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Calculus consists of two main topics -- differentiation and integration. Differentiation is concerned with 'rates of change', for example the rate at which the position of a moving object is changing with respect to time, otherwise known as velocity. Integration is concerning with computing areas,volumes, and lenght by first approximating the region as the sum of many smaller regions which are simpler to compute, and taking the limit as the number of smaller regions increases to infinity.
At first sight, it doesn't seem like these two topics -- differentiation and integration -- have anything to do with one another. In fact, in a calculus course either one could be presented first since it wouldn't require knowledge of the other one (traditionally, differentiation is taught first, then integration, but it isn't necessary to do them in this order.)
However, the amazing fact is that these two seemingly unrelated problems are completely intertwined. The Fundamental Theorem of Calculus, one of the most amazing and profound results in all of mathematics, spells out just how the processes of differentiation and integration are related -- they are essentially reverse operations of one another, or two sides of the same coin. The Fundamental Theorem of Calculus is so named because it ties together the two main themes of the subject. F(x) = f (t)dt is a function then we have = f(t)dt but since Since dt = h we have - f (x) = f (t) - f(x)dt . and using the continuity of F(t), we have the following equality. - f (x) = 0 .Now, the punch line!The function F(x) is differentiable and F '(x) = f (x).
Many calculus books have two parts to the FTC (Fundamental Theorem of Calculus)
Part one states that the area under a section of a curve is the antiderivative evaulated at the upper limit minus the lower limit. That is:
Integral ( f(x) dx) from a to b = F(b) - F(a)
where b is the upper boundary and a is the lower boundary
and
Part two states that the derivate of integration is the integrand:
d/dx integral (f(t) dt) from 0 to x = f(x)
where x is the upper boundary and 0 is the lower boundary.
So what went into the integral that you derive is the result.
Note: it really helps to see the pictures of what is going on.
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State and prove fundamental theorem of cyclic groups?
..?
What does the fundamental theorem of arithmetic state?
Fundamental theorem of arithmetic :- Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique . apart from the other in which factors occur.
What is a fundamental theorem of arithmetic?
The fundamental theorem of arithmetic states that every integer greater than 1 is either a prime number or can be written as a product of prime numbers. In the latter case, the prime numbers are uniquely determined apart from the order in which they appear. The theorem is also known as the unique prime factorisation theorem - for obvious reasons.
What Is The Importance Of Limits And Continuity?
Those are among the most fundamental concepts in calculus; they are used to define derivatives and integrals.
What is the 4 fundamental operation in fraction?
They are the fundamental operations of arithmetic: addition, subtraction, multiplication and division.
What Did Isaac Barrow Do?
He is responsible for the FTC, or fundamental theorem of calculus.
Who started calculus?
there was no sure answer about who started calculus but it was Isaac Newton and Gottfried Wilhelm Leibniz who founded calculus because of their fundamental theorem of calculus.
When was integral calculus invented?
Integral calculus was invented in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz.
Prove fundamental theorem of integral calculus?
The link has the answer to your question. http://www.sosmath.com/calculus/integ/integ03/integ03.html
What is the calculus therom?
The fundamental theorem of calculus is F(b)-F(a) and this allows you to plug in the variables into the integral to find the are under a graph.
How do you solve 5.4e0.06t using the fundamental theorem of calculus?
We need more information. Is there a limit or integral? The theorem states that the deivitive of an integral of a function is the function
How are the proofs of the fundamental theorem of algebra?
look in google if not there, look in wikipedia. fundamental theorem of algebra and their proofs
Why did the calculus student who didn't know the fundamental theorem of calculus fail the entire year of calculus?
I don't know the details about this particular student, but I would hazard a guess that he didn't know quite a few other things about calculus, either. In any case, if you don't know the fundamental theorem - at least, if you don't know how to apply it in practice - you'll have serious problems with many different problems - specifically when you need to do definite integrals.
What is the fundamental theory of calculus?
The fundamental theorum of calculus states that a definite integral from a to b is equivalent to the antiderivative's expression of b minus the antiderivative expression of a.
What is fundamental theorem of arithmetic in 6th grade math?
The Fundamental theorem of arithmetic states that every naturalnumber is either prime or can be uniquely written as a productof primes.
State and prove fundamental theorem of cyclic groups?
..?
What does the fundamental theorem of arithmetic state?
Fundamental theorem of arithmetic :- Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique . apart from the other in which factors occur.
