if you still have difficulty, or not clear, you can see at the address listed at the top of each number.
1. Pengintegralan
from dictionary:
integral=integral
nomina: integral
adjektiva: integral
http://en.wikipedia.org/wiki/Integral
Integral
From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about the concept of integrals in calculus. For the set of numbers, see
integer. For other uses, see Integral (disambiguation).
Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the integral
is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.
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.
The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late seventeenth century. Through the fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: if ƒ is a continuous real-valued function defined on a closed interval [a, b], then, once an antiderivative F of ƒ is known, the definite integral of ƒ over that interval is given by
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. Beginning in the nineteenth century, more sophisticated notions of integral began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. 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 integral first arose from the needs of physics, and they play an important role in the formulation of many physical laws, notably those of electrodynamics. Modern concepts of integration are based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.
exp:
2. penafsiran
from dictionary
tafsir = interpretation
nomina
1. interpretation
2. construction
3. reading
4. commentation
5. commentary
penafsiran = exegesis
nomina
1. interpretation
2. treatment
3. commentation
4. exegesis
5. rede
example :
1,3 = 1
2,55555=3
3. Turunan
from dictionary
turunan = derivative
nomina
1. descendant
2. posterity
3. transcript
4. duplication
5. copy
6. bequest
7. transcription
8. derivative
9. inheritance
10. generation
http://en.wikipedia.org/wiki/Derivative
Derivative
From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about an overview of the term as used in calculus. For a non-technical overview of the subject, see Differential calculus. For other uses, see Derivative (disambiguation).
The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.
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 a quantity is changing at a given point. For example, the derivative of the position (or distance) of a vehicle with respect to time is the instantaneous velocity (respectively, instantaneous speed) at which the vehicle is traveling. Conversely, the integral of the velocity over time is the vehicle's position.
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. The fundamental theorem of calculus states that differentiation is the reverse process to integration.
Example :
The squaring function ƒ(x) = x² is differentiable at x = 3, and its derivative there is 6. This result is established by writing the difference quotient as follows:
Then we obtain the derivative by letting
The last expression shows that the difference quotient equals 6 + h when h is not zero and is undefined when h is zero. (Remember that because of the definition of the difference quotient, the difference quotient is never defined when h is zero.) However, there is a natural way of filling in a value for the difference quotient at zero, namely 6. Hence the slope of the graph of the squaring function at the point (3, 9) is 6, and so its derivative at x = 3 is ƒ '(3) = 6.
More generally, a similar computation shows that the derivative of the squaring function at x = a is ƒ '(a) = 2a.
4. Pembuktian
from dictionary
bukti = evidence
nomina
1. proof
2. evidence
3. the goods
4. substantiation
5. testimony
6. averment
7. warrant
8. token
9. witness
pembuktian=verification
nomina
1. verification
2. argumentation
5. jumlah dan suku banyak
from dictionary
jumlah dan suku banyak dari=number of tribes and many of
exp: (x2+x+2)+(x2+2x+2)= 2x2+3x+4
6. permutasi
permutasi = permutation
http://en.wikipedia.org/wiki/Permutation
Permutation
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For other uses, see Permutation (disambiguation).
In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the elements of a set to other elements of the same set, i.e., exchanging (or "permuting") elements of a set.
Definitions
The general concept of permutation can be defined more formally in different contexts:
In combinatorics
In combinatorics, a permutation is usually understood to be a sequence containing each element from a finite set once, and only once. The concept of sequence is distinct from that of a set, in that the elements of a sequence appear in some order: the sequence has a first element (unless it is empty), a second element (unless its length is less than 2), and so on. In contrast, the elements in a set have no order; {1, 2, 3} and {3, 2, 1} are different ways to denote the same set.
However, there is also a traditional more general meaning of the term "permutation" used in combinatorics. In this more general sense, permutations are those sequences in which, as before, each element occurs at most once, but not all elements of the given set need to be used.
For a related notion in which the ordering of the selected elements form a set, for which the ordering is irrelevant, see Combination.
In group theory
In group theory and related areas, the elements of a permutation need not be arranged in a linear order, or indeed in any order at all. Under this refined definition, a permutation is a bijection from a finite set onto itself. This allows for the definition of groups of permutations; see Permutation group.
Counting permutations
In this section only, the traditional definition from combinatorics is used: a permutation is an ordered sequence of elements selected from a given finite set, without repetitions, and not necessarily using all elements of the given set. For example, given the set of letters {C, E, G, I, N, R}, some permutations are ICE, RING, RICE, NICER, REIGN and CRINGE, but also RNCGI – the sequence need not spell out an existing word. ENGINE, on the other hand, is not a permutation, because it uses the elements E and N twice.
If n denotes the size of the set – the number of elements available for selection – and only permutations are considered that use all n elements, then the total number of possible permutations is equal to n!, where "!" is the factorial operator. This can be shown informally as follows. In constructing a permutation, there are n possible choices for the first element of the sequence. Once it has been chosen, n − 1 elements are left, so for the second element there are only n − 1 possible choices. For the first two elements together, that gives us
n × (n − 1) possible permutations.
For selecting the third element, there are then n − 2 elements left, giving, for the first three elements together,
n × (n − 1) × (n − 2) possible permutations.
Continuing in this way until there are only 2 elements left, there are 2 choices, giving for the number of possible permutations consisting of n − 1 elements:
n × (n − 1) × (n − 2) × ... × 2.
The last choice is now forced, as there is exactly one element left. In a formula, this is the number
n × (n − 1) × (n − 2) × ... × 2 × 1
(which is the same as before because the factor 1 does not make a difference). This number is, by definition, the same as n!.
In general the number of permutations is denoted by P(n, r), nPr, or sometimes
, , where:* n is the number of elements available for selection, and
* r is the number of elements to be selected (0 ≤ r ≤ n).
For the case where r = n it has just been shown that P(n, r) = n!. The general case is given by the formula:
As before, this can be shown informally by considering the construction of an arbitrary permutation, but this time stopping when the length r has been reached. The construction proceeds initially as above, but stops at length r. The number of possible permutations that has then been reached is:
P(n, r) = n × (n − 1) × (n − 2) × ... × (n − r + 1).
So:
n! = n × (n − 1) × (n − 2) × ... × 2 × 1
= n × (n − 1) × (n − 2) × ... × (n − r + 1) × (n − r) × ... × 2 × 1
= P(n, r) × (n − r) × ... × 2 × 1
= P(n, r) × (n − r)!.
But if n! = P(n, r) × (n − r)!, then P(n, r) = n! / (n − r)!.
For example, if there is a total of 10 elements and we are selecting a sequence of three elements from this set, then the first selection is one from 10 elements, the next one from the remaining 9, and finally from the remaining 8, giving 10 × 9 × 8 = 720. In this case, n = 10 and r = 3. Using the formula to calculate P(10,3),
In the special case where n = r the formula above simplifies to:
P(n,r) = \frac{n!}{0!} = \frac{n!}{1} = n!.
The reason why 0! = 1 is that 0! is an empty product, which always equals 1.
In the example given in the header of this article, with 6 integers {1..6}, this would be: P(6,6) = 6! / (6−6)! = (1×2×3×4×5×6) / 0! = 720 / 1 = 720.
Since it may be impractical to calculate n! if the value of n is very large, a more efficient algorithm is to calculate:
P(n, r) = n × (n − 1) × (n − 2) × ... × (n − r + 1).
Other, older notations include nPr, Pn,r, or nPr. A common modern notation is (n)r which is called a falling factorial. However, the same notation is used for the rising factorial (also called Pochhammer symbol)
n(n + 1)(n + 2)⋯(n + r − 1)r.
With the rising factorial notation, the number of permutations is (n − r + 1)r.
7. factorial
from dictionary
factorial=factorial
http://en.wikipedia.org/wiki/Factorial
Factorial
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For the experimental technique, see factorial experiment.
For factorial rings in mathematics, see unique factorization domain
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,
8. akar pangkat tiga dari
from dictionary
akar pangkat= square root
akar dari dua= square root of two
akar pangkat tiga dari = cube root of
exp:
4² = square root of 16 = 16 pangkat 1/2 = 4²
2³ = square root of 3 dari 8 = 8 pangkat 1/3 = 2³
9. pemfaktoran
from dictionary
factor = factor
pemfaktoran = factor
exp:
factor of x2+4x+4 is (x+2)(x+2)
10. aksioma
from dictionary
aksioma=axiom
http://en.wikipedia.org/wiki/Axiom
Axiom
From Wikipedia, the free encyclopedia
In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.
In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems).
Logical axioms are usually statements that are taken to be universally true (e.g., A and B implies A), while non-logical axioms (e.g., a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). When used in that sense, "axiom," "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.
Outside logic and mathematics, the term "axiom" is used loosely for any established principle of some field.
11. sejajar
from dictionary
sejajar=parallel
adjektiva
1. parallel
2. collateral
3. compatible
4. equal
5. flush
6. even
7. square
adverbia
1. on a line
2. abreast
3. in parallel
the symbol is //
12. tegak lurus
from dictionary
tegak lurus = upright
adjektiva
1. perpendicular
2. vertical
3. plumb
4. upright
13. irisan
http://www.ebimbel.co.cc/Himpunan/himpunan-kosong-himpunan-bagian-dan-himpunan-semesta.html
from dictionary
irisan=slices
nomina
1. rasher
2. shred
3. snag
4. incision
adjektiva
1. sliced
Example:
S = (1,2,3,4,5,6,7), A = (1,2,3,4,5,6) and B = (2,3,5,7)
A ∩ B = (2,3,5) is a member of the collective partnership between the A and B
Set of A and B each have the, A = B
14. himpunan semesta
from dictionary
himpunan=collective
nomina
1. community
2. assemblage
3. compilation
himpunan semesta=set of
Example:
If B = (2,4,6,8,10), then the collective might of the universe that is a set of
S = (even numbers)
S = (original numbers)
S = (integer), and so forth
15. himpunan kosong
Set theory
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics.
The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.
Set theory, formalized using first-order logic, is the most common foundational system for mathematics. The language of set theory is used in the definitions of nearly all mathematical objects, such as functions, and concepts of set theory are integrated throughout the mathematics curriculum. Elementary facts about sets and set membership can be introduced in primary school, along with Venn diagrams, to study collections of commonplace physical objects. Elementary operations such as set union and intersection can be studied in this context. More advanced concepts such as cardinality are a standard part of the undergraduate mathematics curriculum.
Beyond its use as a foundational system, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
16. penarikan kesimpulan
from dictionary
penarikan kesimpulan=conclusion drawing
17. pembagian
from dictionary:
pembagian=division
nomina
1. distribution
2. allocation
3. dealing
4. dispensation
5. partition
6. parceling
7. split
8. parcelling
9. disseverance
10. fission
11. division
12. allotment
13. classification
14. disposal
15. disposition
16. share-out
exp: 4:2=2
100:4=25
18. deret aritmatika
from dictionary
deret=array
nomina
1. row
2. line
3. rank
4. battery
5. catena
6. chain
7. string
8. train
9. tier
10. progression
deret aritmatika = arithmetic progression
Exp: 1+2+3+4+…+100
19. deret geometri
from dictionary
deret=array
nomina
1. row
2. line
3. rank
4. battery
5. catena
6. chain
7. string
8. train
9. tier
10. progression
deret geometri= geometry progression
exp: 1+2+4+8+16+…
20. induksi matematika
http://en.wikipedia.org/wiki/Mathematical_induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science.
Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics (see Problem of induction for more information). In fact, mathematical induction is a form of deductive reasoning and is rigorous.
Example
Mathematical induction can be used to prove that the statement
holds for all natural numbers n. It gives a formula for the sum of the natural numbers less than or equal to number n. The proof that the statement is true for all natural numbers n proceeds as follows.
Call this statement P(n).
Basis: Show that the statement holds for n = 0.
P(0) amounts to the statement:
In the left-hand side of the equation, the only term is 0, and so the left-hand side is simply equal to 0.
In the right-hand side of the equation, 0•(0 + 1)/2 = 0.
The two sides are equal, so the statement is true for n = 0. Thus it has been shown that P(0) holds.
Inductive step: Show that if P(n) holds, then also P(n + 1) holds. This can be done as follows.
Assume P(n) holds (for some unspecified value of n). It must be shown that then P(n + 1) holds, that is:
Using the induction hypothesis that P(n) holds, the left-hand side can be rewritten from:
to:
If you are not satisfied with the answers that I have shown you can ask to Dr.Marsigit to http://marsigitenglish.blogspot.com
Sorry if there is an error writing a word or sentence. Thanks.
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