Math Courses for National Exam

Exam preparation and performance can be time consuming and confusing if you don't have the right skills, techniques and tips for doing well. With simple exam preparation skills, by the time you enter the exam, you will be in a position to excel and get the best grade you are capable of. Knowing what is important, revising on relevant topics and doing practice exams are all fundamentally important study tips for exams.



Download Math Courses for National Exam

• Math Courses for Elementary School - SD/MI

Know criteria
This means knowing what the key areas of assessment are and the proportion of marks for each question. Look at your course guide and talk to your teachers to find out this information, so as to allocate the right amount of time in the exam and for exam preparation. From this information you should be able to map out what you need to be focusing on and which questions needs more time. Knowing the criteria and study questions will give you a basic orientation to excel in college exams.

Revise
It is a good idea to start revision at least two weeks before the examination. Participate in study groups to better understand what is required and revise the key areas more effectively. Learn specific areas to be assessed to teach to your study group. This will minimize the time you spend revising and orient you in the best way for the exam. Meet with your study group as often as you can.... once every 3-4 days is a good start but it does depend on your learning style and study preferences.
Create a cheat sheet approximately 1-1.5 pages in length, so as get the bare essentials of the examination area. Remember to be concise in preparing your cheat sheet. It should only jog your memory and orient you in study. How you design and adapt your cheat sheet should depend on your study style.

Practice
Practice exams are vital in making the best of your ability and revision sessions. Do practice exams in real time so as to practice under the time and stress conditions of the actual event. Be sure to go over your practice exams to review areas of concern.

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Math Course for Elementary School - UN SD

In addition to helping students achieve great success, elementary schools have to help keep them from getting lost in the shuffle. It's not unusual at all for several students in one classroom to experience trouble with mathematics. That is why nearly every school has an elementary math intervention program in place. There are many instructional ways to address a student's difficulties with math, and Response to Intervention efforts are excellent options. However, many educators are turning to well-designed online math games to make the process more streamlined, effective and efficient. Read below to find out how these special math games are bolstering educators' efforts to help struggling math students.



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Download Mathematical Theory for the National Exam

UN SMA

1. Download Theory of Mathematics for IPS Program
2. Download Theory of Mathematics for IPA Program

UN SMK

1. Download Theory for SMK Non Teknik
2. Download Theory for SMK Teknik

UN SMP
Download Theory of Matematika SMP Exam 2011 according to grid UN 2010/2011

UN SD

1. A Brief Theory of Mathematics UN SD
2. Content Guide Mathematics UN SD
3. Problem Specification Mathematics UN SD
4. Simulation Problem Mathematics UN SD
5. Summary Material Mathematics UN SD

Differentiated Instruction
It's been a known fact for some time that certain teaching methods just don't click with certain students. How is a teacher supposed to meet the specialized needs of twenty or thirty different students, though? Furthermore, how can an elementary math intervention program be effective if it doesn't address a student's unique problems and needs? One way to get around the problem is by turning to specialized online math games. The best games offer differentiated instruction that assesses a student's weaknesses and strengths and adapts game play accordingly. These games have been shown to make a profound difference for students who are having a hard time with mathematics.

The Benefits of Adaptive Learning
In addition to assessing a student's strengths and weaknesses, a high-quality online math game will adapt according to those things. If a student is experiencing problems with a specific mathematical concept, for example, this type of game will present problems and scenarios that improve confidence and provide alternative ways for a student to arrive at the right answer. If one method doesn't work, this type of game can present different options for the student to consider.

Virtual Manipulatives
At the elementary-school level, students often struggle with understanding basic math concepts. Well-designed online math games incorporate virtual manipulatives that can make a major different. In this situation, a student is able to virtually manipulate objects to form a clearer understanding of a basic math concept. This approach is recommended for a successful elementary math intervention program, but it isn't always easy to achieve in a classroom setting. The right game can provide a unique opportunity to teach a student in a whole new way.
The goal of a well-executed elementary math program is to prevent students from failing or from falling too far behind in their mathematical studies. If students are allowed to move up to higher grades without gaining a clear understanding of math fundamentals, they could be in for trouble throughout the rest of their school careers. There are many advantages to including online math games in the classroom. They don't have to be strictly used for struggling students, but they definitely provide unique benefits in that regard. Jim Wheelin writes about many different educational topics for parents, kids and teachers. He has helped many struggling students to find the best elementary math intervention program to meet their needs. Jim also encourages other to test educational games such as the ones offered at http://www.dreambox.com/math-intervention.

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Bank Soal Matematika Ujian Nasional

Mathematics courses at the National Examination is usually regarded as the most difficult subjects. Generally, the subjects of Mathematics has always been the most "scary" for some school students, does it in the UN? Apparently not the case for most other students. There are some students who put the Indonesian Language is more difficult than mathematics. Indeed, Indonesian Language is sometimes considered to be easy it would be able to make the most of the hassles of high school students while working on it.


In this mathematical blog collection free math problems for you. About this collection is a collection of mathematical problems of the National Exam that will be useful for the preparation of your National Exam, we give some solution.

Download Bank Soal Matematika Ujian Nasional

• Download Soal Matematika UN SMA
• Download Soal Matematika UN SMP
• Download Soal Matematika UN SMK
• Download Soal Matematika UN SD

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Download Soal Matematika UN SD

National Standardization Agency of Education (BSNP) Teuku Ramli Zakaria asserts, there was no change in the schedule of the National Exam 2012. Implementation of national primary school exams (UN-SD) remains on the schedule already decided which will be held on 7-9 May 2012, and subsequent examination will be held on 14-16 May 2012. While the subjects tested there was no change from previous years that Indonesian, Math, and Science (IPA). For the preparation of the National Examination SD / MI students / i where ever you are, we provide one of the Math assessment models made ​​by the teachers / instructors who are competent in their field of mathematics. Thank you, may you succeed the National Exam! Amen!


Soal Matematika UN SD/MI 2010

Paket 1 : View | Download

Paket 2 : View | Download




Soal Matematika UN SD/MI 2011

View | Download

Note: the "view" you can read online without having to download it. If you download it make sure there is a program on your computer Adobe Acrobat Reader, if you do not have it you can download it here: Download Adobe Reader X (10.1.1) (64.69 MB)

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Download Soal Matematika UN SMK

National Standardization Agency of Education (BSNP) Teuku Ramli Zakaria asserts, there was no change in the schedule of the National Exam 2012. Implementation of the national vocational exams were still on the schedule that has been decided that the UN was held at SMK will begin April 16 to 18, 2012, and subsequent examination will be held on 23-25 ​​April 2012. UN to the SMA / MA will begin to be held on April 16 to 19, 2012, and subsequent examination will be held on April 23 to 26. Exam preparation for the National Vocational students / i where ever you are, we provide one of the Math assessment models made ​​by the teachers / instructors who are competent in their field of mathematics. Thank you, may you succeed the National Exam! Amen!


Soal Matematika UN SMK 2010

View | Download



Note: the "view" you can read online without having to download it. If you download it make sure there is a program on your computer Adobe Acrobat Reader, if you do not have it you can download it here: Download Adobe Reader X (10.1.1) (64.69 MB)

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Download Soal Matematika UN SMP

National Standardization Agency of Education (BSNP) Teuku Ramli Zakaria asserts, there was no change in the schedule of the National Exam 2012. SMP implementation of national examinations was still on the schedule that has been decided that the UN will be held on April 23 to 26, 2012, and subsequent examination will take place on 30 to 4 May 2012. As for the level of SD / MI / SDLB UN will be held on 7-9 May 2012, and subsequent examination will be held on 14-16 May 2012. UN to the level of SMA / MA. For the National Exam preparation SMP / MTs students / i where ever you are, we provide one of the Math assessment models made ​​by the teachers / instructors who are competent in their field of mathematics. Thank you, may you succeed the National Exam! Amen!


Soal Matematika UN SMP/MTs 2010

Paket A : View | Download

Paket B : View | Download




Soal Matematika UN SMP/MTs 2011

View | Download

Note: the "view" you can read online without having to download it. If you download it make sure there is a program on your computer Adobe Acrobat Reader, if you do not have it you can download it here: Download Adobe Reader X (10.1.1) (64.69 MB)

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Download Soal Matematika UN SMA

National Standardization Agency of Education (BSNP) Teuku Ramli Zakaria asserts, there was no change in the schedule of the National Exam 2012. Implementation of national examinations remain on the schedule that has been decided that for the UN SMA / MA will begin to be held on April 16 to 19, 2012, and subsequent examination will be held on April 23 to 26. UN to the CMS will begin to be held on April 16 to 18, 2012, and subsequent examination will be held on 23-25 ​​April. For the National Exam preparation SMA / MA students / i where ever you are, we provide one of the Math assessment models made ​​by the teacher / instructor of mathematics who are competent in their field. Thank you, may you succeed the National Exam! Amen!

Soal Matematika UN SMA/MA 2010
Program IPA : View | Download
Program IPS : View | Download

Soal Matematika UN SMA/MA 2011
Program IPA : View | Download
Program IPS : View | Download

Note: the "view" you can read online without having to download it. If you download it make sure there is a program on your computer Adobe Acrobat Reader, if you do not have it you can download it here: Download Adobe Reader X (10.1.1) (64.69 MB)

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Finding Math Courses Online

The Internet has revolutionized almost every aspect of our lives and made education easily accessible online. There are different types of online courses available that vary from under-graduate to post-graduate but that is not all. Mathematics is one such subject that has moved from traditional classroom to online. Finding online maths courses is quite easy these days especially because of the many universities and private institutions offering the same.
One of the biggest advantages of choosing an online maths course is the convenience. Let's take a look at some of the institutions offering online maths courses:

Massachusetts Institute of Technology (mit.edu)

MIT, one of the most prestigious educational institutions of the world offers:

o Algebra I Course
o Introductory Geometry Course
o Probability and Statistics Course

Apart from the above courses, MIT also has almost 100 free online math courses made available to everyone and anyone through their innovative OpenCourseWare project. Under the OpenCourseWare project, you will get to study using slides, lectures, assignments, problem sets, and various other resources that will mainly assist self-study. Above all, you don't need to register for these courses.

University of Massachusetts, Boston (umb.edu)

Another great education institution, UMB offers three main online maths courses:

o Calculus I
o Calculus II

UMB offers 3 free online calculus courses that are bundled with comprehensive instructions. You can download them from their website and it will include suggested problems, multiple lessons, reading assignments and other resources. Above all, you don't need to register for these courses.

College of Eastern Utah (usu.edu)

You can get two standard online maths courses:

o Intermediate Algebra Course
o Quantitative Reasoning Course

The College of Eastern Utah's Mathematics Department has opened its doors to provide access to anyone who is interested in their two different math courses and these can be downloaded for free. You need to have RealPlayer for the Quantitative Reading Course and above all, you don't need to register for these courses.

Dixie State College of Utah (usu.edu)

This Utah College offers two online calculus courses, which are:

o Calculus I Course
o Calculus II Course

Dixie State College of Utah had recently started a pilot program, which provides free downloads of two excellent online math tutorials or courses. The course material includes exams, notes, and various important subject related resources. You don't need to register for any of these courses.

Carnegie Mellon University (cmu.edu)

One of the top universities in the US, Carnegie Melon offers a single online maths course called:

o Statistics Course

The free Statistics Course is available for download and study through Carnegie Mellon's Open Learning Initiative and can be compared to their full semester course that is currently being taught on their University campus. The course material will include virtual simulations, tests, text, interactive learning exercises, and other resources.

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Math Exams

When do you start preparing for school entrance exams? Structured preparation for one year before the exams is a good idea. Carefully paced preparation over a 12 month period should be sufficient, without running the risk of your child becoming bored or 'over prepared'.
Opinions vary on whether children should be intensively coached to maximise their performance in entrance exams. Some favour the view that the selection process should be based on 'natural ability', and others believe that every available resource should be applied to ensure an offer from a first choice school. The majority of parents' opinions fall somewhere between the two extremes, feeling that at least some degree of preparation is both appropriate and beneficial for their children, in the period leading up to entrance exams. At the end of the day, everyone should make an informed decision about what is right for their child, and their particular circumstances.
Children who are already in the independent school sector will more than likely receive suitable preparation at school.
Find out the format of the exams for the particular school you are applying to
Obtain specimen papers or past papers from the school, if they are available
Otherwise, obtain suitable practice papers and/or workbooks for your child to work through
Ensure your child is familiar with the vocabulary used in the specimen papers, past papers and practice papers
Make sure that your child practices with a wide variety of question types
Make sure your child understands the difference between short questions where only an answer is needed, and long questions where makes are awarded for method and workings, not just a final answer
Discuss timing with your child. If a paper is one hour in length and there is a maximum of 100 marks available, that means about 36 seconds per mark as a rough guide
Encourage your child to take extra care with spelling and punctuation
Word games, puzzles and logic problems can be a fun way to practice logical thinking and broaden vocabulary
Mental arithmetic skills are extremely useful. Children should know tables up to 12 reliably, and be able to add and subtract at least 3 digit numbers quickly and accurately.
Whatever style of preparation you opt for, the key to making the exams and the selection process less stressful is for both you and your child to know what to expect. Get as much information from the school as you can, in plenty of time. Many independent schools now require applicants to register and pay an administration fee at least one year before planned entry. The school will normally provide quite a detailed description of how an assessment day or exam day is structured. Claudine M Smith is a Maths tutor with over 12 years experience of tutoring children for common entrance and independent school entrance exams, as well as KS2 SAT, GCSE, IGCSE and A Levels. For more information and advice about preparing for entrance exams and Maths tuition you can visit http://witneymathstutors.co.uk

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Quadratic equations Exam

In mathematics, a quadratic equation is a polynomial equation of the second degree.

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Download Theory of Mathematics for School Exam

Download for Vocational School - Technology Program
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The constants a, b, and c, are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" comes from quadratus, which is the Latin word for "square". Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula (given below).
One common use of quadratic equations is computing trajectories in projectile motion. Another common use is in electronic amplifier design for control of step response and stability.
Quadratic formula
A quadratic equation with real or complex coefficients has two solutions, called roots.

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Quadratic functions Exam

A quadratic function, in mathematics, is a polynomial function. The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis.
The expression ax2 + bx + c in the definition of a quadratic function is a polynomial of degree 2 or second order, or a 2nd degree polynomial, because the highest exponent of x is 2.
If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the equation.

Origin of word

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The adjective quadratic comes from the Latin word quadratum for square. A term like x2 is called a square in algebra because it is the area of a square with side x.
In general, a prefix quadr(i)- indicates the number 4. Examples are quadrilateral and quadrant. Quadratum is the Latin word for square because a square has four sides.

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Sequences and series Exam

In mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function.

For example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,...). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.

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A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.

In mathematics, given an infinite sequence of numbers { an }, a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·. These can be written more compactly using the summation symbol ∑. An example is the famous series from Zeno's dichotomy

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Logarithms Exam

The logarithm of a product of two numbers equals the sum of their logarithms:

This equation forms the basis for multiplying two numbers on a slide rule or using a logarithm table. Since adding is generally easier than multiplying, this led to the rapid adoption of logarithms for calculations after their invention by John Napier in the early 17th century. For such computational purposes, the logarithm to base b = 10 (common logarithm) was primarily used. The natural logarithm uses the constant e (approximately 2.718) as its base, and is especially widespread in calculus. The binary logarithm uses base b = 2 and primarily aids computing applications.

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Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the Richter scale uses the common logarithm to measure the amplitude of seismic events. Logarithms are commonplace in scientific formulas, measure the complexity of algorithms and of fractals, and appear in formulas counting prime numbers. They describe musical intervals, inform some models in psychophysics and can aid in forensic accounting.
The complex logarithm is the inverse of the exponential function applied to complex numbers and generalizes the logarithm to complex numbers. The discrete logarithm is another variant; it has applications in public-key cryptography.

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Three-dimensional Exam

Three-dimensional space is a geometric model of the physical universe in which we live. The three dimensions are commonly called length, width, and depth (or height), although any three directions can be chosen, provided that they do not lie in the same plane.
In physics, our three-dimensional space is viewed as embedded in 4-dimensional space-time, called Minkowski space (see special relativity). The idea behind space-time is that time is hyperbolic-orthogonal to each of the three spatial dimensions.
In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, usually each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled x, y, and z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.
Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods.

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Another mathematical way of viewing three-dimensional space is found in linear algebra, where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three dimensional because every point in space can be described by a linear combination of three independent vectors. In this view, space-time is four dimensional because the location of a point in time is independent of its location in space.
Three-dimensional space has a number of properties that distinguish it from spaces of other dimension numbers. For example, at least 3 dimensions are required to tie a knot in a piece of string. The understanding of three-dimensional space in humans is thought to be learned during infancy using unconscious inference, and is closely related to hand-eye coordination. The visual ability to perceive the world in three dimensions is called depth perception.
In mathematics, a relation is used to describe certain properties of things. That way, certain things may be connected in some way; this is called a relation. Formally, a relation is a set of n-tuples of equal degree. Thus a binary relation is a set of pairs, a ternary relation a set of 3-tuples, and so forth. A ternary relation however is always expressable as two binary relations. Specifically in the context of functions, this is known as currying.
Particularly concerning binary relations, the set of all the starting point is called the domain and the sets of the ending points is the range. The domain is the x's , and the range is the y's.
An example for such a relation might be a function. Functions associate keys with values. The set of all functions is a subset of the set of all relations - a function is a relation where the first value of every tuple is unique through the set.
Other well-known relations are the Equivalence relation and the Order relation. That way, sets of things can be ordered: Take the first element of a set, it is either equal to the element looked for, or there is an order relation that can be used to classify it. That way, the whole set can be classified (compared to some arbitrarily chosen element).
Relations can be transitive. One example of a transitive relation is "smaller-than". If X "is smaller than" Y, and Y is "smaller than" Z, then X "is smaller than" Z
Relations can be symmetric. One example of a symmetric relation is "is equal to".
Relations can be reflexive.
A reflexive relation is "smaller than or equal".

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Derivative Exam

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, the integral of the object's velocity over time is how much the object's position changes from the time when the integral begins to the time when the integral ends.

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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.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.

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Statistics Exam

Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis. The term "mathematical statistics" is closely related to the term "statistical theory" but also embraces modelling for actuarial science and non-statistical probability theory, particularly in Scandinavia.
Statistics deals with gaining information from data. In practice, data often contain some randomness or uncertainty. Statistics handles such data using methods of probability theory.
Statistical science is concerned with the planning of studies, especially with the design of randomized experiments and with the planning of surveys using random sampling. The initial analysis of the data from properly randomized studies often follows the study protocol.
Of course, the data from a randomized study can be analyzed to consider secondary hypotheses or to suggest new ideas. A secondary analysis of the data from a planned study uses tools from data analysis.

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Data analysis is divided into:
descriptive statistics - the part of statistics that describes data, i.e. summarises the data and their typical properties.
inferential statistics - the part of statistics that draws conclusions from data (using some model for the data): For example, inferential statistics involves selecting a model for the data, checking whether the data fulfill the conditions of a particular model, and with quantifying the involved uncertainty (e.g. using confidence intervals).
While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data --- for example, from natural experiments and observational studies, in which case the inference is dependent on the model chosen by the statistician, and so subjective.
Mathematical statistics has been inspired by and has extended many procedures in applied statistics.

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Probability Exam

Probability is a way of expressing knowledge or belief that an event will occur or has occurred. The concept has an exact mathematical meaning in probability theory, which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, Artificial intelligence/Machine learning and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.

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Interpretations
The word probability does not have a consistent direct definition. In fact, there are two broad categories of probability interpretations, whose adherents possess different (and sometimes conflicting) views about the fundamental nature of probability:
Frequentists talk about probabilities only when dealing with experiments that are random and well-defined. The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency "in the long run" of outcomes.
Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved. Probability, for a Bayesian, is a way to represent an individual's degree of belief in a statement, or an objective degree of rational belief, given the evidence.

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Relation and Function Exam

In mathematics, a relation is used to describe certain properties of things. That way, certain things may be connected in some way; this is called a relation. It is clear, that things are either related, or they are not, there are no in-betweens.

Formally, a relation is a set of n-tuples of equal degree. Thus a binary relation is a set of pairs, a ternary relation a set of 3-tuples, and so forth. A ternary relation however is always expressable as two binary relations. Specifically in the context of functions, this is known as currying.

Particularly concerning binary relations, the set of all the starting point is called the domain and the sets of the ending points is the range. The domain is the x's , and the range is the y's.

An example for such a relation might be a function. Functions associate keys with values. The set of all functions is a subset of the set of all relations - a function is a relation where the first value of every tuple is unique through the set.

Other well-known relations are the Equivalence relation and the Order relation. That way, sets of things can be ordered: Take the first element of a set, it is either equal to the element looked for, or there is an order relation that can be used to classify it. That way, the whole set can be classified (compared to some arbitrarily chosen element).

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A function, in a mathematical sense, expresses the idea that one quantity (the argument of the function, also known as the input) completely determines another quantity (the value, or the output). A function assigns exactly one value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain of the function. An example of a function with the real numbers as both its domain and codomain is the function f(x) = 2x, which assigns to every real number the real number with twice its value. In this case, it is written that f(5) = 10.

In addition to elementary functions on numbers, functions include maps between algebraic structures like groups and maps between geometric objects like manifolds. In the abstract set-theoretic approach, a function is a relation between the domain and the codomain that associates each element in the domain with exactly one element in the codomain. An example of a function with domain {A,B,C} and codomain {1,2,3} associates A with 1, B with 2, and C with 3.

There are many ways to describe or represent functions: by a formula, by an algorithm that computes it, by a plot or a graph. A table of values is a common way to specify a function in statistics, physics, chemistry, and other sciences. A function may also be described through its relationship to other functions, for example, as the inverse function or a solution of a differential equation. There are uncountably many different functions from the set of natural numbers to itself, most of which cannot be expressed with a formula or an algorithm. Functions with numerical outputs may be added and multiplied, yielding new functions. Collections of functions with certain properties, such as continuous functions and differentiable functions, usually required to be closed under certain operations, are called function spaces and are studied as objects in their own right, in such disciplines as real analysis and complex analysis. An important operation on functions, which distinguishes them from numbers, is the composition of functions.

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Matrix Exam

In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, such as

An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 6. Entries are often denoted by a variable with two subscripts, as shown on the right; thus in the example above, a2,1 = 20. Matrices of the same size can be added and subtracted entrywise and matrices of compatible sizes can be multiplied. These operations have many of the properties of ordinary arithmetic, except that matrix multiplication is not commutative, that is, AB and BA are not equal in general. Matrices consisting of only one column or row define the components of vectors, while higher-dimensional (e.g., three-dimensional) arrays of numbers define the components of a generalization of a vector called a tensor. Matrices with entries in other fields or rings are also studied.
Matrices are a key tool in linear algebra. One use of matrices is to represent linear transformations, which are higher-dimensional analogs of linear functions of the form f(x) = cx, where c is a constant; matrix multiplication corresponds to composition of linear transformations. Matrices can also keep track of the coefficients in a system of linear equations. For a square matrix, the determinant and inverse matrix (when it exists) govern the behavior of solutions to the corresponding system of linear equations, and eigenvalues and eigenvectors provide insight into the geometry of the associated linear transformation.

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Matrices find many applications. Physics makes use of matrices in various domains, for example in geometrical optics and matrix mechanics; the latter led to studying in more detail matrices with an infinite number of rows and columns. Graph theory uses matrices to keep track of distances between pairs of vertices in a graph. Computer graphics uses matrices to project 3-dimensional space onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives of functions or exponentials to matrices. The latter is a recurring need in solving ordinary differential equations. Serialism and dodecaphonism are musical movements of the 20th century that use a square mathematical matrix to determine the pattern of music intervals.
A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old but still an active area of research. Matrix decomposition methods simplify computations, both theoretically and practically. For sparse matrices, specifically tailored algorithms can provide speedups; such matrices arise in the finite element method, for example.

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Integral Exam

Integral is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral.

The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function ƒ. In this case, it is called an indefinite integral, while the integrals discussed in this article are termed definite integrals. Some authors maintain a distinction between antiderivatives and indefinite integrals.

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Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration [a, b] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. Integrals of differential forms play a fundamental role in modern differential geometry. These generalizations of integrals first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics. There are many modern concepts of integration, among these, the most common is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.

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