Explanation:
Function (mathematics)
A function, in mathematics, associates one
quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as
the output. A function assigns exactly
one output to each input. The argument and the value may be real numbers, but
they can also be elements from any given set. An example of a function is f(x) = 2x, a
function which associates with every number the number twice as large. Thus 5
is associated with 10, and this is written f (5) = 10.
http://en.wikipedia.org/wiki/Function_(mathematics)
Monotonic function
In mathematics, a monotonic function (or monotone function) is a function that preserves the given order. This concept first arose
in calculus, and was later generalized to the more abstract setting of order theory.
Monotonicity
in calculus and analysis
Figure
1. A monotonically increasing function (it is strictly increasing on the left
and right while just non-decreasing in the middle.)
Figure
2. A monotonically decreasing function.
In calculus, a function f defined
on a subset of the real numbers with real values is called monotonic (also monotonically
increasing,
increasing or non-decreasing), if
for all x and y such that x ≤ y one has f(x)
≤ f(y),
so f preserves the order (see Figure 1). Likewise, a function is called monotonically
decreasing (also decreasing or non-increasing) if, whenever x ≤ y, then f(x) ≥ f(y),
so it reverses the order (see Figure 2).
If the order ≤ in the definition of Monotonicity is replaced
by the strict order <, then one obtains a stronger requirement. A function
with this property is called strictly increasing.
Again, by inverting the order symbol, one finds a corresponding concept called strictly
decreasing.
Functions that are strictly increasing or decreasing are one-to-one (because
for x not equal to y,
either x < y or x > y and so, by Monotonicity, either f(x) < f(y)
or f(x)
> f(y),
thus f(x)
is not equal to f(y)).
The terms "non-decreasing" and
"non-increasing" are meant to avoid confusion with "strictly
increasing" and "strictly decreasing", respectively, but should
not be confused with the (much weaker) negative qualifications "not
decreasing" and "not increasing"; see also strict. When functions between
discrete sets are considered in combinatorics,
it is not always obvious that "increasing" and "decreasing"
are taken to include the possibility of repeating the same value at successive arguments,
so one finds the terms weakly
increasing and weakly decreasing to stress this possibility.
The term monotonic
transformation can also
possibly cause some confusion because it refers to a transformation by a strictly increasing function.
Notably, this is the case in economics with respect to the ordinal properties
of a utility function being preserved across a monotonic
transform (see also monotone
preferences)
A function f(x)
is said to be absolutely
monotonic over an interval (a, b) if the derivatives of all
orders of f are nonnegative at all points on the interval.
http://en.wikipedia.org/wiki/Monotonic_function
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes.
Loosely speaking, a derivative can be thought of as how much one quantity is
changing in response to changes in some other quantity; for example, the
derivative of the position of a moving object with respect to time is the
object's instantaneous velocity (conversely, integrating a car's velocity over time yields the
distance traveled).
The derivative
of a function at a chosen input value describes the best linear approximation of the function near that input
value. For a real-valued function of a single real variable, the
derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher
dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely
related notion is the differential of a function.
The process of
finding a derivative is called differentiation.
Computing
the derivative
The derivative of a function can, in principle, be computed from
the definition by considering the difference quotient, and computing its limit.
In practice, once the derivatives of a few simple functions are known, the
derivatives of other functions are more easily computed using rules for obtaining
derivatives of more complicated functions from simpler ones.
Derivatives of elementary
functions
Most
derivative computations eventually require taking the derivative of some common
functions. The following incomplete list gives some of the most frequently used
functions of a single real variable and their derivatives.
•
Derivatives of powers: if
�
where r is
any real number, then
wherever this function is
defined. For example, if f(x)
= x^{1 / 4}, then
and
the derivative function is defined only for positive x, not for x = 0. When r = 0, this rule implies that f′(x) is zero for x ≠
0, which is almost the constant rule.
• Constant
rule: if ƒ(x) is constant, then
(Our solved example in mathguru.com uses this concept)
http://en.wikipedia.org/wiki/Derivative
The
above explanation is copied from Wikipedia, the free encyclopedia and is
remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.
(Our solved example in mathguru.com also uses the below concept.
This is our own explanation, it is not taken from Wikipedia.)
A
function is said to be strictly increasing (or decreasing) in any open interval
(a, b) if its derivative is positive (or negative).