First ClintonTrump Debate 092616 - Betting Markets

The betting market PredictIt on Tuesday morning showed virtually all shares predicting a Clinton victory trending upwards -- meaning that bettors thought the chances of her winning the White House had improved. Shares predicting a Trump victory, by comparison, lost value overnight.  Similarly, the betting site Hypermind showed Clinton’s likelihood of victory rising and Trump’s falling.

The well-regarded Predictwise site, which synthesizes data from multiple betting markets and more, registered one of the more dramatic shifts. Before the debate last night, it had Clinton with a 69 percent to 31 percent advantage is the race to win the White House. (That is measuring probability of an Electoral College victory, not the popular vote.) By Tuesday morning the divide had widened to 74-26 in Clinton’s favor.

https://www.yahoo.com/finance/news/betting-markets-saying-win-election-173000094.html 

That's a major pop aka "statistically significant"

Logistic Function

A logistic function or logistic curve is a common "S" shape (sigmoid curve), with equation:

${\displaystyle f(x)={\frac {L}{1+\mathrm {e} ^{-k(x-x_{0})}}}}$

where

• e = the natural logarithm base (also known as Euler's number),
• x₀ = the x-value of the sigmoid's midpoint,
• L = the curve's maximum value, and
• k = the steepness of the curve.^[1]

For values of x in the range of real numbers from −∞ to +∞, the S-curve shown on the right is obtained (with the graph of f approaching L as x approaches +∞ and approaching zero as x approaches −∞).
The function was named in 1844–1845 by Pierre François Verhulst, who studied it in relation to population growth.^[2] The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.
The logistic function finds applications in a range of fields, including artificial neural networks, biology (especially ecology), biomathematics, chemistry, demography, economics, geoscience, mathematical psychology, probability, sociology, political science, linguistics, and statistics.

https://en.wikipedia.org/wiki/Logistic_function#Applications 

S Curves Everywhere

Tribonaci et al

The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are:

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, … (sequence A000073 in the OEIS)

The tribonacci constant ${\displaystyle {\tfrac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}}$ is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial x³ − x² − x − 1, approximately 1.839286755214161 (sequence A058265 in the OEIS), and also satisfies the equation x + x⁻³ = 2. It is important in the study of the snub cube.
The tribonacci numbers are also given by^[4]

${\displaystyle T(n)=\left\lfloor 3\,b{\frac {\left({\frac {1}{3}}\left(a_{+}+a_{-}+1\right)\right)^{n}}{b^{2}-2b+4}}\right\rceil }$

where ${\displaystyle \lfloor \cdot \rceil }$ denote the nearest integer function and

${\displaystyle a_{\pm }=\left(19\pm 3{\sqrt {33}}\right)^{1/3}}$

${\displaystyle b=\left(586+102{\sqrt {33}}\right)^{1/3}}$.

http://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers

Gabriel's Horn (1/x)

1/X - the most interesting function?

Gabriel's Horn (also called Torricelli's trumpet) is a geometric figure which has infinite surface area, but finite volume. The name refers to the tradition identifying the Archangel Gabriel as the angel who blows the horn to announce Judgment Day, associating the divine, or infinite, with the finite. The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli.

Google

Sequences & Series - Wikipedia et al Links

Sequence
From Wikipedia, the free encyclopedia

Google

In mathematics, informally speaking, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements, or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.

For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers(2, 4, 6,...). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

Fibonacci Sequence

From Wikipedia, the free encyclopedia

A tiling with squares whose side lengths are successive Fibonacci numbers

An approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; this one uses squares of sizes 1, 1, 2, 3, 5, 8, 13, 21, and 34.

In mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence:^[1][2]

 (sequence A000045 in OEIS)

By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two.

Convergent/Divergent Sequence (Limit of a Sequence)

From Wikipedia, the free encyclopedia

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to".^[1] If such a limit exists, the sequence is called convergent. A sequence which does not converge is said to be divergent.^[2] The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests.^[1]

Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.

Fundamental Theorem of Calculus

If, in fact, "The Fundamental Theorem of Calculus" is fundamental, why is it on page 400.  It should be on page 1.  - Can you see the forest for the trees?

Starting calculus with the Fundamental Theorem (FTC) is impossible!  Right?  You must go through the normal progression of functions, limits, derivatives, etc., before introducing such complexity.  Right?  Perhaps not.  The standard approach takes weeks or months before integration is discussed.  In the meantime you're wondering how difficult integrals will be as you struggle with derivatives.  With FTC, on day one, you're dealing with the "forest," the big picture.  It will (should) demystify the entire subject.

Look what you can achieve by examining the above image.  Work backwards.  This is what a function looks like; here's a derivative/slope/tangent; this is a limit and this is how it's used; this is an area function, this is why the derivative of the area function/integral = y = f(x).

Explain the symbology; explain the terminology.  Remember, "The Fundamental Theorem of Calculus" is fundamental!  There's no rush.  Take your time.  Explain each concept while tying them into a cohesive whole.

FTC links