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Merged
APEXCalculus merged 10 commits into from
May 12, 2025
Merged

Non-figure edits from PR #311 #335

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1092511
change two dl to ol
sean-fitzpatrick May 9, 2025
12e0f3a
Greg's edits - derivatives
sean-fitzpatrick May 9, 2025
3662f96
Greg's edits - differentials, concavity
sean-fitzpatrick May 9, 2025
8b0426e
other graph edits
sean-fitzpatrick May 9, 2025
4fbce6c
limit edits
sean-fitzpatrick May 9, 2025
34854d2
don't use smallcaps for intialization, etc
sean-fitzpatrick May 12, 2025
fb25c33
add xref
sean-fitzpatrick May 12, 2025
6386444
remaining edits unrelated to figure placement,etc
sean-fitzpatrick May 12, 2025
71d4793
Merge branch 'main' into may-2025-edits
sean-fitzpatrick May 12, 2025
f4978a0
make consistent use of parentheses for sin etc
sean-fitzpatrick May 12, 2025
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178 changes: 89 additions & 89 deletions ptx/appendix_back_reference.ptx
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2 changes: 1 addition & 1 deletion ptx/sec_ABC.ptx
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Expand Up @@ -1558,7 +1558,7 @@
<statement>
<p>
The functions <m>f(x) = \cos (x)</m> and
<m>g(x) = \sin x</m> intersect infinitely many times,
<m>g(x) = \sin(x)</m> intersect infinitely many times,
forming an infinite number of repeated, enclosed regions.
Find the areas of these regions.
</p>
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2 changes: 1 addition & 1 deletion ptx/sec_FTC.ptx
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Expand Up @@ -2548,7 +2548,7 @@
<webwork xml:id="webwork-ex-FTC-sin-period-explain">
<statement>
<p>
Explain why <m>\ds\int_{a}^{a+2\pi} \sin t\, dt = 0</m> for all values of <m>a</m>.
Explain why <m>\ds\int_{a}^{a+2\pi} \sin(t)\, dt = 0</m> for all values of <m>a</m>.
</p>
<p>
<var form="essay"/>
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14 changes: 7 additions & 7 deletions ptx/sec_Separable.ptx
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Expand Up @@ -424,7 +424,7 @@
<exercise label="ex-ode-separable-determine-3">
<statement>
<p>
<m>\displaystyle (y + 3)\yp + (\ln(x)) \yp - x\sin y = (y+3)\ln(x)</m>
<m>\displaystyle (y + 3)\yp + (\ln(x)) \yp - x\sin(y) = (y+3)\ln(x)</m>
</p>
</statement>
<answer>
Expand All @@ -436,13 +436,13 @@
<exercise label="ex-ode-separable-determine-4">
<statement>
<p>
<m>\displaystyle \yp -x^2\cos y + y = \cos y - x^2 y</m>
<m>\displaystyle \yp -x^2\cos(y) + y = \cos(y) - x^2 y</m>
</p>
</statement>
<answer>
<p>
Separable.
<m>\displaystyle \frac{1}{\cos y - y}\,dy = (x^2+1)\,dx</m>
<m>\displaystyle \frac{1}{\cos(y) - y}\,dy = (x^2+1)\,dx</m>
</p>
</answer>
</exercise>
Expand Down Expand Up @@ -562,13 +562,13 @@
<exercise label="ex-ode-separable-particular-sol-1">
<statement>
<p>
<m>\displaystyle \yp = \frac{\sin(x)}{\cos y}</m>,
<m>\displaystyle \yp = \frac{\sin(x)}{\cos(y)}</m>,
with <m>y(0) = \displaystyle \frac{\pi}{2}</m>
</p>
</statement>
<answer>
<p>
<m>\sin y + \cos(x) = 2</m>
<m>\sin(y) + \cos(x) = 2</m>
</p>
</answer>
</exercise>
Expand Down Expand Up @@ -636,12 +636,12 @@
<exercise label="ex-ode-separable-particular-sol-7">
<statement>
<p>
<m>\displaystyle \yp = (\cos^2x)(\cos^2 2y)</m>, with <m>y(0) = 0</m>
<m>\displaystyle \yp = (\cos^2(x))(\cos^2 (2y))</m>, with <m>y(0) = 0</m>
</p>
</statement>
<answer>
<p>
<m>2\tan 2y = 2x + \sin 2x</m>
<m>2\tan(2y) = 2x + \sin(2x)</m>
</p>
</answer>
</exercise>
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63 changes: 30 additions & 33 deletions ptx/sec_deriv_basic_rules.ptx
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Expand Up @@ -51,8 +51,8 @@
<title>Derivatives of Common Functions</title>
<statement>
<p>
<dl>
<li>
<ol cols="2">
<li xml:id="constant-derivative-rule">
<title>Constant Rule</title>

<idx><h>derivative</h><h>Constant Rule</h></idx>
Expand All @@ -75,24 +75,30 @@
</li>

<li>
<title>Other common functions</title>

<p>
<m>\lzoo{x}{\sin(x)} = \cos(x)</m>
</p>
</li>

<li>
<p>
<m>\lzoo{x}{\cos(x)} = {-\sin(x)}</m>
</p>
</li>

<li>
<p>
<m>\lzoo{x}{e^x} = e^x</m>
</p>
</li>

<li>
<p>
<m>\lzoo{x}{\ln(x)} = \frac{1}{x}</m>, for <m>x \gt 0</m>.
</p>
</li>
</dl>
</ol>

<idx><h>derivative</h><h>basic rules</h></idx>

Expand Down Expand Up @@ -293,14 +299,13 @@
<p>
Let <m>f</m> and <m>g</m> be differentiable on an open interval <m>I</m> and let <m>c</m> be a real number.
Then:
<dl>
<ol>
<li xml:id="sum-difference-derivative-rule">
<title>Sum/Difference Rule</title>
<p>
<md>
<mrow>\lzoo{x}{f(x) \pm g(x)} \amp= \lzoo{x}{f(x)} \pm \lzoo{x}{g(x)}</mrow>
<mrow>\amp= \fp(x)\pm g'(x)</mrow>
</md>
<me>
\lzoo{x}{f(x) \pm g(x)} \lzoo{x}{f(x)} \pm \lzoo{x}{g(x)} = \fp(x)\pm g'(x)
</me>

<idx><h>derivative</h><h>Sum/Difference Rule</h></idx>
<idx><h>Sum/Difference Rule</h><h>of derivatives</h></idx>
Expand All @@ -311,17 +316,16 @@
<li xml:id="constant-multiple-derivative-rule">
<title>Constant Multiple Rule</title>
<p>
<md>
<mrow>\lzoo{x}{c\cdot f(x)} \amp= c\cdot\lzoo{x}{f(x)}</mrow>
<mrow>\amp = c\cdot\fp(x)</mrow>
</md>.
<me>
\lzoo{x}{c\cdot f(x)} = c\cdot\lzoo{x}{f(x)} = c\cdot\fp(x)
</me>.

<idx><h>derivative</h><h>Constant Multiple Rule</h></idx>
<idx><h>Constant Multiple Rule</h><h>of derivatives</h></idx>

</p>
</li>
</dl>
</ol>
</p>
</statement>
</theorem>
Expand Down Expand Up @@ -521,21 +525,7 @@

<subsection>
<title>Higher Order Derivatives</title>
<aside xml:id="aside-derivative-second-order-notation" vshift="2">
<p>
<em>Note:</em> The second derivative notation could be written as
<me>
\frac{d^2y}{dx^2}=\frac{d^2y}{(dx)^2}=\frac{d^2}{(dx)^2}\big(y\big)
</me>.
</p>

<p>
That is, we take the derivative of <m>y</m> twice
(hence <m>d^2</m>),
both times with respect to <m>x</m> (hence <m>(dx)^2=dx^2</m>).
</p>
</aside>


<p>
The derivative of a function <m>f</m> is itself a function,
therefore we can take its derivative.
Expand Down Expand Up @@ -583,14 +573,23 @@
</definition>

<aside xml:id="aside-derivative-second-order-caveat" vshift="6">
<title>Higher Order Derivative Caveat</title>
<title>Higher Order Derivative Notes</title>
<p>
<xref ref="def_Higher_Deriv"/>
comes with the caveat <q>Where the corresponding limits exist.</q>
With <m>f</m> differentiable on <m>I</m>,
it is possible that <m>\fp</m> is <em>not</em>
differentiable on all of <m>I</m>, and so on.
</p>

<p>
Also, the second derivative notation could be written as
<me>
\frac{d^2y}{dx^2} = \frac{d^2y}{(dx)^2} = \frac{d^2}{(dx)^2}\bigl(y\bigr)
</me>.
That is, we take the derivative of <m>y</m> twice (hence <m>d^2</m>),
both times with respect to <m>x</m> (hence <m>(dx)^2 = dx^2</m>).
</p>
</aside>

<figure xml:id="vid_deriv_basic_rules_higher_order_deriv" component="video" vshift="0">
Expand Down Expand Up @@ -669,9 +668,7 @@
What do higher order derivatives <em>mean</em>?
What is the practical interpretation?

<!-- TODO: is ! the right markup here? -->

<idx><h>derivative</h><h>higher order!interpretation</h></idx>
<idx><h>derivative</h><h>higher order</h><h>interpretation</h></idx>
</p>

<p>
Expand Down
162 changes: 79 additions & 83 deletions ptx/sec_deriv_chainrule.ptx
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Expand Up @@ -166,7 +166,7 @@

<p>
The statement of <xref ref="thm_chain_rule"/> takes care to ensure
this problem does not arise, but our focus is more on the derivative result than
this problem does not arise. We will focus more on the derivative result than
on the domain/range conditions.
</p>
</aside>
Expand Down Expand Up @@ -1048,90 +1048,86 @@
That is, the rate at which the <m>u</m> gear makes a revolution is twice as fast as the rate at which the <m>x</m> gear makes a revolution.
</p>

<sidebyside widths="47% 47%" margins="0%">

<stack><!-- Old paragraphs title: -->
<p>
Using the terminology of calculus,
the rate of <m>u</m>-change, with respect to <m>x</m>,
is <m>\lz{u}{x} = 2</m>.
</p>

<p>
Likewise, every revolution of <m>u</m> causes <m>3</m> revolutions of <m>y</m>:
<m>\lz{y}{u} = 3</m>.
How does <m>y</m> change with respect to <m>x</m>?
For each revolution of <m>x</m>,
<m>y</m> revolves <m>6</m> times; that is,
<me>
\frac{dy}{dx} = \frac{dy}{du}\cdot \frac{du}{dx} = 2\cdot 3 = 6
</me>.
</p>

<p>
We can then extend the <xref ref="thm_chain_rule" text="title"/> with more variables by adding more gears to the picture.
</p>
</stack>

<figure xml:id="fig_chainrulegears" vshift="0">
<caption>A series of gears to demonstrate the Chain Rule. Note how <m>\lz{y}{x} = \lz{y}{u}\cdot\lz{u}{x}</m></caption>
<!-- START figures/fig_chainrule_gears.tex -->
<image>
<shortdescription>
3 gears of various sizes demonstrating the chain rule.
</shortdescription>
<description>
Three gears, connected in the order <m>x,u,y</m>.
<m>x</m> is the largest gear, having 36 teeth. It is rotating counter-clockwise.
<m>u</m> is connected to <m>x</m>, and it has 18 teeth. To the left of the connection is <m>\frac{du}{dx} = 2</m>.
<m>y</m> is connected to <m>u</m>, and it has 6 teeth. Below the connection is <m>\frac{dy}{du}=3</m>.
To the right of the gears is the expression <m>\frac{dy}{dx} = 6</m>.
</description>
<latex-image label="img_chainrulegears">

\begin{tikzpicture}[&gt;=latex]

\begin{scope}[shift={(0,-200pt)}]
\begin{scope}
\foreach \x in {0,1,2,...,35}
{%
\draw [rotate around={{\x*10}:(0,0)}] (60pt,0)--(65pt,0) arc (0:{4.}:65pt);
\draw [rotate around={{\x*10+4.}:(0,0)}] (65pt,0) -- (60pt,0) arc (0:6:60pt);
}
\draw [-&gt;] (40pt,0) arc (0:170:40pt);
\draw (0,0) node {$x$};
\end{scope}

\begin{scope}[shift={(4.5pt,-99pt)}]
\foreach \x in {0,1,2,...,17}
{%
\draw [rotate around={{\x*20}:(0,0)}] (30pt,0)--(35pt,0) arc (0:{9}:35pt);
\draw [rotate around={{\x*20+9}:(0,0)}] (35pt,0) -- (30pt,0) arc (0:11:30pt);
}
\draw [-&gt;] (0,25pt) arc (90:-80:25pt);
\draw (0,0) node {$u$};
\draw (45pt,-30pt) node {\small $\ds \frac{dy}{du} = 3$};
\draw (-50pt,30pt) node {\small $\ds \frac{du}{dx} = 2$};
\draw (60pt,40pt) node {\small $\ds \frac{dy}{dx} = 6$};
\end{scope}

\begin{scope}[shift={(53.5pt,-100pt)}]
\foreach \x in {0,1,2,...,5}
{%
\draw [rotate around={{\x*60}:(0,0)}] (10pt,0)--(15pt,0) arc (0:{29}:15pt);
\draw [rotate around={{\x*60+29}:(0,0)}] (15pt,0) -- (10pt,0) arc (0:31:10pt);
}
\draw [-&gt;] (0,-20pt) arc (-90:70:20pt);
\draw (0,0) node {$y$};
\end{scope}
\end{scope}
\end{tikzpicture}

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

<p>
Using the terminology of calculus,
the rate of <m>u</m>-change, with respect to <m>x</m>,
is <m>\lz{u}{x} = 2</m>.
</p>

<figure xml:id="fig_chainrulegears">
<caption>A series of gears to demonstrate the Chain Rule. Note how <m>\lz{y}{x} = \lz{y}{u}\cdot\lz{u}{x}</m></caption>
<!-- START figures/fig_chainrule_gears.tex -->
<image>
<shortdescription>
3 gears of various sizes demonstrating the chain rule.
</shortdescription>
<description>
Three gears, connected in the order <m>x,u,y</m>.
<m>x</m> is the largest gear, having 36 teeth. It is rotating counter-clockwise.
<m>u</m> is connected to <m>x</m>, and it has 18 teeth. To the left of the connection is <m>\frac{du}{dx} = 2</m>.
<m>y</m> is connected to <m>u</m>, and it has 6 teeth. Below the connection is <m>\frac{dy}{du}=3</m>.
To the right of the gears is the expression <m>\frac{dy}{dx} = 6</m>.
</description>
<latex-image label="img_chainrulegears">

\begin{tikzpicture}[&gt;=latex]

\begin{scope}[shift={(0,-200pt)}]
\begin{scope}
\foreach \x in {0,1,2,...,35}
{%
\draw [rotate around={{\x*10}:(0,0)}] (60pt,0)--(65pt,0) arc (0:{4.}:65pt);
\draw [rotate around={{\x*10+4.}:(0,0)}] (65pt,0) -- (60pt,0) arc (0:6:60pt);
}
\draw [-&gt;] (40pt,0) arc (0:170:40pt);
\draw (0,0) node {$x$};
\end{scope}

\begin{scope}[shift={(4.5pt,-99pt)}]
\foreach \x in {0,1,2,...,17}
{%
\draw [rotate around={{\x*20}:(0,0)}] (30pt,0)--(35pt,0) arc (0:{9}:35pt);
\draw [rotate around={{\x*20+9}:(0,0)}] (35pt,0) -- (30pt,0) arc (0:11:30pt);
}
\draw [-&gt;] (0,25pt) arc (90:-80:25pt);
\draw (0,0) node {$u$};
\draw (45pt,-30pt) node {\small $\ds \frac{dy}{du} = 3$};
\draw (-50pt,30pt) node {\small $\ds \frac{du}{dx} = 2$};
\draw (60pt,40pt) node {\small $\ds \frac{dy}{dx} = 6$};
\end{scope}

\begin{scope}[shift={(53.5pt,-100pt)}]
\foreach \x in {0,1,2,...,5}
{%
\draw [rotate around={{\x*60}:(0,0)}] (10pt,0)--(15pt,0) arc (0:{29}:15pt);
\draw [rotate around={{\x*60+29}:(0,0)}] (15pt,0) -- (10pt,0) arc (0:31:10pt);
}
\draw [-&gt;] (0,-20pt) arc (-90:70:20pt);
\draw (0,0) node {$y$};
\end{scope}
\end{scope}
\end{tikzpicture}
<p>
Likewise, every revolution of <m>u</m> causes <m>3</m> revolutions of <m>y</m>:
<m>\lz{y}{u} = 3</m>.
How does <m>y</m> change with respect to <m>x</m>?
For each revolution of <m>x</m>,
<m>y</m> revolves <m>6</m> times; that is,
<me>
\frac{dy}{dx} = \frac{dy}{du}\cdot \frac{du}{dx} = 2\cdot 3 = 6
</me>.
</p>

</latex-image>
</image>
<!-- figures/fig_chainrule_gears.tex END -->
</figure>
</sidebyside>
<p>
We can then extend the <xref ref="thm_chain_rule" text="title"/> with more variables by adding more gears to the picture.
</p>

<p>
It is difficult to overstate the importance of the <xref ref="thm_chain_rule" text="title"/>.
Expand Down
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