue imperdiet placerat.
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          laoreet urna a, eleifend enim.
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          egestas velit nec, tristique sem.
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          erat nisi consectetur turpis,
          id varius dolor lorem vitae mauris.
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          Cum sociis natoque penatibus et magnis dis parturient montes,
          nascetur ridiculus mus.
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          eu dignissim est consequat sed.
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          Proin in leo in lorem porttitor varius.
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          In imperdiet blandit congue.
        </p>
      </li>
      <li>
        <p>
          Ut nec sem vitae ipsum interdum vestibulum sit amet sed velit.
          Aliquam tempor nibh vitae augue pulvinar,
          at ultricies urna commodo.
          Donec in porta lectus, ac sagittis felis.
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          In lobortis lacus vel ornare vehicula.
          Duis aliquet, ligula semper sodales adipiscing,
          augue nibh ornare ante, quis pulvinar justo mi eget mi.
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          Duis dignissim magna quis velit mattis,
          in cursus lectus vehicula.
          Morbi quis tempus felis, ut gravida nisi.
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          Vivamus eu commodo est, pretium fringilla dolor.
          Curabitur vel sollicitudin libero.
          Integer sit amet auctor felis.
          Maecenas sagittis erat at ante feugiat,
          in tincidunt ligula pretium.
          Integer eget auctor ipsum, quis volutpat felis.
          Morbi id dignissim eros.
          Suspendisse aliquet pulvinar lorem gravida egestas.
          Cum sociis natoque penatibus et magnis dis parturient montes,
          nascetur ridiculus mus.
          Praesent nec massa dui.
          Suspendisse convallis lacus sit amet adipiscing varius.
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          in condimentum metus pharetra.
          Curabitur sem dolor,
          auctor vitae sagittis vestibulum, posuere imperdiet metus.
          Etiam pretium lacus urna, vel auctor diam tincidunt non.
          Etiam viverra sodales iaculis.
        </p>
      </li>
      <li>
        <p>
          Sed varius leo urna.
          Phasellus tempus mollis ultricies.
          Curabitur non neque aliquet, facilisis tortor in, sodales dui.
          Donec hendrerit ultricies nulla mollis rhoncus.
          In vel lobortis est.
          Vestibulum consectetur lacus vel sem dignissim vestibulum.
          Etiam sed elementum ligula, vel congue turpis.
          Morbi nec diam mattis, venenatis eros et, elementum tellus.
          Integer sed orci ornare, elementum elit id, lacinia augue.
          Vestibulum ante ipsum primis in faucibus orci luctus et ultrices posuere cubilia Curae; In et libero id turpis pharetra faucibus.
          Integer consequat dignissim semper.
          Donec pretium magna at ullamcorper ultricies.
          Nam quis suscipit elit.
          Donec cursus tellus et venenatis feugiat.
          Mauris dictum molestie leo, vitae aliquet metus luctus vitae.
        </p>
        <p>
          Ut id iaculis leo.
          Sed nec vestibulum mi.
          Mauris est mauris, porta in nulla eget, bibendum luctus nisl.
          Praesent et posuere felis, molestie vehicula velit.
          Nulla a nunc venenatis, aliquam orci nec, congue felis.
          Vestibulum a dolor nisi.
          Morbi sed nisi nulla.
          Nam iaculis felis a enim blandit, at venenatis diam congue.
          Nulla augue diam, egestas eget fermentum nec, posuere eget risus.
          Praesent egestas nulla eros, eget accumsan augue euismod vel.
          Pellentesque pellentesque non erat vitae posuere.
          Curabitur lacus arcu, varius sed risus ut,
          ullamcorper tincidunt lorem.
          Sed et lacus dignissim, tincidunt nisl ac, porttitor sapien.
        </p>
      </li>
    </ul>
  </p>

</section>