Chapter 1. UC Calc Demonstrations-Prototypes

1.1 UC Brightcove Test

Here is a Brightcove video link. This will launch in a new tab/window in the user's browser. Next, please see Figure 1.1, which is below.

In the figure below, we reference the UC Brightcove player in an IFRAME:

This is a test paragraph with the word Figure in blah blah the middle.

This is a new paragraph with Figure in the middle.

This is a new paragraph with Section in the middle.

1.2 MathJax Implementation Test \(e^x\)

This paragraph has the expression \(x > {-b \pm \sqrt{b^2-4ac} \over 2a}\) "inline" in the text. Arthur C. Clarke said "Any sufficiently advanced technology is indistinguishable from magic." This next bit uses a different delimiter for display (standalone) formatting: \[\begin{aligned} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, +\, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{aligned}\] and then we continue our discussion.

Things to bear in mind about using MathJax:

• While we are testing, the MathJax support is only available in this particular Digfir project (or those you clone from it). If you want to test again the "real" chapters, we can turn it on, but I didn't want to do so and have it cause problems with previewing your editing work.
• We probably want to put stand-alone expressions in their own paragraph block rather than using the square-bracket "display" delimeters to allow for CSS formatting to be applied to that type of block (i.e. how much spacing to put between it and adjoining paragraphs, borders/shading, etc.). That is how the expressions imported from the existing eBook are formatted (as indicated by green line in Digfir editor).
• For multiple choice questions with math in the answers, we'll probably need to experiment with some formatting solutions for the problem shown in Question 1.1 below where "Salem" shifts down as part of making room fo the expression, but the letter "A." does not.
• Double curly braces are used within Digfir questions to embed fill-in-the-blank answers See question 1.2 below. It's possible we might need to prevent Digfir from accidentally parsing the braces where LaTeX is placed. I my experiment below, I think it didn't cause a problem because there are no opening double curly braces, only closing ones.
• See MathJax TeX and LaTeX Support for general guidelines and details.
• You can do lots of cool things when you right-click (Control-click on Macs) on a MathJax expression in a published page, like see the original LaTeX, zoom, etc.

This is a test of aligned: \(\begin{aligned} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z + xy \end{aligned}\)

1.3 Specifying Answer Precision/Alternate Answers

For numeric answers, the precision required in the answer follows from the number of digits specified. In the following example, ".50" is the first answer listed. The student can respond with .5, .501, .495, and 0.5, but not .51, .505, or .49. A secondary answer, appended with an underscore ("_"), is also acceptable. In this case "1/2" is specified and will be marked correct by the system. This latter answer is only processed as a string, not a true fraction, so "25/50" would not be accepted.

Double click in the light green shaded area of the question below to edit the answer values after the asterisk.

1.4 Showing/Hiding Items

To set up content that is initially collapsed on the page, create a new Box block, and put the content blocks in that box. The following block types can be hidden:

Paragraphs
Figures
Tables
Lists
Boxes
Icons

The same kinds of blocks listed can serve as "triggers" that, when clicked, cause hidden content to be revealed. To enable the functionality, using the Raw XML editor, add the attribute ' onclick="show" ' to the tags of blocks on the containing box that will initially be hidden. Those blocks will display a peach background color in the Editor as an indicator. Add the attribute ' onlick_trigger="true" ' to the tags of blocks that when clicked will reveal the content. They will not have any special appearance.

Demonstration in box below. The box inside the containing box (block_type="note") has two items set as click triggers. You could also just set the note box itself as the trigger.

Technical note: To avoid the "Dynamic Figures" icon image overhanging the border of the collapsed box, it is necessary to include special CSS styling. This is handled by setting the container box's block_type to "clearfixed". Also, the figure that's revealed needs the block_type "clearboth" to avoid sliding under the icon.

1.5 Math Jax Test by Frank

The logarithm function (or simply, the logarithm) was discovered by Scotsman John Napier (1550-1617) in the sixteenth century and soon thereafter simplified by the English mathematician Henry Briggs (1561-1631) who introduced base 10. In his "Arithmetica Logarithmica" (1625), he introduced the first logarithm tables. The scientific impact of this work cannot be understated, leading quickly to the invention of the slide rule (a calculating device used in schools well into the 1960's) and Briggs' tables were instrumental to the work of astronomers like Johannes Kepler (1571-1630). Since logarithms allow one to reduce multiplication to addition and division to subtraction, namely \[\log_{10} (AB) = \log_{10}A + \log_{10} B\]

\[\log_{10} (\frac{A}{B}) = \log_{10}A - \log_{10} B\]

for positive \(A\) and \(B\), difficult calculations with very large numbers become possible. One should also note that, at the time, the Indian system of numeration using 0,1,2,..,9 (known also as the Arabic system) had not yet really taken hold in Europe. He was born in Basel, Switzerland, and studied under Johann Bernoulli. Euler contributed to many fields of mathematics and engineering, including calculus, number theory and was a founder of the fields of topology and the calculus of variations. In the year 1728, when Euler was only 21, he wrote a scientific paper titled "Meditation Upon Experiments Made Recently on the Firing of a Canon." This paper was only published in 1862 in Euler's Opera Posthuma II. In this paper he introduces, for the first time, the number \(e \approx 2.71828\). At a later time he introduces the logarithm to the base \(e\). Euler defines e as the limiting value of the numbers as \(n\) gets larger and larger, but did not prove that this limiting value existed (perhaps it was obvious to a genius of his magnitude.) Moreover, it is not clear to us how he actually arrived at this number. We present here a likely scenario. As a student of the Swiss mathematician Johann Bernoulli, who was the author of the first ever calculus book, Euler quickly became a master of this new discipline. Clearly, the early masters would have wanted to calculate the more difficult derivatives of the transcendental functions, particularly the derivative of \(y= \log_{10}x\), which had been around for over one hundred years. So let us look at a likely birth of \(e\). The important point we wish to make is that the existence of this incredible number is most assuredly not the result of a sudden burst of inspiration, but the result of seeking the answer to a natural question, for example like finding the derivative \(\frac{dy}{dx}\) of \(y= \log_{10}x\). So let us, as Euler likely did, find this derivative from the definition:

\begin{eqnarray} \frac{dy}{dx} & = & \lim \limits_{h \longrightarrow 0} \frac{f(x+h)-f(x)}{h} \\ & = & \lim \limits_{h \longrightarrow 0} \frac{1}{h} \Big(\log_{10} (x+h)- \log_{10}(x) \Big) \\ & = & \lim \limits_{h \longrightarrow 0} \frac{1}{h} \Big(\log_{10} (\frac{x+h}{x}) \Big) \\ & = & \lim \limits_{h \longrightarrow 0} \frac{1}{h} \Big(\log_{10} (1+\frac{h}{x}) \Big) \\ & = & \lim \limits_{h \longrightarrow 0} \frac{1}{x} \Big(\frac{x}{h} \cdot \log_{10} (1+\frac{h}{x}) \Big) \\ & = & \lim \limits_{h \longrightarrow 0} \frac{1}{x} \log_{10} (1+\frac{h}{x})^\frac{x}{h} \\ & = & \frac{1}{x} \cdot \lim \limits_{h \longrightarrow 0} \log_{10} (1+\frac{h}{x})^\frac{x}{h}\end{eqnarray}

Here we must consider taking \(h\) small, but either positive or negative. Let us set \(h=\frac{x}{n}\) or \(h=-\frac{x}{n}\). Then the expression \((1+\frac{h}{x})^\frac{x}{h}\) becomes either \((1+\frac{1}{n})^n\) or \((1-\frac{1}{n})^{-n}\).

It turns out that, as \(n\) gets larger and larger (\(n \longrightarrow \infty\)), both of these expressions get closer and closer to the same number, which Euler called \(e\) (for Euler?). We show first that the numbers \(e_n := (1+\frac{1}{n})^n\) get closer to some number (called \(e\)) as \(n\) gets larger. This is the claim that Euler makes in his 1728 paper, when he introduces the expression \((1+\frac{1}{n})^n\). From the binomial theorem one can show two facts:

A very deep property of the real numbers asserts that there must be a real number \(e\) such that gets arbitrarily close to e as n gets larger. We write this as:

\[\lim \limits_{n \longrightarrow \infty} e_n = e\]

Thus in the process of finding the derivative of, we (or Euler) discovered a totally new number \(e\). Before we proceed, we should show that.....

The “prime” notation y′ and f′(x) was introduced by the French mathematician Joseph Louis Lagrange (1736–1813). There is another standard notation for the derivative that we owe to Leibniz (Figure 2):

In Example 2, we showed that the derivative of y = x⁻² is y′ = −2x⁻³. In Leibniz notation, we would write

To specify the value of the derivative for a fixed value of x, say, x = 4, we write

Thinking of \(\frac{dy}{dx}\) as a fraction forces us to confront the question: what actually are \(dy\) and \(dx\) ? We cannot answer this question precisely. Leibniz thought of them as "infinitely small" changes in \(y\) and \(x\) . The notation, however, proved to be so incredibly useful (especially to physicists) that in the first half of the 20th century mathematicians figured out a rigorous explanation which is now known as the theory of differential forms which you may encounter in your advanced studies.

A main goal of this chapter is to develop the basic rules of differentiation. These rules enable us to find derivatives without computing limits.

We make a few remarks before proceeding:

• It may be helpful to remember the Power Rule in words: To differentiate xⁿ, “bring down the exponent and subtract one (from the exponent).”

• • The Power Rule is valid for all exponents, whether negative, fractional, or irrational:

• The Power Rule can be applied with any variable, not just x. For example,

Next, we state the Linearity Rules for derivatives, which are analogous to the linearity laws for limits.

The derivative f′(x) gives us important information about the graph of f(x). For example, the sign of f′(x) tells us whether the tangent line has positive or negative slope, and the magnitude of f′(x) reveals how steep the slope is.

1.6 Question Demos - Embedded Queries etc.

Advanced question setups: Google It.

Dynamic Figure test.

1.7 Video Link to UC Brightcove Video Test

Here is a Brightcove video link.