Fundamental theorem of calculus- Thanks to wikipedia
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Fundamental theorem Limits of functions Continuity Mean value theorem
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The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration[1] can be reversed by a differentiation. The first part is also important because it guarantees the existence of antiderivatives for continuous functions.[2]
The second part, sometimes called the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals.
The first published statement and proof of a restricted version of the fundamental theorem was by James Gregory (1638–1675).[3] Isaac Barrow (1630–1677) proved the first completely general version of the theorem, while Barrow's student Isaac Newton (1643–1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities.
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[edit] Physical intuition
Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time (or some other quantity) add up to the net change in the quantity.In the case of a particle traveling in a straight line, its position, x, is given by x(t) where t is time and x(t) means that x is a function of t. The derivative of this function is equal to the infinitesimal change in quantity, dx, per infinitesimal change in time, dt (of course, the derivative itself is dependent on time). This change in displacement per change in time is the velocity v of the particle. In Leibniz's notation:
[edit] Geometric intuition
For a continuous function y = ƒ(x) whose graph is plotted as a curve, each value of x has a corresponding area function A(x), representing the area beneath the curve between 0 and x. The function A(x) may not be defined, but it is given that it represents the area under the curve.The area under the curve between x and x + h could be computed by finding the area between 0 and x + h, then subtracting the area between 0 and x. In other words, the area of this “sliver” would be A(x + h) − A(x).
There is another way to estimate the area of this same sliver. h is multiplied by ƒ(x) to find the area of a rectangle that is approximately the same size as this sliver. It is intuitive that the approximation improves as h grows smaller.
At this point, it is true A(x + h) − A(x) is approximately equal to ƒ(x)·h. In other words, ƒ(x)·h ≈ A(x + h) − A(x), with this approximation becoming an equality as h approaches 0 in the limit.
When both sides of the equation are divided by h:
It can thus be shown, in an informal way, that ƒ(x) = A’(x). That is, the derivative of the area function A(x) is the original function ƒ(x); or, the area function is simply the antiderivative of the original function.
Computing the derivative of a function and “finding the area” under its curve are "opposite" operations. This is the crux of the Fundamental Theorem of Calculus. Most of the theorem's proof is devoted to showing that the area function A(x) exists in the first place.
[edit] Formal statements
There are two parts to the Fundamental Theorem of Calculus. Loosely put, the first part deals with the derivative of an antiderivative, and the second part deals with the relationship between antiderivatives and definite integrals.[edit] First part
This part of the theorem is sometimes referred to as the First Fundamental Theorem of Calculus.[4]A real-valued function F is defined on a closed interval [a, b] by setting, for all x in [a, b],
[edit] Proof
For a given f(t), define the function F(x) as- Notice that the expression on the left side of the equation is Newton's difference quotient for F at x1.
Also, and
Therefore, according to the squeeze theorem,
(Leithold et al., 1996)
[edit] Corollary
The fundamental theorem is often employed to compute the definite integral of a function ƒ for which an antiderivative F is known. Specifically, if ƒ is a real-valued continuous function on [a, b], and F is an antiderivative of ƒ in [a, b], then[edit] Proof
Let with ƒ continuous on [a, b]. If g is an antiderivative of ƒ, then g and F have the same derivative, by the first part of the theorem. It follows that there is a number c such that F(x) = g(x) + c, for all x in [a, b]. Letting x = a,[edit] Second part
This part is sometimes referred to as the Second Fundamental Theorem of Calculus[5] or the Newton-Leibniz Axiom.Let ƒ be a real-valued function defined on a closed interval [a, b] that admits an antiderivative F on [a, b]. That is, ƒ and F are functions such that for all x in [a, b],
When an antiderivative F exists, there are infinitely many antiderivatives for ƒ, obtained by adding an arbitrary constant to F (which is lost in differentiation). Also, by the first part of the theorem, antiderivatives of ƒ always exist when ƒ is continuous.
[edit] Proof
This is a limit proof by Riemann sums. Let ƒ be (Riemann) integrable on the interval [a, b], and let ƒ admit an antiderivative F on [a, b]. Begin with the quantity F(b) − F(a). Let there be numbers x1, ..., xn such thatLet F be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). Then there exists some c in (a, b) such that
By taking the limit of the expression as the norm of the partitions approaches zero, we arrive at the Riemann integral. We know that this limit exists because ƒ was assumed to be integrable. That is, we take the limit as the largest of the partitions approaches zero in size, so that all other partitions are smaller and the number of partitions approaches infinity.
So, we take the limit on both sides of (2). This gives us
It almost looks like the first part of the theorem follows directly from the second, because the equation where g is an antiderivative of ƒ, implies that has the same derivative as g, and therefore F ′ = ƒ. This argument only works if we already know that ƒ has an antiderivative, and the only way we know that all continuous functions have antiderivatives is by the first part of the Fundamental Theorem[6]. For example if ƒ(x) = e−x2, then ƒ has an antiderivative, namely
[edit] Examples
As an example, suppose the following is to be calculated:[edit] Generalizations
We don't need to assume continuity of ƒ on the whole interval. Part I of the theorem then says: if ƒ is any Lebesgue integrable function on [a, b] and x0 is a number in [a, b] such that ƒ is continuous at x0, thenIn higher dimensions Lebesgue's differentiation theorem generalizes the Fundamental theorem of calculus by stating that for almost every x, the average value of a function ƒ over a ball of radius r centered at x will tend to ƒ(x) as r tends to 0.
Part II of the theorem is true for any Lebesgue integrable function ƒ which has an antiderivative F (not all integrable functions do, though). In other words, if a real function F on [a, b] admits a derivative ƒ(x) at every point x of [a, b] and if this derivative ƒ is Lebesgue integrable on [a, b], then
- Rudin (1987, th. 7.21)
The conditions of this theorem may again be relaxed by considering the integrals involved as Henstock-Kurzweil integrals. Specifically, if a continuous function F(x) admits a derivative ƒ(x) at all but countably many points, then ƒ(x) is Henstock-Kurzweil integrable and F(b) − F(a) is equal to the integral of ƒ on [a, b]. The difference here is that the integrability of ƒ does not need to be assumed. (Bartle 2001, Thm. 4.7)
The version of Taylor's theorem which expresses the error term as an integral can be seen as a generalization of the Fundamental Theorem.
There is a version of the theorem for complex functions: suppose U is an open set in C and ƒ : U → C is a function which has a holomorphic antiderivative F on U. Then for every curve γ : [a, b] → U, the curve integral can be computed as
One of the most powerful statements in this direction is Stokes' theorem: Let M be an oriented piecewise smooth manifold of dimension n and let ω be an n−1 form that is a compactly supported differential form on M of class C1. If ∂M denotes the boundary of M with its induced orientation, then
The theorem is often used in situations where M is an embedded oriented submanifold of some bigger manifold on which the form ω is defined.
[edit] See also
[edit] Notes
- ^ More exactly, the theorem deals with definite integration with variable upper limit and arbitrarily selected lower limit. This particular kind of definite integration allows us to compute one of the infinitely many antiderivatives of a function (except for those which do not have a zero). Hence, it is almost equivalent to indefinite integration, defined by most authors as an operation which yields any one of the possible antiderivatives of a function, including those without a zero.
- ^ Spivak, Michael (1980), Calculus (2nd ed.), Houstan, Texas: Publish or Perish Inc.
- ^ See, e.g., Marlow Anderson, Victor J. Katz, Robin J. Wilson, Sherlock Holmes in Babylon and Other Tales of Mathematical History, Mathematical Association of America, 2004, p. 114.
- ^ Apostol 1967, §5.1
- ^ Apostol 1967, §5.3
- ^ Spivak, Michael (1980), Calculus (2nd ed.), Houston, Texas: Publish or Perish Inc.
[edit] References
- Apostol, Tom M. (1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-00005-1.
- Bartle, Robert (2001), A Modern Theory of Integration, AMS, ISBN 0821808451.
- Larson, Ron; Edwards, Bruce H.; Heyd, David E. (2002), Calculus of a single variable (7th ed.), Boston: Houghton Mifflin Company.
- Leithold, L. (1996), The calculus of a single variable (6th ed.), New York: HarperCollins College Publishers.
- Malet, A, Studies on James Gregorie (1638-1675) (PhD Thesis, Princeton, 1989).
- Rudin, Walter (1987), Real and Complex Analysis (third ed.), New York: McGraw-Hill Book Co.
- Stewart, J. (2003), "Fundamental Theorem of Calculus", Calculus: early transcendentals, Belmont, California: Thomson/Brooks/Cole.
- Turnbull, H. W., ed. (1939), The James Gregory Tercentenary Memorial Volume, London.
- Spivak, Michael (1980), Calculus (2nd ed.), Houston, Texas: Publish or Perish Inc..
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