Skip to content

Commit

Permalink
fix descriptions in Section 1.1
Browse files Browse the repository at this point in the history
  • Loading branch information
@sean-fitzpatrick
sean-fitzpatrick committed Oct 2, 2024
1 parent b1a1abf commit 283ff8f
Showing 1 changed file with 82 additions and 63 deletions.
145 changes: 82 additions & 63 deletions ptx/sec_limit_intro.ptx
View file
Original file line number Diff line number Diff line change
Expand Up @@ -38,13 +38,13 @@
<image>
<description>
<p>
Graph of <m>\sin(x)/x</m> showing the domain and range of
<m>-7</m> to <m>7</m> and <m>0</m> to
<m>1</m> respectively. The <m>x</m> intercepts are at
Graph of <m>\sin(x)/x</m>, shown for <m>x</m> between
<m>-7</m> and <m>7</m>, and <m>y</m> between <m>0</m> and
<m>1</m>. The <m>x</m> intercepts are at
<m>x=-2\pi, -\pi, \pi</m>, and <m>2\pi</m>, and a <m>y</m> intercept is at <m>y = 1</m>.
The function has a downward curve for <m>-\pi \lt x \lt \pi</m> and an
upward curve for <m>-2\pi \gt x \lt -\pi</m>, and
<m>\pi \gt x \lt 2\pi</m>. The graph is undefined for <m>x = 0</m>.
The graph has a downward curve for <m>-\pi \lt x \lt \pi</m> and an
upward curve for <m>-2\pi \lt x \lt -\pi</m>, and
<m>\pi \lt x \lt 2\pi</m>. The graph is undefined for <m>x = 0</m>.
</p>
</description>
<shortdescription>
Expand Down Expand Up @@ -75,10 +75,10 @@
<image>
<description>
<p>
Graph of <m>\sin(x)/x</m> zoomed in on values where <m>x</m> is near <m>1</m>. The
domain of the graph is <m>0.5</m> to <m>1.5</m>.
Graph of <m>\sin(x)/x</m> zoomed in on values where <m>x</m> is near <m>1</m>.
This view of the graph shows <m>x</m> from <m>0.5</m> to <m>1.5</m>.
The graph has only a slight downward curve. It shows that for <m>x = 1</m>,
<m>\sin(x)/x</m> is approx. <m>0.84</m>
<m>\sin(x)/x</m> is approximately <m>0.84</m>
</p>
</description>
<shortdescription>
Expand Down Expand Up @@ -136,10 +136,11 @@
<image width="47%">
<description>
<p>
Graph of <m>\sin(x)/x</m> zoomed in on values where <m>x</m> is near <m>0</m>. The
domain of the graph is <m>-1</m> to <m>1</m>. The graph
has a downward curve and symmetric, peaking at <m>y = 1</m>. There is a dot at
<m>x = 0</m> showing the equation is undefined for values of <m>x = 0</m>,
Graph of <m>\sin(x)/x</m> zoomed in on values where <m>x</m> is near <m>0</m>.
The image shows the portion of the graph where <m>x</m> is from <m>-1</m> to <m>1</m>.
The graph has a downward curve and is symmetric about <m>x=0</m>.
The height of the graph approaches <m>y = 1</m> when <m>x</m> is near <m>0</m>.
A hollow dot at the point <m>(0,1)</m> shows that the function is undefined when <m>x = 0</m>;
that is, <m>f(0) =</m> undefined.
</p>
</description>
Expand Down Expand Up @@ -367,16 +368,19 @@
<image>
<description>
<p>
Graph of <m>(x^2 - x - 6)/(6x^2 - 19x + 3)</m>,
zoomed on values near <m>x = 3</m>. The domain is approximately
<m>2.5</m> to <m>2.5</m>.
There is a slight upward curve to the graph. Shows that the limit of the
equation as <m>x</m> approaches <m>3</m> is <m>0.294</m>. The graph also
shows that the equation is undefined for <m>x = 3</m>.
Graph of <m>f(x)=\frac{x^2 - x - 6}{6x^2 - 19x + 3}</m>,
zoomed on values near <m>x = 3</m>, and showing the portion of the graph
for <m>x</m> from <m>2.5</m> to <m>3.5</m>.
</p>
<p>
There is a slight upward curve to the graph.
The graph suggests that the limit of the function
as <m>x</m> approaches <m>3</m> is <m>0.294</m>. The graph also
shows that the function is undefined for <m>x = 3</m>.
</p>
</description>
<shortdescription>
Graph of the equation, shows that when x = 3 the y = undefined,
Graph of the function for this example, which shows that when x = 3, f(x) is undefined,
but near 0.294.
</shortdescription>
<latex-image label="img_limit1">
Expand Down Expand Up @@ -533,9 +537,11 @@
<p>
Graph of the piecewise-defined function in <xref ref="ex_limit2"/>.
For values of <m>x \lt 0</m> the graph is straight
with a slope of <m>1</m> and for values <m>x \gt 0</m> the graph is
curved downward with a negative slope. Shows that at <m>x = 0</m>,
<m>y</m> is undefined, but near <m>1</m>.
with a slope of <m>1</m> and for values of <m>x \gt 0</m> the graph
curves downward. A hollow dot at the point <m>(0,1)</m> shows that at <m>x = 0</m>,
<m>f(x)</m> is undefined.
However, both parts of the graph, for <m>x\lt 0</m> and for <m>x\gt 0</m>,
get close to the point <m>(0,1)</m> as <m>x</m> gets close to <m>0</m>.
</p>
</description>
<shortdescription>
Expand Down Expand Up @@ -697,9 +703,19 @@
<p>
Graph of piecewise function in <xref ref="ex_no_limit1"/>.
For values of <m>x \leq 1</m> the graph
has a upward curve, and where <m>x = 1</m> <m>y = 2</m>.
For values of <m>x \gt 1</m> the graph is straight with a
positive slope, where <m>x = 1</m>, <m>y</m> is undefined.
has a upward curve, and the graph ends at the point <m>(1,2)</m>,
illustrating the fact that <m>f(1)=2</m>.
</p>
<p>
For values of <m>x \gt 1</m> the graph is a straight line with a positive slope.
Moving left to right, the line begins at the point <m>(1,1)</m>,
at which there is a hollow dot, indicating that to the right of <m>x=1</m>,
the value of <m>f(x)</m> approaches 1.
</p>
<p>
The most important feature of the graph is that it shows how <m>f(x)</m>
approaches two different values as <m>x</m> approaches <m>1</m>,
depending on whether <m>x\lt 1</m> or <m>x\gt 1</m>.
</p>
</description>
<shortdescription>
Expand Down Expand Up @@ -788,12 +804,12 @@
Graph of the function for <xref ref="ex_no_limit2"/>.
The graph hows a horizontal
asymptote at <m>y = 0</m> and a vertical asymptote at <m>x = 1</m>.
Because of the vertical asymptote at <m>x = 1</m> the equation
Because of the vertical asymptote at <m>x = 1</m> the function
has no limit as <m>x</m> approaches <m>1</m>.
</p>
</description>
<shortdescription>
Graph of the equation showing that as x approaches 1 f(x) asymptotes.
Graph of the function f(x), showing that as x approaches 1, there is a vertical asymptote.
</shortdescription>
<latex-image label="img_nolimit2">
\begin{tikzpicture}
Expand Down Expand Up @@ -906,16 +922,16 @@
<image>
<description>
<p>
Graph of <m>sin(1/x)</m> displaying the <m>x</m> and <m>y</m>
intervals <m>-1</m> to <m>1</m>. As <m>x</m> gets close to
<m>0</m> the cycles shorten, rendering a wide, vertical line.
This line is however not solid as it is just a bunch of lines
really close to each other.
The graph of <m>f(x)=sin(1/x)</m> is shown, for <m>x</m> values between <m>-1</m> and <m>1</m>.
Like any sinusoidal graph, the curve oscillates back and forth between <m>y=1</m> and <m>y=-1</m>.
However, as <m>x</m> gets close to <m>0</m>, the argument of the sine function increases rapidly,
causing the distance between successive peaks to get smaller and smaller as the graph nears the <m>y</m> axis.
As <m>x</m> gets close to zero, the oscillations get so close together that it is no longer possible to distinguish them,
and the curve appears to become a solid, vertical strip.
</p>
</description>
<shortdescription>
Graph of the equation showing a thick line at x = 0, where the thick
line is just the oscillation of a single thin line on short cycles.
Graph of the function sin(1/x), showing oscillations that become so rapid near the origin that they blur together.
</shortdescription>
<latex-image label="img_nolimit3a">
\begin{tikzpicture}
Expand Down Expand Up @@ -979,15 +995,18 @@
<image>
<description>
<p>
Graph of <m>sin(1/x)</m> displaying the <m>y</m> interval <m>-1</m> to
<m>1</m> and <m>x</m> interval <m>-0.1</m> to <m>0.1</m>. As <m>x</m> gets close
to <m>0</m> the cycles shorten, rendering a wide, vertical line.
This line is however not solid as it is just a bunch of lines
really close to each other.
Another graph of <m>f(x)=\sin(1/x)</m> is shown,
this time zoomed in to show only the <m>x</m> interval from <m>-0.1</m> to <m>0.1</m>.
The features of the graph are the similar to what is visible over the larger interval:
further from the origin, we see the graph oscillating (rapidly) between <m>y=1</m> and <m>y=-1</m>.
Near the orgin, the oscillations become so rapid that we can no longer tell them apart.
What we conclude from the graph is that on any interval containing <m>x=0</m>,
<m>f(x)=\sin(1/x)</m> takes on every <m>y</m> value between <m>-1</m> and <m>1</m>.
(In fact, <m>f(x)</m> attains every value infinitely many times!)
</p>
</description>
<shortdescription>
Graph of the same equation, shown with a smaller x interval, -0.1 to 0.1.
Graph of the same function, sin(1/x), shown with a smaller x interval, -0.1 to 0.1.
</shortdescription>
<latex-image label="img_nolimit3b">
\begin{tikzpicture}
Expand Down Expand Up @@ -1175,18 +1194,17 @@
<image width="47%">
<description>
<p>
Graph showing a downward curved equation with the domain and range
<m><var name="$xmin"/></m> to <m><var name="$xmax"/></m> and
<m><var name="$ymin"/></m> to <m><var name="$ymax"/></m> respectively.
There are two dots plotted on the line of the equation at
<m>(1, 10)</m> and <m>(5, 20)</m> with a dotted line intercepting the
points. The dotted line has a positive slope. The line of the
equation intercepts the <m>x</m> axis at <m>(0, 0)</m>
The image shows the graph of a function, along with a line that intersects the graph at two points.
The graph has the shape of a parabola that opens downward,
and is displayed over the region <m>0\leq x\leq 6</m>,
with a <m>y</m> range from 0 to 25.
There are two points plotted on the graph at coordinates
<m>(1, 10)</m> and <m>(5, 20)</m>, and the line through these points is an example of a secant line.
</p>
</description>
<shortdescription>
Graph showing a downward curved equation, with points at (1,10) and (5,20), with a
dotted line intercepting both points and a positive slope.
A downward curved graph, with marked points at (1,10) and (5,20), and a
line intercepting both points.
</shortdescription>
<latex-image label="img_diffquot1">
\begin{tikzpicture}
Expand Down Expand Up @@ -1254,13 +1272,13 @@
<image>
<description>
<p>
Graph of the same equation from 1.1.26, with the points <m>(1,10)</m>
and <m>(3,21)</m>. The secant line has a steeper slope than in
figure 1.1.26. Here the value of h is <m>2</m>.
Graph of the function from <xref ref="fig_diffquot1"/>, with the points on the graph <m>(1,10)</m>
and <m>(3,21)</m> marked. A secant line is drawn through these points; it has a steeper slope than in
<xref ref="fig_diffquot1"/>. Here the value of <m>h</m> is <m>2</m>.
</p>
</description>
<shortdescription>
Graph of the previous equation, with the points (1,10)
Graph of the same function as the previous figure, with the points (1,10)
and (3,21). The secant line has a steeper slope, equal to 5.5.
</shortdescription>
<latex-image label="img_diff_quot_smallha">
Expand All @@ -1287,14 +1305,15 @@
<image>
<description>
<p>
Graph of the same equation from <xref ref="fig_diff_quot_smallha"/>, but
with the points <m>(1,10)</m> and <m>(2,17)</m>. Shows the dotted line with a steeper
slope than in figure 1.1.26. Here the value of h is <m>1</m>.
Graph of the function from <xref ref="fig_diff_quot_smallha"/>, but
with the points <m>(1,10)</m> and <m>(2,17)</m> on the graph marked.
These points correspond to a value of <m>h=1</m>,
and the secant line through these points has a steeper slope than in <xref ref="fig_diff_quot_smallha"/>.
</p>
</description>
<shortdescription>
Graph of the same equation, with the points (1,10)
and (2,17). The secant line has a steeper slope equal to 7.
Graph of the same function as the previous figure, with the points (1,10)
and (2,17) marked. The secant line has a steeper slope equal to 7.
</shortdescription>
<latex-image label="img_diff_quot_smallhb">
\begin{tikzpicture}
Expand All @@ -1320,13 +1339,13 @@
<image>
<description>
<p>
Graph of the same equation from <xref ref="fig_diff_quot_smallhb"/>, but
with the points <m>(1,10)</m> and <m>(1.5,13.875)</m>. Shows the dotted line with a
steeper slope than in figure 1.1.26. Here the value of h is <m>0.5</m>.
Graph of the function from <xref ref="fig_diff_quot_smallhb"/>, but
with the points <m>(1,10)</m> and <m>(1.5,13.875)</m> on the graph marked, corresponding to the value <m>h=0.5</m>.
The secant line through these points again has a steeper slope than in the previous figures.
</p>
</description>
<shortdescription>
Graph of the same equation, with the points (1,10) and
Graph of the same function, with the points (1,10) and
(1.5,13.875). The secant line has a steeper slope equal to 7.75.
</shortdescription>
<latex-image label="img_diff_quot_smallhc">
Expand Down

0 comments on commit 283ff8f

Please sign in to comment.