U.S.S. Mathematica

Exploring Maths... and science too!

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Basic integration formulas.

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INTEGRATION BY PARTS INTRO - great job, Sal (Khan Academy)

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A good, broad overview to Calculus II and Integrals

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Calculus I in 20 minutes - a good overview

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This is what makes e so important (in the calculus of functions of real or complex variables).

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e^x is special (In CALC) because its derivative at zero is equal to 1

Functions f(x) = ax are shown for several values of a. e is the unique value of a, such that the derivative of f(x) = ax at the point x = 0 is equal to 1. The blue curve illustrates this case, ex. For comparison, functions 2x (dotted curve) and 4x (dashed curve) are shown; they are not tangent to the line of slope 1 and y-intercept 1 (red).

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the number "e"

The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms.[11] A general exponential function y = ax has derivative given as the limit:

The limit on the right-hand side is independent of the variable x: it depends only on the base a. When the base is e, this limit is equal to one, and so e is symbolically defined by the equation:

Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e, as opposed to some other number, as the base of the exponential function makes calculations involving the derivative much simpler.

[ussmathematica]

e (mathematical constant)

From Wikipedia, the free encyclopedia

  "Euler's number" redirects here. For γ, a constant in number theory, see Euler's constant. For other uses, see List of topics named after Leonhard Euler#Euler—numbers.

  The number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm.[1] It is the limit of (1 + 1/n)n as n becomes large, an expression that arises in the study of compound interest, and can also be calculated as the sum of the infinite series[2]

The constant can be defined in many ways; for example, e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is equal to 1.[3] The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case, e is the number whose natural logarithm is 1. There are also more alternative characterizations.

Sometimes called Euler's number after the Swiss mathematician Leonhard Euler, e is not to be confused with γ—the Euler–Mascheroni constant, sometimes called simply Euler's constant. The number e is also known as Napier's constant, but Euler's choice of this symbol is said to have been retained in his honor.[4] The number e is of eminent importance in mathematics,[5] alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler's identity. Like the constant π, e is irrational: it is not a ratio of integers; and it is transcendental: it is not a root of any non-zero polynomial with rational coefficients. The numerical value of e truncated to 50 decimal places is

2.71828182845904523536028747135266249775724709369995... (sequence A001113 in OEIS).

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Sin Curve and Transformations

In day to day life the graphs of some data versus time are sinusoids. In physics the graphs of electric current, fourier series, ballistic trajectiles are sinusoidal. Navigation, space flights and astronomy depend on calculations of sine and cosine. Radio and Television signals are transmitted as sine or cosine waves. Music waves with various frequencies and amplitude are similar to sine or cosine waves. Heart beats, breathing recorded by medical equipments are sinusoidal waves. GPS and cell phones depend on sine/cosine functions.

http://math.tutorvista.com/trigonometry/sinusoidal-graph.html

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the power of the unit circle.

Trig essentials.

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Solid of Revolution (part 4): More volumes around the x-axis.

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Solids of Rev. pt 2 khan

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Solids of Revolutions 1 (Khan Academy)

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The universe is believed to be mostly composed of dark energy and dark matter, both of which are poorly understood at present. Less than 5% of the universe is ordinary matter, a relatively small contribution.

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

Found this in attempting to find a suitable background image for my tumblr...

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

A very, very interesting picture indeed....

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Intro: To Boldly Go...

I'm starting this tumblr to help me (and others) learn math. By posting interesting things here I hope to develop a different take on understanding maths, rather than being rushed through them in lecture halls. . .