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Finished edits to Chapter 10 #443

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80 changes: 38 additions & 42 deletions ptx/sec_par_calc.ptx
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Original file line number Diff line number Diff line change
Expand Up @@ -371,7 +371,7 @@
any line that passes through the center of a circle intersects the circle at right angles.
</p>

<figure xml:id="fig_parcalc2" vshift="1">
<figure xml:id="fig_parcalc2" vshift="3">
<caption>Illustrating how a circle's normal lines pass through its center</caption>
<!-- START figures/fig_parcalc2.tex -->
<image width="47%">
Expand Down Expand Up @@ -661,7 +661,7 @@
</md>
</p>

<figure xml:id="fig_parcalc4" vshift="0">
<figure xml:id="fig_parcalc4" vshift="1">
<caption>Graphing the parametric equations in <xref ref="ex_parcalc4"/> to demonstrate concavity</caption>
<!-- START figures/fig_parcalc4.tex -->
<image width="47%">
Expand Down Expand Up @@ -789,38 +789,12 @@
</md>.
</p>

<p>
The points of inflection are found by setting <m>\frac{d^2y}{dx^2}=0</m>.
This is not trivial,
as equations that mix polynomials and trigonometric functions generally do not have <q>nice</q> solutions.
</p>

<p>
In <xref ref="fig_parcalc5a"/> we see a plot of the second derivative.
It shows that it has zeros at approximately <m>t=0.5,\,3.5,\,6.5,\,9.5,\,12.5</m> and <m>16</m>.
These approximations are not very good,
made only by looking at the graph.
Newton's Method provides more accurate approximations.
Accurate to 2 decimal places, we have:
<md>
t=0.65,\,3.29,\,6.36,\,9.48,\,12.61\,\text{ and } \,15.74
</md>.
</p>

<p>
The corresponding points have been plotted on the graph of the parametric equations in <xref ref="fig_parcalc5b"/>.
Note how most occur near the <m>x</m>-axis,
but not exactly on the axis.
</p>

<figure xml:id="fig_parcalc5">
<caption>In (a), a graph of <m>\frac{d^2y}{dx^2}</m>, showing where it is approximately 0. In (b), graph of the parametric equations in <xref ref="ex_parcalc5"/> along with the points of inflection</caption>
<sidebyside widths="47% 47%" margins="0%" valign="bottom">
<figure xml:id="fig_parcalc5a">
<caption/>
<figure xml:id="fig_parcalc5a" vshift="3">
<caption>Graphing <m>\frac{d^2y}{dx^2}</m> in <xref ref="ex_parcalc5"/>, showing where it is approximately 0</caption>
<!-- START figures/fig_parcalc4.tex -->
<image width="47%">
<!-- START figures/fig_parcalc5b.tex -->
<image>
<shortdescription>Graph of the second derivative is a sinusoid with increasing amplitude.</shortdescription>
<shortdescription>Graph of the second derivative is a sinusoid with increasing amplitude.</shortdescription>
<description>
<p>
The image shows the graph <m>y = 2\cos(t)-4t\sin(t)</m>, which is a graph of <m>\frac{d^2y}{dx^2}</m>.
Expand Down Expand Up @@ -848,15 +822,40 @@
\end{tikzpicture}

</latex-image>
</image>
</image>
<!-- figures/fig_parcalc5b.tex END -->
</figure>

<figure xml:id="fig_parcalc5b">
<caption/>
<p>
The points of inflection are found by setting <m>\frac{d^2y}{dx^2}=0</m>.
This is not trivial,
as equations that mix polynomials and trigonometric functions generally do not have <q>nice</q> solutions.
</p>

<p>
In <xref ref="fig_parcalc5a"/> we see a plot of the second derivative.
It shows that it has zeros at approximately <m>t=0.5,\,3.5,\,6.5,\,9.5,\,12.5</m> and <m>16</m>.
These approximations are not very good,
made only by looking at the graph.
Newton's Method provides more accurate approximations.
Accurate to 2 decimal places, we have:
<md>
t=0.65,\,3.29,\,6.36,\,9.48,\,12.61\,\text{ and } \,15.74
</md>.
</p>

<p>
The corresponding points have been plotted on the graph of the parametric equations in <xref ref="fig_parcalc5b"/>.
Note how most occur near the <m>x</m>-axis,
but not exactly on the axis.
</p>

<figure xml:id="fig_parcalc5b" vshift="2">
<caption>A graph of the parametric equations in <xref ref="ex_parcalc5"/> along with the points of inflection</caption>
<!-- START figures/fig_parcalc4.tex -->
<image width="47%">
<!-- START figures/fig_parcalc5.tex -->
<image>
<shortdescription>Graph of the parametric curve in this example, with points of inflection marked.</shortdescription>
<shortdescription>Graph of the parametric curve in this example, with points of inflection marked.</shortdescription>
<description>
<p>
The graph shows a curve that appears to be sinusoidal, but with a frequency that increases with <m>x</m>.
Expand Down Expand Up @@ -892,9 +891,6 @@
</image>
<!-- figures/fig_parcalc5.tex END -->
</figure>
</sidebyside>

</figure>
</solution>
</example>

Expand Down Expand Up @@ -1281,7 +1277,7 @@
Find the surface area if this shape is rotated about the <m>x</m>-axis,
as shown in <xref ref="fig_parcalc8"/>.
</p>
<figure xml:id="fig_parcalc8" vshift="-1">
<figure xml:id="fig_parcalc8" vshift="3">
<caption>Rotating a teardrop shape about the <m>x</m>-axis in <xref ref="ex_parcalc8"/></caption>

<!-- START figures/figparcalc8_3D.asy -->
Expand Down
8 changes: 4 additions & 4 deletions ptx/sec_param_eqs.ptx
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Expand Up @@ -187,7 +187,7 @@
<caption>A table of values of the parametric equations in <xref ref="ex_pareq1"/> along with a sketch of their graph</caption>
<sidebyside valign="bottom" widths="47% 47%">
<figure xml:id="fig_pareq1a"><caption/>
<tabular>
<tabular halign="center">
<row>
<cell><m>t</m></cell>
<cell><m>x</m></cell>
Expand Down Expand Up @@ -306,7 +306,7 @@
<caption>A table of values of the parametric equations in <xref ref="ex_pareq2"/> along with a sketch of their graph</caption>
<sidebyside valign="bottom" widths="47% 47%">
<figure xml:id="fig_pareq2a"><caption/>
<tabular>
<tabular halign="center">
<row>
<cell><m>t</m></cell>
<cell><m>x</m></cell>
Expand Down Expand Up @@ -1072,7 +1072,7 @@
</md>
</p>

<figure xml:id="fig_pareq8" vshift="0">
<figure xml:id="fig_pareq8" vshift="2">
<caption>Graphing the parametric equations <m>x=4\cos(t) +3</m>, <m>y=2\sin(t) +1</m> in <xref ref="ex_pareq8"/></caption>
<!-- START figures/fig_pareq8.tex -->
<image width="47%">
Expand Down Expand Up @@ -1538,7 +1538,7 @@
illustrating the cusp at <m>(1,4)</m>.
</p>

<figure xml:id="fig_pareq9" vshift="0">
<figure xml:id="fig_pareq9" vshift="2">
<caption>Graphing the curve in <xref ref="ex_pareq9"/>; note it is not smooth at <m>(1,4)</m></caption>
<!-- START figures/fig_pareq9.tex -->
<image width="47%">
Expand Down
4 changes: 2 additions & 2 deletions ptx/sec_polar.ptx
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Expand Up @@ -48,7 +48,7 @@
<idx><h>coordinates</h><h>polar</h></idx>
</p>

<figure xml:id="fig_polar_intro1" vshift="0">
<figure xml:id="fig_polar_intro1" vshift="2">
<caption>Illustrating polar coordinates</caption>
<image width="33%">
<shortdescription>Illustration of polar coordinates relative to a pole and initial ray.</shortdescription>
Expand Down Expand Up @@ -706,7 +706,7 @@
This graph is also plotted in <xref ref="fig_polar3"/>.
</p>

<figure xml:id="fig_polar3" vshift="-1">
<figure xml:id="fig_polar3" vshift="3">
<caption>Plotting standard polar plots</caption>
<!-- START figures/fig_polar3.tex -->
<image width="47%">
Expand Down
8 changes: 4 additions & 4 deletions ptx/sec_polarcalc.ptx
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Expand Up @@ -126,7 +126,7 @@
</md>.
</p>

<figure xml:id="fig_polcalc1" vshift="0">
<figure xml:id="fig_polcalc1" vshift="3">
<caption>The limaçon in <xref ref="ex_polcalc1"/> with its tangent line at <m>\theta=\pi/4</m> and points of vertical and horizontal tangency</caption>
<!-- START figures/fig_polcalc1.tex -->
<image width="47%">
Expand Down Expand Up @@ -940,7 +940,7 @@
Find the area bounded between the polar curves <m>r=1</m> and <m>r=2\cos(2\theta)</m>,
as shown in <xref ref="fig_polcalc6a"/>.
</p>
<figure xml:id="fig_polcalc6a" vshift="-2">
<figure xml:id="fig_polcalc6a" vshift="0">
<caption>The region bounded by the functions in <xref ref="ex_polcalc6"/></caption>
<image width="47%">
<shortdescription>A zoomed in view of a region bounded by a circle, a rose curve, and the x axis.</shortdescription>
Expand Down Expand Up @@ -995,7 +995,7 @@
</md>.
</p>

<figure xml:id="fig_polcalc6b" vshift="-11.5">
<figure xml:id="fig_polcalc6b" vshift="-4.5">
<caption>Breaking the region bounded by the functions in <xref ref="ex_polcalc6"/> into its component parts</caption>
<image width="47%">
<shortdescription>A zoomed in view of a polar region, showing it divided into two parts.</shortdescription>
Expand Down Expand Up @@ -1157,7 +1157,7 @@
</md>.
</p>

<figure xml:id="fig_polcalc7" vshift="0">
<figure xml:id="fig_polcalc7" vshift="2">
<caption>The limaçon in <xref ref="ex_polcalc7"/> whose arc length is measured</caption>
<!-- START figures/fig_polcalc7.tex -->
<image width="47%">
Expand Down