Derivatives and Integrals: An Annotated Discourse

<section xml:id="section-sage-cells" label="section-sage-cells">
    <title>Computing Integrals with Sage (<m>\int</m>)</title>


    <plaintitle>Computing Integrals with Sage (∫)</plaintitle>

    <idx><h>Sage</h><h>integration</h></idx>
    <idx><h>Sage</h><h>integration</h><h>cell</h></idx>
    <idx><h>Sage</h><h>integration</h><h>numerical</h></idx>
    <idx><h>numerical</h><see>Sage integration</see></idx>
    <idx><h>numerical</h><see>Sage</see></idx>

    <idx>numerics</idx>
    <idx><h>numerics</h><seealso>Sage</seealso></idx>
    <idx><h>numerical integration</h><h>Sage</h><h>cell</h><see>Sage</see></idx>

    <idx><h>A</h><see>F</see></idx>
    <idx><h>A</h><see>G</see></idx>
    <idx><h>A</h><see>H</see></idx>
    <idx><h>A</h><h>B</h></idx>
    <idx><h>A</h><h>B</h><h>C</h></idx>
    <idx><h>A</h><h>B</h><h>C</h><seealso>D</seealso></idx>
    <idx><h>A</h><h>B</h><h>C</h><seealso>A</seealso></idx>


    <idx><h>X</h></idx>
    <idx><h>X</h><seealso>F</seealso></idx>
    <idx><h>X</h><seealso>G</seealso></idx>
    <idx><h>X</h><seealso>H</seealso></idx>
    <idx><h>X</h><h>Y</h></idx>
    <idx><h>X</h><h>Y</h><h>Z</h><see>D</see></idx>
    <idx><h>X</h><h>Y</h><h>Z</h><see>A</see></idx>


    <idx>mixed-content <em>emphasized</em></idx>
    <idx><h>structured-content <em>emphasized</em></h></idx>
    <idx><h sortby="cat">sorted as if
    <q>Cat</q>
    </h></idx>
    <idx sortby="quorum">sorted as if
    <q>Quorum</q>
    </idx>
    <idx><h>units</h><h sortby="A">Z (
    <alert>sort as</alert>
    A)</h></idx>
    <idx><h>units</h><h sortby="Z">A (
    <alert>sort as</alert>
    Z)</h></idx>
    <idx><c>verbatim text</c>, use sortby</idx>

    <idx sortby="R"><m>\rho</m><ndash />fibers</idx>




    <idx>Cayley graph</idx>
    <idx>cayley graph</idx>
    <idx>CAYLEY GRAPH</idx>



    <idx><h sortby="x"><m>x</m> as a variable</h></idx>
    <p>
      Sage can compute definite integrals.
      The output contains the approximate numerical value of the definite integral,
      followed by an upper bound of the error in the approximation.
    </p>
<sage>
<input>numerical_integral(sin(x)^2, (0, 2))</input>

<output>(1.189200623826982, 1.320277913471315e-14)</output>
</sage>
    <p>
      Given the Fundamental Theorem,
      we would find the antiderivative useful.
          <idx>Cayley graph</idx>
          <idx><h sortby="x"><m>x</m> as a variable</h></idx>
    </p>
<sage>
<input>integral(sin(x)^2, x)</input>
<output>1/2*x - 1/4*sin(2*x)</output>
</sage>
    <p>
      The same command can be used to employ the antiderivative in the application of the Fundamental Theorem.
      Notice that the answer is <em>exact</em>
      and any further manipulation is likely to be simply producing a numerical approximation.
          <idx>cayley graph</idx>
    </p>

<sage type="full">
<input>integral(sin(x)^2, (x, 0, 2))</input>
<output>-1/4*sin(4) + 1</output>
</sage>
    <p>
      There are integrals you really do not want to evaluate,
      or you do not want your reader to evaluate.
      A Sage cell can be configured for display purposes only<mdash />you can look but you cannot touch.
          <idx>CAYLEY GRAPH</idx>
    </p>
<sage type="display">
<input>integral(e^(x^2), x)</input>
<output>-1/2*I*sqrt(pi)*erf(I*x)</output>
</sage>
    <p>
      You can give a Sage element a <c>doctest</c>
          <idx>doctest</idx>
          <idx><h>attributes</h><h>doctest</h></idx>
      attribute, whose value mirrors the optional hash tags used in Sage doctests.
      Possible values are <c>random</c>,
      <c>long time</c>,
      <c>not implemented</c>,
      <c>not tested</c>,
      <c>known bug</c>,
      <c>absolute</c>,
      <c>relative</c>,
      and <c>optional</c>.
      The values <c>absolute</c> and <c>relative</c> refer to floating-point tolerances for equality and require a second attribute <c>tolerance</c> to specify a floating point value.
      The value <c>optional</c> refers to the test requiring that an optional Sage package be present.
      The name of that package should be provided in the <c>package</c> attribute.
    </p>

    <p>
      The next cell is marked in the source as <c>doctest="random"</c>,
      and so is specified as unpredictable and not tested.
      But there is some
      <q>sample</q>
      output which will appear in the <latex /> version
      (and always be the same).
          <idx>c</idx>
          <idx><h sortby="c">A bug test, sorted as
      <q>c</q>
      </h><h>b</h><h>c one</h></idx> <idx><h sortby="c">A bug test,
      sorted as
      <q>c</q>
      </h><h>x</h><h>c two</h></idx> <idx><h sortby="c">A bug test,
      sorted as
      <q>c</q>
      </h><h>b</h><h>c three</h></idx>
    </p>
<sage doctest="random">
<input>random()</input>
<output>0.11736021338650582</output>
</sage>
    <p>
      While the next cell is random,
      the returned value will never be more than <m>0.01</m> away from <m>12</m>,
      since the <c>random()</c> function stays between <m>0</m> and <m>1</m>.
      So we provide <m>12.005</m> as the expected answer,
      but test with an absolute tolerance of <m>\epsilon=0.006</m>.
    </p>
<sage doctest="absolute" tolerance="0.006">
<input>12 + 0.01*random()</input>
<output>12.005</output>
</sage>
    <p>
      Sage has some functions which affect output,
      generally making mathematics look more like mathematics via <latex /> syntax.
      This is a simple test,
      and you should see the variable and superscript in italics,
      properly formatted as output when viewed within HTML output.
      We have provided expected output for doctesting,
      but it is sort of silly to have this as part of <latex /> output,
      even if it is instructive.
    </p>
<sage>
<input>pretty_print(html("$a^2$"))</input>
<output>&lt;script type="math/tex"&gt;a^2&lt;/script&gt;</output>
</sage>

    <p>
      Sage, and by extension, the Sage Cell Server,
      can interpret several languages.
      The next example has code in the <c>R</c> language,
          <idx><h>R</h></idx>
      a popular open source language for statistics.
      As an author,
      you add the attribute <c>language="r"</c> to your <c>sage</c> element.
      (The default language is Sage,
      so you do not need to indicate that repeatedly.)
      Note that a language like <c>R</c> likes to use a
      <q>less than</q>
      character, &lt;, special character in XML. You need to escape it by writing <c>&amp;lt;</c> as we have done in the source for this example.
      (See the discussion in <xref ref="subsection-xml-escape" text="type-global" />.)
    </p>
    <p>
      As a reader you learn that the
      <q>Evaluate</q>
      button for a pre-loaded Sage cell will indicate the language in use.
    </p>
<sage doctest="not tested" language="r">
<input>
ruth &lt;- c(22, 25, 34, 35, 41, 41, 46, 46, 46, 47, 49, 54, 54, 59, 60)
bonds &lt;- c(16, 25, 24, 19, 33, 25, 34, 46, 37, 33, 42, 40, 37, 34, 49, 73, 46, 45, 45, 5, 26, 28)
dimaggio &lt;- c(12, 14, 20, 21, 25, 29, 30, 30, 31, 32, 32, 39, 46)
summary(ruth)
summary(bonds)
summary(dimaggio)
boxplot(ruth, bonds, dimaggio)
</input>
</sage>
    <p>
      The Sage Cell Server supports the following languages:
      <c>sage</c>,
      <c>gap</c>,
      <c>gp</c>, <c>html</c>,
      <c>maxima</c>,
      <c>octave</c>,
      <c>python</c>,
      <c>r</c>,
      and <c>singular</c>.
    </p>
    <p>
      Here is another <c>R</c> cell.
      Unfortunately,
      it seems Sage's <c>doctest</c> facility cannot be used easily with code from other languages.
      In the source for this example,
      we have employed the <init>XML</init> escape sequence,
      <c>&amp;lt;</c> several times
      (see <xref ref="subsection-xml-escape" text="type-global" />).
    </p>
<sage language="r">
<input>
age &lt;- c(25, 30, 56)
gender &lt;- c("male", "female", "male")
weight &lt;- c(160, 110, 220)
mydata &lt;- data.frame(age,gender,weight)
summary(mydata)
cor(mydata$age,mydata$weight)
mean(mydata$age)
sd(mydata$age)
plot(mydata$age,mydata$weight)
</input>
</sage>

    <p>
      The Sage Cell server imports a few important R packages.
      As of 2022-06-04 these are <c>deSolve</c>,
      <c>ggplot2</c>,
      <c>pracma</c>,
      <c>survey</c>,
      <c>swirl</c>,
      and <c>tidyverse</c>.
      This next example uses the <c>ggplot</c> library for both a data set and the plotting capabilities.
      Note the initial use of the <c>library()</c> function.
      This is a modified version of the <articletitle>Bubble plot</articletitle> example at
      <url href="http://r-statistics.co/Top50-Ggplot2-Visualizations-MasterList-R-Code.html">Top 50 <c>ggplot2</c> Visualizations <mdash />The Master List</url>.
    </p>
<sage language="r">
<input>
# load package and data
library(ggplot2)
data(mpg, package="ggplot2")
mpg_select &lt;- mpg[mpg$manufacturer %in% c("audi", "ford", "honda", "hyundai"), ]
# Scatterplot
g &lt;- ggplot(mpg_select, aes(displ, cty)) +
labs(subtitle="mpg: Displacement vs City Mileage",
title="Bubble chart")
g + geom_jitter(aes(col=manufacturer, size=hwy)) +
geom_smooth(aes(col=manufacturer), method="lm", se=F)
</input>
</sage>
    <p>
      Here is a blank Sage cell that you may use for practice and experimentation with the commands above.
      Note that this cell allows a choice of languages,
      and is not linked with any of the previous cells,
      so a reader would need to start fresh,
      or cut/paste definitions from other cells.
    </p>
    <sage type="practice" />
    <p>
      On the other hand a <tag>sage</tag> element with no content will also create an empty Sage cell for the reader's use,
      but now it will be specific to a particular language and linked to others of the same language.
      Run the <c>R</c> cell above that defines the variable <c>ruth</c> and then try typing <c>summary(ruth)</c> in the cell below.
    </p>
    <sage language="r" />
    <p>
      You can make Sage blocks which are of <c>type="invisible"</c>,
      which will never be shown to a reader,
      but which get doctested.
      Why do this?
      If some code produces an error,
      and you hope it is fixed someday,
      use an invisible block to raise the error.
      Once fixed, the doctest will fail,
      and you can adjust your commentary to suit.
      There is such a block right now,
      <em>but</em> you will need to go to the source to see it.
    </p>

<sage type="invisible">
<input>graphs(augment='vertices').next()</input>
<output>
Traceback (most recent call last):
...
NotImplementedError
</output>
</sage>
    <p>
      Our maximum width for text, designed for readability,
      suggests you should format your Sage code with a maximum of about 54 characters.
      On a mobile device,
      the number of displayed characters might be as low as 28 in portrait orientation,
      and again around 50 in landscape.
      You can use a variety of techniques to shorten long lines,
      such as using intermediate variables.
      Since Sage is just a huge Python library,
      you can use any of Python's facilities for handling long lines.
      These include a continuation character
      (which I try to avoid using)
      or natural places where you can break long lines,
      such as between entries of a list.
      Also, if writing loops or functions,
      you may wish to have your indentation be only two characters wide
      (rather than, say, four).
    </p>
    <p>
      Sage output can sometimes be quite long,
      though this has improved with some changes in Sage's output routines.
      Output code should be included primarily for doctesting purposes,
      and in this use,
      you may break at almost whitespace character and the doctesting framework will adjust accordingly.
      You may wish to show sample output in a static format,
      like a PDF, so you can consider formatting your output to fit the width constraints of that medium.
      Or you may even adjust exactly what is output,
      to keep it from being too verbose.
      Sage doctesting also allows for a wild-card style syntax which allows you to skip over huge chunks of meaningless or unpredictable output,
      such as tracebacks on error messages.
    </p>
    <p>
      This paragraph is just a placeholder.
      It has handful of index entries,
      all starting with the letters
      <q>gas</q>, taken from
      <pubtitle>Indexing for Editors and Authors: A Practical Guide to Understanding Indexes</pubtitle>
      by Leise, Mertes, and Badgett.
      The intent is to test letter-by-letter versus word-by-word sorting of index entries.
      We use a word-by-word order, resulting in:
      <ol>
        <li>gas</li>
        <li>gas masks</li>
        <li>gas production</li>
        <li>gas works</li>
        <li>gasoline</li>
        <li>gastritis</li>
      </ol>
          <idx>gasoline</idx>
          <idx>gas works</idx>
          <idx>gas</idx>
          <idx>gas masks</idx>
          <idx>gas production</idx>
          <idx>gastritis</idx>
    </p>
    <paragraphs>
      <title>Titled Sage Cells</title>
      <idx><h>Sage cell</h><h>with a title</h></idx>
<sage>
<input>integral(sin(x)^2, x)</input>
<output>1/2*x - 1/4*sin(2*x)</output>
</sage>
      <p>
        You can place Sage cells inside of a <c>paragraphs</c> if you want to give them a title,
        but no numbers, <etc />.
      </p>
    </paragraphs>
  </section>