Calculus uses methods originally based on the summation of infinitesimal differences.
It includes the examination of changes in an expression by smaller and smaller differences.
Mathematics is about numbers (counting), quantity, and coordinates.
Def. "[a]n abstract representational system used in the study of numbers, shapes, structure and change and the relationships between these concepts" is called mathematics.
Notation: let the symbol represent change in.
Notation: let the symbol represent an infinitesimal change in.
Notation: let the symbol represent an infinitesimal change in one of more than one.
Def. a result of an "operation of deducing one function from another according to some fixed law" is called a derivative.
be a function where values of may be any real number and values resulting in are also any real number.
- is a small finite change in which when put into the function produces a .
These small changes can be manipulated with the operations of arithmetic: addition (), subtraction (), multiplication (), and division ().
Dividing by and taking the limit as → 0, produces the slope of a line tangent to f(x) at the point x.
as and go towards zero,
This ratio is called the derivative.
where z is held constant and
where x is held contstant.
Notation: let the symbol be the gradient, i.e., derivatives for multivariable functions.
Area under a curve
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
Any particular individual rectangle for a sum of rectangular areas is
The approximate area under the curve is the sum of all the individual (i) areas from i = 0 to as many as the area needed (n):
Def. a "number, the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed" is called an integral.
Notation: let the symbol represent the integral.
This can be within a finite interval [a,b]
when i = 0 the integral is evaluated at and i = n the integral is evaluated at . 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.
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."
"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)."
Def. an "integral the domain of whose integrand is a curve" is called a line integral.
"The pulsar dispersion measures [(DM)] provide directly the value of
along the line of sight to the pulsar, while the interstellar Hα intensity (at high Galactic latitudes where optical extinction is minimal) is proportional to the emission measure"
- Calculus can be described using set theory.
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). In comparative experiments, members of the complementary group, the control group, receive either no treatment or a standard treatment."
Def. a “short and/or incomplete realization of a certain method or idea to demonstrate its feasibility" 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
- findings, and
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