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Topic: Mean value theorem

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	PlanetMath: mean-value theorem
		The geometrical meaning of this theorem is illustrated in the picture:
		The mean-value theorem is often used in the integral context: There is a
		This is version 6 of mean-value theorem, born on 2002-02-15, modified 2004-07-21.
	planetmath.org /encyclopedia/MeanValueTheorem.html (96 words)

	MeanValueTheorm
		Many times when we use a theorem in solving a problem, we take for granted that the hypotheses given in the theorem are satisfied and we never check to see if that is in fact true.
		The mean value theorem states that under the specified hypotheses, there is a point in the interval of interest such that the slope of the tangent line at that point is equal to the slope of the secant line connecting the two endpoints of the graph of the function.
		To find the value of c given in the Mean Value Theorem, we need to find a tangent line to the curve that has the same slope as the secant line.
	www2.umassd.edu /temath/TEMATH2/Examples/MeanValueTheorem.html (1370 words)

	Mean value theorem - Wikipedia, the free encyclopedia
		In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the "average" derivative of the section.
		The mean value theorem can be used to prove Taylor's theorem, of which it is a special case.
		Thus the Mean value theorem says that given any chord of a smooth curve, we can find a point lying between the end-points of the chord such that the tangent at that point is parallel to the chord.
	en.wikipedia.org /wiki/Mean_value_theorem (821 words)

	* Rolle's Theorem and the Mean Value Theorem
		Theorem 1.40 (Rolle's Theorem) Suppose that f (x) is a differentiable function whose derivative is a continuous function.
		Figure 1.10: Mean Value Theorem: somewhere between a and b, the graph must be parallel to the chord AB.
		So the theorem is saying that there must be a point c between a and b at which the slope of the graph is equal to the slope of the chord AB.
	www.maths.abdn.ac.uk /~igc/tch/ma1002/diff/node39.html (563 words)

	Mean value theorem Summary
		In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the "average" derivative of the section.
		The mean value theorem can be used to prove Taylor's theorem, of which it is a special case.
		Thus the Mean value theorem says that given any chord of a smooth curve, we can find a point lying between the end-points of the chord such that the tangent at that point is parallel to the chord.
	www.bookrags.com /Mean_value_theorem (1355 words)

	Rolle and the Mean Value Theorem
		Theorem 5.15 (Rolle's Theorem) Let f be continuous on [a, b], and differentiable on (a, b), and suppose that
		Theorem 5.18 (The Mean Value Theorem) Let f be continuous on [a, b], and differentiable on (a, b).
		Theorem 5.21 (The Cauchy Mean Value Theorem) Let f and g be both continuous on [a, b] and differentiable on (a, b).
	www.maths.abdn.ac.uk /~igc/tch/ma2001/notes/node42.html (862 words)

	MATH 1401 #11: MEAN VALUE THEOREM
		The Mean Value Theorem is used to prove many of the theorems and properties developed in Calculus.
		Rolle's Theorem is a simplified version of the Mean Value Theorem and is used to prove it.
		Geometric interpretation of the Mean Value Theorem: Where the theorem applies, there is a value of c on the interval (a,b) where the tangent line to the function at x = c is parallel to the secant line through the points on the curve at x = a and x = b.
	www-math.cudenver.edu /~rbyrne/online/140w11.htm (479 words)

	math lessons - Mean value theorem
		In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the gradient (slope) of the curve is equal to the "average" gradient of the section.
		Some mathematicians consider this theorem to be the most important theorem of calculus (see also: the fundamental theorem of calculus).
		The theorem is not often used to solve mathematical problems; rather, it is more commonly used to prove other theorems.
	www.mathdaily.com /lessons/Mean_value_theorem (689 words)

	The Mean Value Theorem
		Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval.
		I expect that in your textbook the conditions given for the Mean Value Theorem are either that "f(x) is differentiable for all x in [a,b]" or that "f(x) is differentiable for all x in (a,b) and continuous for all a in [a,b]".
		The conclusion of the Mean Value Theorem is that somewhere on the curve between P and Q there is a point where the tangent line is parallel to the line joining P and Q. The first coordinate of this point is the number "c" in your problem.
	mathcentral.uregina.ca /QQ/database/QQ.09.05/candace2.html (371 words)

	The Mean Value Theorem
		The Mean Value Theorem relates the slope of a secant line to the slope of a tangent line.
		The Mean Value Theorem says that somewhere in between a and b, there is a point c on the curve where the tangent line has the same slope as the secant line.
		This means that f is constant on the interval.
	marauder.millersville.edu /~bikenaga/calculus/mvt/mvt.html (730 words)

	Intermediate value theorem - Wikipedia, the free encyclopedia
		The intermediate value theorem states the following: Suppose that I is an interval [a, b] in the real numbers R and that f : I → R is a continuous function.
		Darboux's theorem states that all functions that result from the differentiation of some other function on some interval have the intermediate value property (even though they need not be continuous).
		The theorem implies that on any great circle around the world, the temperature, pressure, elevation, carbon dioxide concentration, or anything else that varies continuously, there will always exist two antipodal points that share the same value for that variable.
	en.wikipedia.org /wiki/Intermediate_value_theorem (1043 words)

	Rolle's and Mean Value Theorems
		The Mean Value Theorem (MVT, for short) is one of the most frequent subjects in mathematics education literature.
		Thus Rolle's theorem claims the exsitentce of a point at which the tangent to the graph is parallel to the secant, provided the latter is horizontal.)
		Rolle's theorem is clearly a particular case of the MVT in which f satisfies an additional condition,
	www.cut-the-knot.org /Curriculum/Calculus/MVT.shtml (721 words)

	The Mean Value Theorem
		The answers are "yes", "of course", and "yes" (this is the first derivative test), but ALL of these answers depend on a theorem called the Mean Value Theorem (MVT).
		The Mean Value Theorem says that there is a point c in (a,b) at which the function's instantaneous rate of change is the same as its average rate of change over the entire interval [a,b].
		At x = 0, the value of the derivative is 1.
	oregonstate.edu /instruct/mth251/cq/Stage7/Lesson/MVT.html (603 words)

	Math Help - Mean Value Theorem
		Mean Value Theorem is an elementary math result in mathematical analysis due to Lagrange that states that:
		Therefore, there is a point on any arc of the graph of the function at which the tangent is parallel to the chord joining the end points of the arc.
		The generalized mean value theorem known as Cauchy's mean value theorem extends this to show that given two such functions, f and g, one can solve
	www.math-help.info /Mean_Value_Theorem.html (235 words)

	The Mean Value Theorem
		The previous theorem guarantees the existence of at least one maximum and one minimum points, but there may be several maximum or minimum points.
		The theorem, is given that name in honor of its discoverer, Michel Rolle, the French mathematician.
		Geometrically, the mean value theorem says that if the graph of a continuous function has a tangent (which is not vertical) between two points A and B, then there is at least one point at which the tangent is parallel to the line
	library.thinkquest.org /C006002/Pages/The_Mean_Value_Theorem.htm (436 words)

	Calculus@Internet
		The Mean Value Theorem - The mean value theorem refers to the mean or average rate of the change of f in the interval [a,b].
		The Mean Value Theorem - The mean value theorem refers to the mean or average rate of the change of f in the interval [a b].
		The Mean Value Theorem - An introduction to the Mean Value Theorem.
	www.calculus.net /ci2/search/?request=category&code=1214&off=0&tag=9200438920658 (244 words)

	The Mean Value Theorem - HMC Calculus Tutorial
		Though the theorem seems logical, we cannot be sure that it is always true without a proof.
		This is formalized in the Mean Value Theorem.
		The Mean Value Theorem is behind many of the important results in calculus.
	www.math.hmc.edu /calculus/tutorials/mean_value (404 words)

	Chapter 17 (Site not responding. Last check: 2007-10-20)
		The Mean value theorem is a rather simple and obvious theorem, yet the same can not be said about its implications in Calculus.
		This tells us that there exists a value c between 4 and 6, such that the derivative evaluated at this point c, gives the slope of the line parallel to the line connecting the two endpoints on the anti-derivative.
		This is the mean value theorem; there exists a number c between a and b on the graph of f’(x) such that f’(c) gives the slope of a tangent through f(c) that is parallel to the line through (a, f(a)) and (b, f(b)).
	www.understandingcalculus.com /chapters/20/mean_value_theorem.htm (478 words)

	Karl's Calculus Tutor - 5.4 Applications of Derivatives: Like a Steam Locomotive
		You don't need to know the proof of The Extreme Value Theorem (unless your instructor has, in his or her cruelty, said you need to know it), and you probably don't need to know the proof of the Intermediate Value Theorem either, but you do need to understand what each of them asserts.
		The Extreme Value Theorem asserts that if a function is continuous over a closed interval, it has a maximum and a minimum on that interval.
		And that is the contradiction that proves the theorem.
	www.karlscalculus.org /calc5_3.html (1840 words)

	UCES Methods and Analysis Chap. 2.2: Numerical Integration / Mean Value Theorem
		The mean value theorem and its variations will be developed and used to establish the order of convergence for a variety of algorithms.
		The mean value theorem, like the intermediate value theorem, will appear to be clearly true once one draws the picture associated with it.
		The next theorem makes the rate of convergence more precise, but now we assume that the integrand has a continuous derivative.
	www.krellinst.org /UCES/archive/classes/CNA/dir2.2/uces2.2.html (1496 words)

	Mean Value Theorem Question
		Use the Intermediate Value Theorem and/or the Mean Value Theorem and/or properties of
		Intermediate value theorem: G(0)<0, G(2)>0, so there is at least one root between.
		I know the x value is within the interval, due to the intermediate value theorem, as you stated.
	www.physicsforums.com /showthread.php?p=1164535#post1164535 (523 words)

	Intermediate Mean Value Theorem (Site not responding. Last check: 2007-10-20)
		It is useful in many areas of science to be able to find out not just the mean value of a function, but also to glean some idea of the absolute size of it too.
		It is clear that the mean of two numbers could be zero, but the size, which is more often called the norm, of two numbers cannot.
		Although the mean value is, perhaps, more intuitive and physically appealling, it is the RMS value which arises more frequently in applications.
	metric.ma.ic.ac.uk /integration/applications/imvt/index.html (342 words)

	Mean value theorem
		The definition of continuity hold that the function has limits such as any value of y in the interval [a,b] could be represented as the funciton of a point (be it rational or irrational).
		By continuity of the function, i/b-a must be one of the value f(x) assumes between in the interval [a,b].
		Which is the proof of the Mean Value Theorem.
	www.physicsforums.com /showthread.php?threadid=135528 (596 words)

	Taylor's Theorem
		Theorem 5.29 (Taylors Theorem - Lagrange form of Remainder) Let f be continuous on [a, x], and assume that each of f',
		We now explore the meaning and content of the theorem with a number of examples.
		This is another way to get the Binomial theorem described in Section 1.8.
	www.maths.abdn.ac.uk /~igc/tch/ma2001/notes/node46.html (733 words)

	Visual Calculus - Mean Value Theorem (Site not responding. Last check: 2007-10-20)
		Objectives: In this tutorial, we discuss Rolle's Theorem and the Mean Value Theorem.
		We look at some applications of the Mean Value Theorem that include the relationship of the derivative of a function with whether the function is increasing or decreasing.
		Drill problems on using the Mean Value Theorem.
	archives.math.utk.edu /visual.calculus/3/mvt.3/index.html (271 words)

	Calculus I (Math 2413) - Applications of Derivatives - The Mean Value Theorem (Site not responding. Last check: 2007-10-20)
		Now, to find the numbers that satisfy the conclusions of the Mean Value Theorem all we need to do is plug this into the formula given by the Mean Value Theorem.
		Here are a couple of nice facts that can be proved using the Mean Value Theorem. Note that in both of these facts we are assuming the functions are continuous and differentiable on the interval
		in the interval and this is exactly what it means for a function to be constant on the interval and so we’ve proven the fact.
	tutorial.math.lamar.edu /AllBrowsers/2413/MeanValueTheorem.asp (511 words)

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