Calculus

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Calculus uses methods originally based on the summation of infinitesimal differences.

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It includes the examination of changes in an expression by smaller and smaller differences.

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Notation[edit]

Notation: let the symbol Def. indicate that a definition is following.

Notation: let the symbols between [ and ] be replacement for that portion of a quoted text.

Notation: let the symbol ... indicate unneeded portion of a quoted text.

Sometimes these are combined as [...] to indicate that text has been replaced by ....

Universals[edit]

Def. a "characteristic or property that particular things have in common"[1] is called a universal.

"When we examine common words, we find that, broadly speaking, proper names stand for particulars, while other substantives, adjectives, prepositions, and verbs stand for universals."[2]

Such words as "entity", "object", "thing", and perhaps "body", words "connoting universal properties, ... constitute the very highest genus or "summum genus"" of a classification of universals.[3] To propose a definition for say a plant whose flowers open at dawn on a warm day to be pollinated during the day time using the word "thing", "entity", "object", or "body" seems too general and is.

To help with definitions, their meanings and intents, there is the learning resource theory of definition.

Control groups[edit]

This is an image of a Lewis rat. Credit: Charles River Laboratories.

The findings demonstrate a statistically systematic change from the status quo or the control group.

“In the design of experiments, treatments [or special properties or characteristics] are applied to [or observed in] experimental units in the treatment group(s).[4] In comparative experiments, members of the complementary group, the control group, receive either no treatment or a standard treatment.[5]"[6]

Proof of concept[edit]

Def. a “short and/or incomplete realization of a certain method or idea to demonstrate its feasibility"[7] is called a proof of concept.

Def. evidence that demonstrates that a concept is possible is called proof of concept.

The proof-of-concept structure consists of

  1. background,
  2. procedures,
  3. findings, and
  4. interpretation.[8]

Mathematics[edit]

Variations[edit]

Differences[edit]

Notation: let the symbol \Delta represent change in.

Notation: let the symbol d represent an infinitesimal change in.

Notation: let the symbol \partial represent an infinitesimal change in one of more than one.

Derivatives[edit]

Let

y = f(x)

be a function where values of x may be any real number and values resulting in y are also any real number.

\Delta x is a small finite change in x which when put into the function f(x) produces a \Delta y.

These small changes can be manipulated with the operations of arithmetic: addition (+), subtraction (-), multiplication (*), and division (/).

\Delta y = f(x + \Delta x) - f(x)

Dividing \Delta y by \Delta x and taking the limit as \Delta x → 0, produces the slope of a line tangent to f(x) at the point x.

For example,

f(x) = x^2
f(x + \Delta x) = (x + \Delta x)^2 = x^2 + 2x\Delta x + \Delta x^2
\Delta y = (x^2 + 2x\Delta x + \Delta x^2) - x^2
\Delta y/\Delta x = (2x\Delta x + \Delta x^2)/\Delta x
\Delta y/\Delta x = 2x + \Delta x

as \Delta x and\Delta y go towards zero,

dy/dx = 2x + dx = limit_{\Delta x\to 0}{f(x+\Delta x)-f(x)\over \Delta x} = 2x.

This ratio is called the derivative.

Partial derivatives[edit]

Let

y = f(x,z)

then

\partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z
\partial y/ \partial x = \partial f(x,z)

where z is held constant and

\partial y / \partial z = \partial f(x,z)

where x is held contstant.

Gradients[edit]

Notation: let the symbol \nabla be the gradient, i.e., derivatives for multivariable functions.

\nabla f(x,z) = \partial y = \partial f(x,z) = \partial f(x,z) \partial x + \partial f(x,z) \partial z.

Area under a curve[edit]

For

\Delta x * \Delta y = [f(x + \Delta x) - f(x)] * \Delta x

the area under the curve shown in the diagram at right is the light purple rectangle plus the dark purple rectangle in the top figure

\Delta x * \Delta y + f(x) * \Delta x = f(x + \Delta x) * \Delta x.

Any particular individual rectangle for a sum of rectangular areas is

f(x_i + \Delta x_i) * \Delta x_i.

The approximate area under the curve is the sum \sum of all the individual (i) areas from i = 0 to as many as the area needed (n):

\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i.

Integrals[edit]

Notation: let the symbol \int represent the integral.

limit_{\Delta x\to 0}\sum_{i=0}^{n} f(x_i + \Delta x_i) * \Delta x_i = \int f(x)dx.

This can be within a finite interval [a,b]

\int_a^b f(x) \; dx

when i = 0 the integral is evaluated at a and i = n the integral is evaluated at b. Or, an indefinite integral (without notation on the integral symbol) as n goes to infinity and i = 0 is the integral evaluated at x = 0.

Theoretical calculus[edit]

Def. a branch of mathematics that deals with the finding and properties ... of infinitesimal differences [or changes] is called a calculus.

"Calculus [focuses] on limits, functions, derivatives, integrals, and infinite series."[9]

"Although calculus (in the sense of analysis) is usually synonymous with infinitesimal calculus, not all historical formulations have relied on infinitesimals (infinitely small numbers that are nevertheless not zero)."[10]

See also[edit]

References[edit]

  1. "universal, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. May 28, 2014. Retrieved 2014-06-04. 
  2. Bertrand Russel (1912). Chapter 9, In: The Problems of Philosophy. 
  3. Irving M. Copi (1955). Introduction to Logic. New York: The MacMillan Company. pp. 472. 
  4. Klaus Hinkelmann, Oscar Kempthorne (2008). Design and Analysis of Experiments, Volume I: Introduction to Experimental Design (2nd ed.). Wiley. ISBN 978-0-471-72756-9. http://books.google.com/?id=T3wWj2kVYZgC&printsec=frontcover. 
  5. R. A. Bailey (2008). Design of comparative experiments. Cambridge University Press. ISBN 978-0-521-68357-9. http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521683579. 
  6. "Treatment and control groups, In: Wikipedia". San Francisco, California: Wikimedia Foundation, Inc. May 18, 2012. Retrieved 2012-05-31. 
  7. "proof of concept, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. November 10, 2012. Retrieved 2013-01-13. 
  8. Ginger Lehrman and Ian B Hogue, Sarah Palmer, Cheryl Jennings, Celsa A Spina, Ann Wiegand, Alan L Landay, Robert W Coombs, Douglas D Richman, John W Mellors, John M Coffin, Ronald J Bosch, David M Margolis (August 13, 2005). "Depletion of latent HIV-1 infection in vivo: a proof-of-concept study". Lancet 366 (9485): 549-55. doi:10.1016/S0140-6736(05)67098-5. Retrieved on 2012-05-09. 
  9. "Calculus, In: Wikipedia". San Francisco, California: Wikimedia Foundation, Inc. October 13, 2012. Retrieved 2012-10-14. 
  10. "infinitesimal calculus, In: Wiktionary". San Francisco, California: Wikimedia Foundation, Inc. Setember 19, 2012. Retrieved 2013-01-31. 

Further reading[edit]

External links[edit]