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Merged
StevenClontz merged 9 commits into from
Nov 13, 2024
Merged

added sage code in comments under each figure; svgs inside PR comments #393

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32f90e6
added sage code in comments under each figure; svgs inside PR comments
cg2wilson Oct 30, 2024
e4ff0bf
uncommented sage, per discussion in meeting today
cg2wilson Nov 1, 2024
440a196
added LT2 and LT3 images
cg2wilson Nov 3, 2024
bb78208
add sage images for LT4 and LT5
cg2wilson Nov 8, 2024
2f793c3
add sage imgs for LT6
cg2wilson Nov 8, 2024
72e8961
Merge remote-tracking branch 'origin/main' into cg2wilson-LT-sageimgs
StevenClontz Nov 13, 2024
b06c655
give images same aspect ratio
StevenClontz Nov 13, 2024
ad78257
LT3 aspect ratio
StevenClontz Nov 13, 2024
8665243
remove unuseful alt text
StevenClontz Nov 13, 2024
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73 changes: 63 additions & 10 deletions source/calculus/source/01-LT/01.ptx
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Expand Up @@ -17,10 +17,18 @@
In <xref ref="limit_censored" /> the graph of a function is given, but something is wrong. The graphic card failed and one portion did not render properly. We can’t see what is happening in the neighborhood of <m>x=2</m>.
</p>
<figure xml:id="limit_censored">
<image width="50%" source="limit_censored.png" />
<image width="50%">
<sageplot>
x = var('x')
f(x) = -x^3+5*x^2-5.5*x
p = plot(f,(x,-1,3.5),ymin=-2,ymax=3, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
q = line([(2,0.1),(2,3)],thickness=10,color='black')
sh = p+q
sh
</sageplot>
</image>
<caption>A graph of a function that has not been rendered properly.</caption>
</figure>

</introduction>
<task>
<p>
Expand Down Expand Up @@ -53,10 +61,18 @@
In <xref ref="limit_uncensored" /> the graphic card is working again and we can see more clearly what is happening in the neighborhood of <m>x=2</m>.
</p>
<figure xml:id="limit_uncensored">
<image width="50%" source="limit_uncensored.png" />
<image width="50%">
<sageplot>
x = var('x')
f(x) = -x^3+5*x^2-5.5*x
p = plot(f,(x,-1,3.5),ymin=-2,ymax=3, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
q = circle((2,1), .1,thickness=2, fill=True, facecolor="white")
sh=p+q
sh
</sageplot>
</image>
<caption>A graph of a function that has rendered properly</caption>
</figure>

</introduction>
<task>
<p>
Expand Down Expand Up @@ -130,10 +146,23 @@

</p>
<figure xml:id="limit-left-and-right">
<image width="50%" source="limit-left-and-right.png" />
<image width="50%">
<sageplot>
x = var('x')
f(x) = piecewise([((-3,-0.05),-1),((0.05,3),1)])
p = plot(f,(x,-3,3),ymin=-2,ymax=2, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2)
q = circle((0,-1), .1,thickness=1, fill=True, facecolor="white")
r = circle((0,1), .1,thickness=1, fill=True, facecolor="white")
s = circle((0,0), .1,thickness=1, fill=True, facecolor="blue")
sh = p+q+r+s
sh
</sageplot>
</image>
<caption>A piecewise-defined function</caption>
</figure>

<!--

-->
</introduction>

<p>
Expand Down Expand Up @@ -205,11 +234,20 @@
Consider the following graph:
</p>
<figure xml:id="limits-one-sided-fluency">
<image width="50%" source="limits-one-sided-fluency.png" />
<image width="50%">
<sageplot>
x=var('x')
f(x) = piecewise([((-7,-1),-0.5*x+1.5),((-0.95,1.95),1/(x+1)),((1.96,7),-3*cos(pi*x))])
c1 = circle((-1,2), 0.1,thickness=1,fill=True, facecolor="white")
c2 = circle((2,1/3),0.1,thickness=1,fill=True, facecolor="white")
c3 = circle((2,-3),0.1,thickness=1, fill=True, facecolor="blue")
p=plot(f, (x,-7,7), ymin=-4, ymax=6, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2)
sh=p+c1+c2+c3
sh
</sageplot>
</image>
<caption>Another piecewise-defined function</caption>
</figure>


</introduction>
<task>
<p>
Expand Down Expand Up @@ -293,7 +331,22 @@
<statement>
<p> Let’s see an example of the application of this theorem. First examine the following picture. Explain why, from the picture, it seems that both assumptions of the theorem hold. </p>
<figure xml:id="squeeze">
<image width="100%" source="squeeze.png" />
<image>
<sageplot>
x=var('x')
h(x) = 1
g(x) = cos(0.5*x)
f(x) = cos(x)
p1=plot(f, (x,-pi,pi), ymin=-1.5, ymax=1.5, gridlines=True, axes_labels=('$x$',''), thickness=2, color="blue")
p2=plot(g, (x,-pi,pi), ymin=-1.5, ymax=1.5, gridlines=True, axes_labels=('$x$',''), thickness=2, color="red")
p3=plot(h, (x,-pi,pi), ymin=-1.5, ymax=1.5, gridlines=True, axes_labels=('$x$',''), thickness=2, color="green")
label1 = text('$h(x)$', (2,1.25), color="green", fontsize="xx-large")
label2 = text('$g(x)$', (2,.75), color = "red", fontsize="xx-large")
label3 = text('$f(x)$', (2,-.75),color = "blue", fontsize="xx-large")
sh = p1+p2+p3+label1+label2+label3
sh
</sageplot>
</image>
<caption>A pictorial example of the Squeeze Theorem.</caption>
</figure>
</statement>
Expand Down
18 changes: 14 additions & 4 deletions source/calculus/source/01-LT/02.ptx
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Expand Up @@ -260,11 +260,21 @@ the function as <m>x</m> tends to 2?


<activity xml:id = "activity-limits-numerically5">
<introduction>
<p> In <xref ref="figure-limit-numerical-5" /> is the graph for <m>f(x) = \sin\left(\dfrac{1}{x}\right)</m>. Several values for <m>f(x)</m> in the neighborhood of <m>x = 0</m> are approximated in <xref ref = "table-limit-numerical5"/>. </p>
<introduction>
<p>
In <xref ref="figure-limit-numerical-5" /> is the graph for <m>f(x) = \sin\left(\dfrac{1}{x}\right)</m>. Several values for <m>f(x)</m> in the neighborhood of <m>x = 0</m> are approximated in <xref ref = "table-limit-numerical5"/>.
</p>
<figure xml:id="figure-limit-numerical-5">
<image width="50%" source="limits_numerical5_graph.png" />
<caption>Graph of <m>f(x) = \sin(1/x)</m>.</caption>
<image width="50%">
<sageplot>
x = var('x')
f = sin(1/x)
p1 = plot(f,(x,-1,-0.01), ymin=-1,ymax=1, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=1, color='blue', aspect_ratio=1)
p2 = plot(f,(x,0.01,1), ymin=-1,ymax=1, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=1, color='blue', aspect_ratio=1)
p1+p2
</sageplot>
</image>
<caption>Graph of <m>f(x) = \sin(1/x)</m>.</caption>
</figure>
<table xml:id="table-limit-numerical5">
<tabular halign="center">
Expand Down
89 changes: 33 additions & 56 deletions source/calculus/source/01-LT/03.ptx
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Expand Up @@ -202,66 +202,43 @@
</p>
<sidebyside>
<figure>
<image width="50%" xml:id="two-functions-limits">
<latex-image>
\begin{tikzpicture}[scale=1]
\begin{axis}[ %human,
axis lines=middle,
grid=major,
xmin=-2, xmax=8.5,
ymin=-3, ymax=3,
xtick={-2,-1,...,8},
ytick={-4,-3,...,4},
tick style={thick},
% x label style={at={(axis description cs:1,0.7)}},
% y label style={at={(axis description cs:0.4,1)}},
ylabel=$y$,
xlabel=$x$,
]
\addplot[domain=-2:1, blue, ultra thick] {(1/3)*(x)-1/3};
\addplot[domain=1:5, blue, ultra thick] {-0.5*x +0.5};
\addplot[domain=5:7, blue, ultra thick] {x-5};
\addplot[mark=none, color=blue, nodes near coords={$f(x)$}] coordinates {(2,1)};
\addplot [only marks, blue] table {
5 0 \\
1 2 \\
8 2 \\
};

\addplot[only marks, color=blue, mark=*, fill=white] coordinates{(1,0) };

\addplot[only marks, color=blue, mark=*, fill=white] coordinates {(5,-2) };

\addplot[only marks, color=blue, mark=*, fill=white] coordinates {(7,2) };


\end{axis}
\end{tikzpicture}
</latex-image>
</image>
<image width="50%">
<sageplot>
x=var('x')
f = (1/3)*x-1/3
g= -0.5*x+0.5
h = x-5
b = polygon2d([[-3,-5],[-3,4],[9,4],[9,-5]], fill=False, thickness=0)
p1 = plot(f,(x,-3,1), gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2,color='blue',aspect_ratio=1, xmin=-3, xmax=9, ymin=-5, ymax=4)
p2 = plot(g,(x,1,5), gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2,color='blue',aspect_ratio=1)
p3 = plot(h,(x,5,7), gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2,color='blue',aspect_ratio=1)
c1 = circle((1,0),0.1,fill=True,facecolor='white',thickness=1)
c2 = circle((1,2),0.1,fill=True,facecolor='blue',thickness=1)
c3 = circle((5,-2),0.1,fill=True,facecolor='white',thickness=1)
c4 = circle((5,0),0.1,fill=True,facecolor='blue', thickness=1)
c5 = circle((7,2),0.1,fill=True,facecolor='white',thickness=1)
c6 = circle((8,2),0.1,fill=True,facecolor='blue',thickness=1)
b+p1+p2+p3+c1+c2+c3+c4+c5+c6
</sageplot>
</image>
<caption> The graph of <m>f(x)</m>.</caption>
</figure>
<figure>
<image width="50%" xml:id="graph-piecewise">
<latex-image>
\begin{tikzpicture}[scale=1]

\begin{axis}[ axis lines=middle, %human,
grid=major, grid,xtick={-3,-2,...,3}, ytick={-2,-1,...,4}, ymin=-2.2,xmin=-3.5,ymax=4.5,xmax=3.5, ylabel={}, xlabel={$x$}]
\addplot[domain=-3:0, color=blue, samples=100]{0.5*x};
\addplot[domain=0:2, color=blue, samples=100]{2^x};
\addplot[domain=2:4, color=blue, samples=50]{0.5*x+3};

\addplot[only marks, color=blue, mark=*, fill=blue] coordinates{(0,1) };

\addplot[only marks, color=blue, mark=*, fill=white] coordinates {(0,0) };


\addplot[mark=none, color=blue, nodes near coords={$g(x)$}] coordinates {(2,1)};
\end{axis}
\end{tikzpicture}
</latex-image>
</image>
<sageplot>
x=var('x')
f = 0.5*x
g = 2^x
h = 0.5*x+3
b = polygon2d([[-6,-3],[-6,6],[6,6],[6,-3]], fill=False, thickness=0)
p1 = plot(f,(x,-6,0), gridlines=True, axes_labels=('$x$','$g(x)$'), thickness=2,color='blue',aspect_ratio=1, xmin=-6, xmax=6, ymin=-3, ymax=6)
p2 = plot(g,(x,0,2), gridlines=True, axes_labels=('$x$','$g(x)$'), thickness=2,color='blue',aspect_ratio=1)
p3 = plot(h,(x,2,6), gridlines=True, axes_labels=('$x$','$g(x)$'), thickness=2,color='blue',aspect_ratio=1)
c1 = circle((0,0),0.1, fill=True, facecolor='white', thickness=1)
c2 = circle((0,1),0.1, fill=True, facecolor='blue', thickness=1)
b+p1+p2+p3+c1+c2
</sageplot>
</image>
<caption> The graph of <m>g(x)</m>.</caption>
</figure>
</sidebyside>
Expand Down
105 changes: 61 additions & 44 deletions source/calculus/source/01-LT/04.ptx
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Expand Up @@ -60,8 +60,26 @@
A function <m>f</m> defined on <m>-4 \lt x \lt 4</m> has the graph pictured below. Use the graph to answer each of the following questions.
</p>
<figure xml:id="figure-limits-different-kinds">
<image source="limits-different-kinds.png" width="50%">
<description>I need help with alt text for this that doesn't give the answers away.</description>
<image width="50%">
<sageplot>
x = var('x')
f = 2*x^2 + 16*x + 33
g = -x
h = x^2 + 0.5*x - 4
j = -2.5
k = sin(pi/(x-2))-3
p1 = plot(f,(x,-4,-3),ymin=-4,ymax=4, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
p2 = plot(g,(x,-3,-2),ymin=-4,ymax=4, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
p3 = plot(h,(x,-2,1),ymin=-4,ymax=4, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
p4 = plot(j,(x,1,2),ymin=-4,ymax=4, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
p5 = plot(k,(x,2.01,4),ymin=-4,ymax=4, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
c1 = circle((-2,2),0.1,fill=True,facecolor='white',thickness=1)
c2 = circle((-2,-1),0.1,fill=True,facecolor='blue',thickness=1)
c3 = circle((-1,1),0.1,fill=True,facecolor='blue',thickness=1)
c4 = circle((-1,-3.5),0.1,fill=True,facecolor='white',thickness=1)
c5 = circle((3,-3),0.1,fill=True,facecolor='white',thickness=1)
p1+p2+p3+p4+p5+c1+c2+c3+c4+c5
</sageplot>
</image>
</figure>
</introduction>
Expand Down Expand Up @@ -154,8 +172,26 @@
questions below, consider the values <m>a = -3</m>, <m>-2</m>, <m>-1</m>, <m>0</m>, <m>1</m>, <m>2</m>, <m>3</m>.
</p>
<figure xml:id="figure-continuous-graph2">
<image source="limits-different-kinds.png" width="50%">
<description>(for accessibility)</description>
<image width="50%">
<sageplot>
x = var('x')
f = 2*x^2 + 16*x + 33
g = -x
h = x^2 + 0.5*x - 4
j = -2.5
k = sin(pi/(x-2))-3
p1 = plot(f,(x,-4,-3),ymin=-4,ymax=4, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
p2 = plot(g,(x,-3,-2),ymin=-4,ymax=4, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
p3 = plot(h,(x,-2,1),ymin=-4,ymax=4, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
p4 = plot(j,(x,1,2),ymin=-4,ymax=4, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
p5 = plot(k,(x,2.01,4),ymin=-4,ymax=4, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2, aspect_ratio=1)
c1 = circle((-2,2),0.1,fill=True,facecolor='white',thickness=1)
c2 = circle((-2,-1),0.1,fill=True,facecolor='blue',thickness=1)
c3 = circle((-1,1),0.1,fill=True,facecolor='blue',thickness=1)
c4 = circle((-1,-3.5),0.1,fill=True,facecolor='white',thickness=1)
c5 = circle((3,-3),0.1,fill=True,facecolor='white',thickness=1)
p1+p2+p3+p4+p5+c1+c2+c3+c4+c5
</sageplot>
</image>
</figure>
</introduction>
Expand Down Expand Up @@ -236,46 +272,27 @@
<p>
Consider the function <m>f</m> whose graph is pictured below.
</p>
<figure>
<image width="50%" xml:id="graph-continuity-and-discontinuities">
<latex-image>
\begin{tikzpicture}[scale=1]
\begin{axis}[ %human,
axis lines=middle,
grid=major,
xmin=-2, xmax=8.5,
ymin=-3, ymax=3,
xtick={-2,-1,...,8},
ytick={-4,-3,...,4},
tick style={thick},
% x label style={at={(axis description cs:1,0.7)}},
% y label style={at={(axis description cs:0.4,1)}},
ylabel=$y$,
xlabel=$x$,
]
\addplot[domain=-2:1, blue, ultra thick] {(1/3)*(x)-1/3};
\addplot[domain=1:5, blue, ultra thick] {-0.5*x +0.5};
\addplot[domain=5:7, blue, ultra thick] {x-5};
\addplot[mark=none, color=blue, nodes near coords={$f(x)$}] coordinates {(2,1)};
\addplot [only marks, blue] table {
5 0 \\
1 2 \\
8 2 \\
};

\addplot[only marks, color=blue, mark=*, fill=white] coordinates{(1,0) };

\addplot[only marks, color=blue, mark=*, fill=white] coordinates {(5,-2) };

\addplot[only marks, color=blue, mark=*, fill=white] coordinates {(7,2) };


\end{axis}
\end{tikzpicture}
</latex-image>
</image>
<caption> The graph of <m>f(x)</m>.</caption>
</figure>
<figure>
<image width="50%">
<sageplot>
x=var('x')
f = (1/3)*x-1/3
g= -0.5*x+0.5
h = x-5
p1 = plot(f,(x,-2,1), ymin=-3,ymax=3, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2,color='blue',aspect_ratio=1)
p2 = plot(g,(x,1,5), ymin=-3,ymax=3, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2,color='blue',aspect_ratio=1)
p3 = plot(h,(x,5,7), ymin=-3,ymax=3, gridlines=True, axes_labels=('$x$','$f(x)$'), thickness=2,color='blue',aspect_ratio=1)
c1 = circle((1,0),0.1,fill=True,facecolor='white',thickness=1)
c2 = circle((1,2),0.1,fill=True,facecolor='blue',thickness=1)
c3 = circle((5,-2),0.1,fill=True,facecolor='white',thickness=1)
c4 = circle((5,0),0.1,fill=True,facecolor='blue', thickness=1)
c5 = circle((7,2),0.1,fill=True,facecolor='white',thickness=1)
c6 = circle((8,2),0.1,fill=True,facecolor='blue',thickness=1)
p1+p2+p3+c1+c2+c3+c4+c5+c6
</sageplot>
</image>
<caption> The graph of <m>f(x)</m>.</caption>
</figure>
<p>
Give a list of <m>x</m>-values where <m>f(x)</m> is not continuous. Be prepared to defend your answer based on <xref ref="definition-continuity"/>.
</p>
Expand Down
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