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make pdf file generation work

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jverzani
2022-10-10 14:28:05 -04:00
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12
quarto/limits/intermediate_value_theorem.qmd
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@@ -672,9 +672,13 @@ caption = """
Elevation profile of the Hardrock 100 ultramarathon. Treating the elevation profile as a function, the absolute maximum is just about 14,000 feet and the absolute minimum about 7600 feet. These are of interest to the runner for different reasons. Also of interest would be each local maxima and local minima - the peaks and valleys of the graph - and the total elevation climbed - the latter so important/unforgettable its value makes it into the chart's title.
"""
ImageFile(:limits, imgfile, caption)
# ImageFile(:limits, imgfile, caption)
nothing
```
[![Elevation profile of the Hardrock 100 ultramarathon. Treating the elevation profile as a function, the absolute maximum is just about 14,000 feet and the absolute minimum about 7600 feet. These are of interest to the runner for different reasons. Also of interest would be each local maxima and local minima - the peaks and valleys of the graph - and the total elevation climbed - the latter so important/unforgettable its value makes it into the chart's title.
](figures/hardrock-100.png)](https://hardrock100.com)
The extreme value theorem discusses an assumption that ensures absolute maximum and absolute minimum values exist.
@@ -885,9 +889,13 @@ figure= "figures/cannonball.jpg"
caption = """
Trajectories of potential cannonball fires with air-resistance included. (http://ej.iop.org/images/0143-0807/33/1/149/Full/ejp405251f1_online.jpg)
"""
ImageFile(:limits, figure, caption)
# ImageFile(:limits, figure, caption)
nothing
```
![Trajectories of potential cannonball fires with air-resistance included. (http://ej.iop.org/images/0143-0807/33/1/149/Full/ejp405251f1_online.jpg)
](./figures/cannonball.jpg)
In 1638, according to Amir D. [Aczel](http://books.google.com/books?id=kvGt2OlUnQ4C&pg=PA28&lpg=PA28&dq=mersenne+cannon+ball+tests&source=bl&ots=wEUd7e0jFk&sig=LpFuPoUvODzJdaoug4CJsIGZZHw&hl=en&sa=X&ei=KUGcU6OAKJCfyASnioCoBA&ved=0CCEQ6AEwAA#v=onepage&q=mersenne%20cannon%20ball%20tests&f=false), an experiment was performed in the French Countryside. A monk, Marin Mersenne, launched a cannonball straight up into the air in an attempt to help Descartes prove facts about the rotation of the earth. Though the experiment was not successful, Mersenne later observed that the time for the cannonball to go up was greater than the time to come down. ["Vertical Projection in a Resisting Medium: Reflections on Observations of Mersenne".](http://www.maa.org/publications/periodicals/american-mathematical-monthly/american-mathematical-monthly-contents-junejuly-2014)

8
quarto/limits/limits.qmd
View File

@@ -262,14 +262,14 @@ xs = [1/10^i for i in 1:5]
This progression can be seen to be increasing. Cauchy, in his treatise, can see this through:
$$
\begin{align*}
(1 + \frac{1}{m})^n &= 1 + \frac{1}{1} + \frac{1}{1\cdot 2}(1 = \frac{1}{m}) + \\
& \frac{1}{1\cdot 2\cdot 3}(1 - \frac{1}{m})(1 - \frac{2}{m}) + \cdots \\
&+
\frac{1}{1 \cdot 2 \cdot \cdots \cdot m}(1 - \frac{1}{m}) \cdot \cdots \cdot (1 - \frac{m-1}{m}).
\end{align*}
$$
These values are clearly increasing as $m$ increases. Cauchy showed the value was bounded between $2$ and $3$ and had the approximate value above. Then he showed the restriction to integers was not necessary. Later we will use this definition for the exponential function:
@@ -836,7 +836,7 @@ This accurately shows the limit does not exist mathematically, but `limit(ceil(x
The `limit` function doesn't compute limits from the definition, rather it applies some known facts about functions within a set of rules. Some of these rules are the following. Suppose the individual limits of $f$ and $g$ always exist (and are finite) below.
$$
\begin{align*}
\lim_{x \rightarrow c} (a \cdot f(x) + b \cdot g(x)) &= a \cdot
\lim_{x \rightarrow c} f(x) + b \cdot \lim_{x \rightarrow c} g(x)
@@ -850,7 +850,7 @@ $$
\frac{\lim_{x \rightarrow c} f(x)}{\lim_{x \rightarrow c} g(x)}
&(\text{provided }\lim_{x \rightarrow c} g(x) \neq 0)\\
\end{align*}
$$
These are verbally described as follows, when the individual limits exist and are finite then: