11 Lecture in math
Projects
Absolute value
Inequalities
Functions
Models
Presentations of the projects
Write your names,
student numbers,
majors, topics for the
presentations
Stretches, translations
for inequalities,
functions graphs
Simultaneous inequalities
Non-linear inequalities
Quadratic inequalities
Cubic inequalities
Functions
In mathematics, a function is a relation between a
set of inputs and a set of permissible outputs with
the property that each input is related to exactly
one output. An example is the function that relates
each real number x to its square x2. The output of a
function f corresponding to an input x is denoted by
f(x) (read "f of x"). In this example, if the input is −3,
then the output is 9, and we may write f(−3) = 9.
The input variable(s) are sometimes referred to as
the argument(s) of the function.
Functions (continued)
Functions of various kinds are "the central objects
of investigation" in most fields of modern
mathematics. There are many ways to describe or
represent a function. Some functions may be
defined by a formula or algorithm that tells how to
compute the output for a given input. Others are
given by a picture, called the graph of the function.
In science, functions are sometimes defined by a
table that gives the outputs for selected inputs. A
function could be described implicitly, for example
as the inverse to another function or as a solution
of a differential equation.
Functions (continued)
The input and output of a function can be
expressed as an ordered pair, ordered so that the
first element is the input (or tuple of inputs, if the
function takes more than one input), and the
second is the output. In the example above, f(x) =
x2, we have the ordered pair (−3, 9). If both input
and output are real numbers, this ordered pair can
be viewed as the Cartesian coordinates of a point
on the graph of the function. But no picture can
exactly define every point in an infinite set.
Functions (continued)
In modern mathematics, a function is defined by its set of inputs,
called the domain; a set containing the set of outputs, and possibly
additional elements, as members, called its codomain; and the set of
all input-output pairs, called its graph. (Sometimes the codomain is
called the function's "range", but warning: the word "range" is
sometimes used to mean, instead, specifically the set of outputs. An
unambiguous word for the latter meaning is the function's "image". To
avoid ambiguity, the words "codomain" and "image" are the preferred
language for their concepts.) For example, we could define a function
using the rule f(x) = x2 by saying that the domain and codomain are the
real numbers, and that the graph consists of all pairs of real numbers
(x, x2). Collections of functions with the same domain and the same
codomain are called function spaces, the properties of which are
studied in such mathematical disciplines as real analysis, complex
analysis, and functional analysis.
Functions (continued)
In analogy with arithmetic, it is possible to
define addition, subtraction, multiplication, and
division of functions, in those cases where the
output is a number. Another important
operation defined on functions is function
composition, where the output from one
function becomes the input to another function.
Even and odd functions
In mathematics, even functions and odd
functions are functions which satisfy particular
symmetry relations, with respect to taking
additive inverses. They are important in many
areas of mathematical analysis, especially the
theory of power series and Fourier series.
One-to-one function
A function for which every element of the range
of the function corresponds to exactly one
element of the domain.
Inverse function
In mathematics, an inverse function is a
function that "reverses" another function
Linear function
Quadratic function
Power function
Exponential function
Logarithmic function
Math modeling
A mathematical model is a description of a system using
mathematical concepts and language. The process of
developing a mathematical model is termed mathematical
modeling. Mathematical models are used not only in the
natural sciences (such as physics, biology, earth science,
meteorology) and engineering disciplines (e.g. computer
science, artificial intelligence), but also in the social sciences
(such as economics, psychology, sociology and political
science); physicists, engineers, statisticians, operations
research analysts and economists use mathematical models
most extensively. A model may help to explain a system and to
study the effects of different components, and to make
predictions about behaviour.
Math modeling (continued)
Mathematical models can take many forms, including but not
limited to dynamical systems, statistical models, differential
equations, or game theoretic models. These and other types
of models can overlap, with a given model involving a variety
of abstract structures. In general, mathematical models may
include logical models, as far as logic is taken as a part of
mathematics. In many cases, the quality of a scientific field
depends on how well the mathematical models developed on
the theoretical side agree with results of repeatable
experiments. Lack of agreement between theoretical
mathematical models and experimental measurements often
leads to important advances as better theories are developed.
Least squares
The method of least squares is a standard
approach to the approximate solution of
overdetermined systems, i.e., sets of equations
in which there are more equations than
unknowns. "Least squares" means that the
overall solution minimizes the sum of the
squares of the errors made in the results of
every single equation.
Least squares (continued)
The most important application is in data fitting.
The best fit in the least-squares sense minimizes the
sum of squared residuals, a residual being the
difference between an observed value and the
fitted value provided by a model. When the
problem has substantial uncertainties in the
independent variable (the 'x' variable), then simple
regression and least squares methods have
problems; in such cases, the methodology required
for fitting errors-in-variables models may be
considered instead of that for least squares.
Least squares (continued)
Least squares problems fall into two categories: linear or
ordinary least squares and non-linear least squares,
depending on whether or not the residuals are linear in
all unknowns. The linear least-squares problem occurs in
statistical regression analysis; it has a closed-form
solution. A closed-form solution (or closed-form
expression) is any formula that can be evaluated in a
finite number of standard operations. The non-linear
problem has no closed-form solution and is usually solved
by iterative refinement; at each iteration the system is
approximated by a linear one, and thus the core
calculation is similar in both cases.
Least squares (continued)
When the observations come from an
exponential family and mild conditions are
satisfied, least-squares estimates and maximumlikelihood estimates are identical. The method
of least squares can also be derived as a method
of moments estimator.
Least squares (continued)
The following discussion is mostly presented in
terms of linear functions but the use of leastsquares is valid and practical for more general
families of functions. Also, by iteratively
applying local quadratic approximation to the
likelihood (through the Fisher information), the
least-squares method may be used to fit a
generalized linear model.
Least squares (continued)
For the topic of approximating a function by a
sum of others using an objective function based
on squared distances, see least squares
(function approximation).
The result of fitting a set of data points with a
quadratic function.
The least-squares method is usually credited to
Carl Friedrich Gauss (1795), but it was first
published by Adrien-Marie Legendre.
Population growth model
A population model is a type of mathematical model that is applied to the study of
population dynamics.
Models allow a better understanding of how complex interactions and processes
work. Modeling of dynamic interactions in nature can provide a manageable way of
understanding how numbers change over time or in relation to each other. Ecological
population modeling is concerned with the changes in population size and age
distribution within a population as a consequence of interactions of organisms with
the physical environment, with individuals of their own species, and with organisms of
other species. The world is full of interactions that range from simple to dynamic.
Many, if not all, of Earth’s processes affect human life. The Earth’s processes are
greatly stochastic and seem chaotic to the naked eye. However, a plethora of patterns
can be noticed and are brought forth by using population modeling as a tool.
Population models are used to determine maximum harvest for agriculturists, to
understand the dynamics of biological invasions, and have numerous environmental
conservation implications. Population models are also used to understand the spread
of parasites, viruses, and disease. The realization of our dependence on environmental
health has created a need to understand the dynamic interactions of the earth’s flora
and fauna. Methods in population modeling have greatly improved our understanding
of ecology and the natural world.
Predator - pray model
The Lotka–Volterra equations, also known as
the predator–prey equations, are a pair of firstorder, non-linear, differential equations
frequently used to describe the dynamics of
biological systems in which two species interact,
one as a predator and the other as prey.
Logistic equation
A logistic function or logistic curve is a common
"S" shape (sigmoid curve)
Learning curve
A learning curve is a graphical representation of
the increase of learning (vertical axis) with
experience (horizontal axis).
Limit
In mathematics, the limit of a function is a fundamental concept in
calculus and analysis concerning the behavior of that function near a
particular input.
Informally, a function f assigns an output f(x) to every input x. We say
the function has a limit L at an input p: this means f(x) gets closer and
closer to L as x moves closer and closer to p. More specifically, when f
is applied to any input sufficiently close to p, the output value is forced
arbitrarily close to L. On the other hand, if some inputs very close to p
are taken to outputs that stay a fixed distance apart, we say the limit
does not exist.
The notion of a limit has many applications in modern calculus. In
particular, the many definitions of continuity employ the limit: roughly,
a function is continuous if all of its limits agree with the values of the
function. It also appears in the definition of the derivative: in the
calculus of one variable, this is the limiting value of the slope of secant
lines to the graph of a function.
Continuity
In mathematics, a continuous function is, roughly speaking, a function for
which small changes in the input result in small changes in the output.
Otherwise, a function is said to be a discontinuous function. A continuous
function with a continuous inverse function is called a homeomorphism.
Continuity of functions is one of the core concepts of topology, which is
treated in full generality below. The introductory portion of this article
focuses on the special case where the inputs and outputs of functions are real
numbers. In addition, this article discusses the definition for the more general
case of functions between two metric spaces. In order theory, especially in
domain theory, one considers a notion of continuity known as Scott
continuity. Other forms of continuity do exist but they are not discussed in
this article.
As an example, consider the function h(t), which describes the height of a
growing flower at time t. This function is continuous. By contrast, if M(t)
denotes the amount of money in a bank account at time t, then the function
jumps whenever money is deposited or withdrawn, so the function M(t) is
discontinuous.
Derivative
The derivative of a function of a real variable measures
the sensitivity to change of a quantity (a function or
dependent variable) which is determined by another
quantity (the independent variable). It is a fundamental
tool of calculus. For example, the derivative of the
position of a moving object with respect to time is the
object's velocity: this measures how quickly the position
of the object changes when time is advanced. The
derivative measures the instantaneous rate of change of
the function, as distinct from its average rate of change,
and is defined as the limit of the average rate of change in
the function as the length of the interval on which the
average is computed tends to zero.
Derivative (continued)
The derivative of a function at a chosen input value describes the best
linear approximation of the function near that input value. In fact, the
derivative at a point of a function of a single variable is the slope of the
tangent line to the graph of the function at that point.
The notion of derivative may be generalized to functions of several real
variables. The generalized derivative is a linear map called the
differential. Its matrix representation is the Jacobian matrix, which
reduces to the gradient vector in the case of real-valued function of
several variables.
The process of finding a derivative is called differentiation. The reverse
process is called antidifferentiation. The fundamental theorem of
calculus states that antidifferentiation is the same as integration.
Differentiation and integration constitute the two fundamental
operations in single-variable calculus.
Integral