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Topic: Fundamental theorem of calculus

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Antiderivative

Mean value theorem

Integration by parts

Integral calculus

Chain rule

Derivative

Integral

Product rule

Substitution rule

Table of derivatives

Trigonometric substitution

Calculus

Derivative (examples)

Derivative

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	Fundamental theorem of calculus - Biocrawler (Site not responding. Last check: 2007-10-09)
		The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other.
		This theorem is of such central importance in calculus that it deserves to be called the fundamental theorem for the entire field of study.
		An important consequence of this, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated.
	www.biocrawler.com /encyclopedia/Fundamental_theorem_of_calculus (1168 words)

	Fundamental Theorem of Calculus \| World of Mathematics
		The Fundamental Theorem of Calculus states that the area under the graph of a function over an interval can be calculated by evaluating any antiderivative of the function at the endpoints of the interval.
		This remarkable theorem is called the Fundamental Theorem of Calculus because it provides the link between the two branches of calculus, differential calculus and integral calculus.
		Integral calculus is the study of areas under curves, which are defined in terms of limits of Riemann sums -- the area under a curve is approximated by a collection of rectangles, and the widths of these rectangles are taken to be smaller and smaller to obtain better and better estimates of the actual area.
	www.bookrags.com /research/fundamental-theorem-of-calculus-wom (647 words)

	Visual Calculus - Fundamental Theorem of Calculus (Site not responding. Last check: 2007-10-09)
		Illustration of the Fundamental Theorem of Calculus using Maple and a LiveMath Notebook.
		Graphing the function A from the second part of the theorem using a graphing calculator or a LiveMath Notebook.
		The fundamental theorem of calculus and the chain rule:
	archives.math.utk.edu /visual.calculus/4/ftc.9/index.html (232 words)

	Fundamental theorem of calculus - Wikipedia, the free encyclopedia
		The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another.
		It is of such central importance in calculus that it is called the fundamental theorem for the entire field of study.
		In his 2003 book (page 340), James Stewart credits the idea that led to the fundamental theorem to the English mathematician Isaac Barrow although the first known proof of the fundamental theorem was due to the Scottish mathematician James Gregory.
	en.wikipedia.org /wiki/Fundamental_theorem_of_calculus (1219 words)

	History of the Calculus -- Differential and Integral Calculus
		Newton are usually designated the inventors of calculus, mainly for their separate discoveries of the fundamental theorem of calculus and work on notation.
		fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations.
		Fundamental Theorem of Calculus: If a function f is continuous on the interval [a, b] and F is an antiderivative of f on the interval [a, b], then
	www.edinformatics.com /inventions_inventors/calculus.htm (1543 words)

	Calculus - Wikipedia, the free encyclopedia
		Today, calculus is used in every branch of the physical sciences, in computer science, in statistics, and in engineering; in economics, business, and medicine; and as a general method whenever the goal is an optimal solution to a problem that can be given in mathematical form.
		Calculus avoids division by zero by using the concept of the limit which, roughly speaking, is a method of controlling an otherwise uncontrollable output, such as division by zero or multiplication by infinity.
		Calculus continues to be further generalized, such as with the development of the Lebesgue integral in 1900.
	en.wikipedia.org /wiki/Calculus (2449 words)

	What’s Wrong with Calculus
		We need not belabor this point, because the evidence over the past decade has been overwhelming in showing that student's are not learning much calculus in our calculus courses, including studies on retention of the material, ability to adapt their calculus experience to new settings, and so on.
		In spite of a calculus course that is saturated with “applications,” laden with references to physics and chemistry, and packed with numerical techniques, most of our colleagues in other disciplines see very little relationship between their fields and the calculus course.
		Clearly, our calculus course does not prepare scientists in other fields to recognize, understand, and utilize the calculus that many of their fields are based upon.
	faculty.etsu.edu /knisleyj/calculus/Crisis.htm (1402 words)

	PlanetMath: proof of the fundamental theorem of calculus
		This proves the first part of the theorem.
		"proof of the fundamental theorem of calculus" is owned by paolini.
		This is version 6 of proof of the fundamental theorem of calculus, born on 2003-07-17, modified 2006-09-01.
	planetmath.org /encyclopedia/ProofOfTheSecondFundamentalTheoremOfCalculus.html (133 words)

	Webquest Historical Roots of Calculus (Site not responding. Last check: 2007-10-09)
		Calculus made modern science possible and no physical theory has ever broken the link to the calculus.
		The subject is defined by a fantastic leading idea, one basic axiom, a calm and profound intellectual invention, a deep property, two crucial definitions, one ancillary definition, one major theorem, and the fundamental theorem of the calculus.
		In its largest, its most architectural aspect, the calculus is a great, even spectacular theory of space and time, a demonstration that in the real numbers there is an instrument adequate to their representation.
	coe.west.asu.edu /students/msyrkel/webquestcalculus.htm (997 words)

	SparkNotes: Definite Integral: Antiderivatives and the Fundamental Theorem of Calculus
		In this section we present the fundamental theorem of calculus.
		The first part of the theorem roughly states that in order to find the total change in the function F(x) from a to b, we must take a kind of sum (an integral) of the instantaneous rates of change (the derivatives) between a and b.
		The second part of the theorem says the instantaneous rate at which area is being added to the region under the graph as the right boundary of the region is extended is equal to the value of the function at the right boundary.
	www.sparknotes.com /math/calcbc1/definiteintegral/section3.rhtml (480 words)

	An introduction to calculus (Site not responding. Last check: 2007-10-09)
		Calculus is the study of the effect on functions of small changes in their variables.
		One can study either the effect of one small change (differential calculus) or the accumulation of many small changes (integral calculus).
		One amazing result, the Fundamental Theorem of Calculus, shows how these two areas of calculus are related.
	www.uwm.edu /Dept/Math/Resources/Calculus/Key/node2.htm (109 words)

	Fundamental Theorem (Site not responding. Last check: 2007-10-09)
		The Fundamental Theorem of Calculus (FTC) is a theorem par excellence and no calculus student should go throug a first course in Calculus without gaining some insight into nature of this theorem and its applications.
		There are actually two versions of the FTC and the order in which they are proved is strictly a matter of taste.
		In this module, what we call version 2 theorem is in fact the version that students will find more useful.
	www.uncwil.edu /courses/webcalc/calc1/INTEG/ftc.htm (216 words)

	Fundamental Theorems of Calculus -- from Wolfram MathWorld
		This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic
		The second fundamental theorem of calculus holds for
		Indefinite Integral, Integral, Second Fundamental Theorem of Calculus.
	mathworld.wolfram.com /FundamentalTheoremsofCalculus.html (131 words)

	Fundamental theorem - Wikipedia, the free encyclopedia
		The names are mostly traditional; so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory.
		Theorems may be called fundamental because they are results from which further, more complicated theorems follow, without reaching back to axioms.
		The mathematical literature will sometimes refer to the fundamental lemma of a field; this is often, but not always, the same as the fundamental theorem of that field.
	en.wikipedia.org /wiki/Fundamental_theorem (161 words)

	Calculus Theorems (Site not responding. Last check: 2007-10-09)
		Once you understand Limits and the definition of "Continuous Function", then you are ready to study five theorems that lead up to the Fundamental Theorem of Calculus.
		Intermediate Value Theorem -- if k is between f(a) and f(b), then there exists a c in [a,b] such that f(c)=k.
		Fundamental Theorem of Calculus -- if f is the derivative of F, then the integral from a to b of f(x)dx is F(b)-F(a)
	mcraefamily.com /MathHelp/CalculusTheorem.htm (254 words)

	Karl's Calculus Tutor - Living Backwards -- The Fundamental Theorem of Calculus
		To that end your goal ought to be to learn calculus backwards and forwards, and I can tell you that up till now, you only know it forwards.
		Which is the thrust of the Fundamental Theorem of Calculus.
		And that is the proof of the Fundamental Theorem of Calculus.
	www.karlscalculus.org /calc10_1.html (2794 words)

	The fundamental theorem of calculus
		The fundamental theorem of calculus states, in part, that
		We shall now show why Theorem 5 is plausible.
		An important theorem from real analysis says that for a function f which is continous on [a,b] there is only one number which satisfies the double inequality (8), and this number is called the integral of f on [a,b], and is denoted by
	www.uwm.edu /Dept/Math/Resources/Calculus/Key/node68.htm (387 words)

	Karl's Calculus Tutor - The Nitty Gritty of the Fundamental Theorem
		Karl's Calculus Tutor - The Nitty Gritty of the Fundamental Theorem
		On first blush it seems that we have proved the Fundamental Theorem of Calculus outright in the main text.
		There is a theorem (whose proof is well beyond first year students) that when a domain interval is closed and bounded (that is it includes the endpoints and has endpoints at both ends), then any function that is continuous on that entire interval is also uniformly continuous on that entire interval.
	www.karlscalculus.org /l10_2a.html (780 words)

	Fundamental Theorem of Calculus
		The geometric ratio of the fundamental theorem of calculus.
		A fundamental theorem of dyadic calculus for the unit square.
		A remark on the fundamental theorem of integral calculus.
	math.fullerton.edu /mathews/c2003/FunTheoremCalculusBib/Links/FunTheoremCalculusBib_lnk_2.html (551 words)

	18 Integration - First Fundamental Theorem of Calculus (Site not responding. Last check: 2007-10-09)
		The area of the region A under the curve y = h(x) between x = a and x = b could be approximated by covering the region by squares, all of the same size, and then allowing the size (measured by their width) approach zero.
		The First Fundamental Theorem of Calculus says that if h(x) is continuous between and at the endpoints x = a and x = b, that is continuous on the closed interval [a,b], then all the rectangle-based approximations approach a single finite limiting value as the width of the base tends to zero.
		A proof of the First Fundamental Theorem of Calculus is given in the appendices.
	whyslopes.com /etc/CalculusAndBeyond/ch18.html (809 words)

	Calculus III (Math 2415) - Line Integrals - Fundamental Theorem for Line Integrals
		Fundamental Theorem of Calculus that told us how to evaluate definite integrals. This told us,
		C. Also, we did not specify the number of variables for the function since it is really immaterial to the theorem. The theorem will hold regardless of the number of variables in the function.
		This is easy enough to justify since all we need to so is look at the theorem above. The theorem tells us that in order to evaluate this integral all we need are the initial and final points of the curve. This in turn tells us that the line integral must be independent of path.
	tutorial.math.lamar.edu /AllBrowsers/2415/FundThmLineIntegrals.asp (623 words)

	The Fundamental Theorem of Calculus
		The first part of the Fundamental Theorem of Calculus says that our earlier observation is indeed correct.
		The Fundamental Theorem of Calculus, Part II So far we have found a very interesting relationship between the definite integral and differentiation.
		The Fundamental Theorem of Calculus is remarkable for it expresses a relationship between integration, which is a sophisticated kind of addition, and differentiation.
	www.ugrad.math.ubc.ca /coursedoc/math101/notes/integration/ftc.html (1010 words)

	Lesson #99 The Fundamental Theorem of Calculus and Properties of the Definite Integral
		The Fundamental Theorem of Calculus Each branch of mathematics has a fundamental theorem associated with it.
		The Fundamental Theorem of Calculus — there are actually two parts to this theorem: The First Fundamental Theorem of Calculus: The derivative of the integral of a function is equal to the function.
		The Second Fundamental Theorem of Calculus: The integral of the derivative of a function is is equal to the function evaluated at its endpoints.
	www.pen.k12.va.us /Div/Winchester/jhhs/math/lessons/calc2004/day99.html (264 words)

	Fundamental Theorem of Calculus - HMC Calculus Tutorial
		We are all used to evaluating definite integrals without giving the reason for the procedure much thought.
		The Fundamental Theorem of Calculus justifies our procedure of evaluating an antiderivative at the upper and lower limits of integration and taking the difference.
		Then it may be proven that G(x) is an antiderivative for f on [a,b].
	www.math.hmc.edu /calculus/tutorials/fundamental_thm (240 words)

	Calculus@Internet
		Fundamental Theorem of Calculus - As the name suggests the Fundamental Theorem of Calculus (FTC) is an important theorem.
		Fundamental Theorem of Calculus - The first part of this theorem tells us how to evaluate a definite integral provided that f has an indefinite integral.
		The Fundamental Theorem of Calculus - Definition and application of the fundamental theorem of calculus.
	www.calculus.net /ci2/search/?request=category&code=136&off=0&tag=9200438920658 (91 words)

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