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Mean Value Theorem

Mean Value Theorem

Mean Value Theorem

If a signal is quantized, the mean value is the MDV.

MEAN

For the theorem in harmonic function theory, see Harmonic function § The mean value property. (Source: en.wikipedia.org In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval. (Source:en.wikipedia.org))

5.1 Proof of Cauchy's mean value theorem (Source: en.wikipedia.org 8 Mean value theorem for vector-valued functions (Source:en.wikipedia.org e9 Mean value theorems for definite integrals (Source:n.wikipedia.org)))

9.1 First mean value theorem for definite integrals (Source: en.wikipedia.org)

9.2 Proof of the first mean value theorem for definite integrals (Source: en.wikipedia.org 9.3 Second mean value theorem for definite integrals (Source:en.wikipedia.org))

10 A probabilistic analogue of the mean value theorem (Source: en.wikipedia.org 9.4 Mean value theorem for integration fails for vector-valued functions (Source:en.wikipedia.org eA restricted form of the theorem was proved by Michel Rolle in 1691; the result was what is now known as Rolle's theorem, and was proved only for polynomials, without the techniques of calculus. The mean value theorem in its modern form was stated and proved by Augustin Louis Cauchy in 1823. (Source:n.wikipedia.org)))

. Thus the mean value theorem says that given any chord of a smooth curve, we can find a point on the curve lying between the end-points of the chord such that the tangent of the curve at that point is parallel to the chord. The following proof illustrates this idea. (Source: en.wikipedia.org Proof: Assume the derivative of f at every interior point of the interval I exists and is zero. Let (a, b) be an arbitrary open interval in I. By the mean value theorem, there exists a point c in (a,b) such that (Source:en.wikipedia.org))

Cauchy's mean value theorem can be used to prove L'Hôpital's rule. The mean value theorem is the special case of Cauchy's mean value theorem when (Source: en.wikipedia.org Cauchy's mean value theorem, also known as the extended mean value theorem, (Source:en.wikipedia.org eThe proof of Cauchy's mean value theorem is based on the same idea as the proof of the mean value theorem. (Source:n.wikipedia.org)))

There is no exact analog of the mean value theorem for vector-valued functions. (Source: en.wikipedia.org The mean value theorem generalizes to real functions of multiple variables. The trick is to use parametrization to create a real function of one variable, and then apply the one-variable theorem. (Source:en.wikipedia.org e

Jean Dieudonné in his classic treatise Foundations of Modern Analysis discards the mean value theorem and replaces it by mean inequality as the proof is not constructive and one cannot find the mean value and in applications one only needs mean inequality. Serge Lang in Analysis I uses the mean value theorem, in integral form, as an instant reflex but this use requires the continuity of the derivative. If one uses the Henstock–Kurzweil integral one can have the mean value theorem in integral form without the additional assumption that derivative should be continuous as every derivative is Henstock–Kurzweil integrable. The problem is roughly speaking the following: If f : U → R (Source: en.wikipedia.org)In Principles of Mathematical Analysis, Rudin gives an inequality which can be applied to many of the same situations to which the mean value theorem is applicable in the one dimensional case: (Source:n.wikipedia.org)))

However a certain type of generalization of the mean value theorem to vector-valued functions is obtained as follows: Let f be a continuously differentiable real-valued function defined on an open interval I, and let x as well as x + h be points of I. The mean value theorem in one variable tells us that there exists some t* between 0 and 1 such that (Source: en.wikipedia.org Mean Value Inequality — If the norm of Df(x + th) is bounded by some constant M for t in [0, 1], then (Source:en.wikipedia.org eThere are various slightly different theorems called the second mean value theorem for definite integrals. A commonly found version is as follows: (Source:n.wikipedia.org)))

Sahoo, Prasanna. (1998). Mean value theorems and functional equations. Riedel, T. (Thomas), 1962-. Singapore: World Scientific. ISBN (Source: en.wikipedia.org Lozada-Cruz, German (2020-10-02). "Some variants of Cauchy's mean value theorem". International Journal of Mathematical Education in Science and Technology. 51 (7): 1155–1163. doi:10.1080/0020739X.2019.1703150. ISSN 0020-739X. (Source:en.wikipedia.org eHobson, E. W. (1909). "On the Second Mean-Value Theorem of the Integral Calculus". Proc. London Math. Soc. S2–7 (1): 14–23. doi:10.1112/plms/s2-7.1.14. MR 1575669. (Source:n.wikipedia.org)))

Weisstein, Eric W. "Cauchy's Mean-Value Theorem". MathWorld. (Source: en.wikipedia.org Di Crescenzo, A. (1999). "A Probabilistic Analogue of the Mean Value Theorem and Its Applications to Reliability Theory". J. Appl. Probab. 36 (3): 706–719. doi:10.1239/jap/1032374628. JSTOR 3215435. (Source:en.wikipedia.org o

4.4.3 State three important consequences of the Mean Value Theorem. (Source: openstax.org The Mean Value Theorem is one of the most important theorems in calculus. We look at some of its implications at the end of this section. First, let’s start with a special case of the Mean Value Theorem, called Rolle’s theorem. (Source:openstax.org))4.4.2 Describe the significance of the Mean Value Theorem. (Source:penstax.org)))

openstax.org openstax.org openstax.org openstax.org)))). The Mean Value Theorem generalizes Rolle’s theorem by considering functions that do not necessarily have equal value at the endpoints. Consequently, we can view the Mean Value Theorem as a slanted version of Rolle’s theorem (Figure 4.25). The Mean Value Theorem states thRolle’s theorem is a special case of the Mean Value Theorem. In Rolle’s theorem, we consider differentiable functions (Source:Let’s now look at three corollaries of the Mean Value Theorem. These results have important consequences, which we use in upcoming sections. (Source:At this point, we know the derivative of any constant function is zero. The Mean Value Theorem allows us to conclude that the converse is also true. In particular, if (Source:at if (Source:

For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer. (Source: openstax.org This fact is important because it means that for a given function (Source:openstax.org o

State three important consequences of the Mean Value Theorem. (Source: math.libretexts.org Describe the significance of the Mean Value Theorem. (Source:math.libretexts.org))Section URL: https://openstax.org/books/calculus-volume-1/pages/4-4-the-mean-value-theorem (Source:penstax.org)))

Rolle’s theorem is a special case of the Mean Value Theorem. In Rolle’s theorem, we consider differentiable functions \(f\) that are zero at the endpoints. The Mean Value Theorem generalizes Rolle’s theorem by considering functions that are not necessarily zero at the endpoints. Consequently, we can view the Mean Value Theorem as a slanted version of Rolle’s theorem (Figure \(\PageIndex{5}\)). The Mean Value Theorem states that if \(f\) is continuous over the closed interval \([a,b]\) and differentiable over the open interval \((a,b)\), then there exists a point \(c∈(a,b)\) such that the tangent line to the graph of \(f\) at \(c\) is parallel to the secant line connecting \((a,f(a))\) and \((b,f(b)).\) (Source: math.libretexts.org Rolle’s theorem is a special case of the Mean Value Theorem. In Rolle’s theorem, we consider differentiable functions \(f\) that are zero at the endpoints. The Mean Value Theorem generalizes Rolle’s theorem by considering functions that are not necessarily zero at the endpoints. Consequently, we can view the Mean Value Theorem as a slanted version of Rolle’s theorem (Figure \(\PageIndex{5}\)). The Mean Value Theorem states that if \(f\) is continuous over the closed interval \([a,b]\) and differentiable over the open interval \((a,b)\), then there exists a point \(c∈(a,b)\) such that the tangent line to the graph of \(f\) at \(c\) is parallel to the secant line connecting \((a,f(a))\) and \((b,f(b)).\) (Source:math.libretexts.org))

For \(f(x)=\sqrt{x}\) over the interval \([0,9]\), show that \(f\) satisfies the hypothesis of the Mean Value Theorem, and therefore there exists at least one value \(c∈(0,9)\) such that \(f′(c)\) is equal to the slope of the line connecting \((0,f(0))\) and \((9,f(9))\). Find these values \(c\) guaranteed by the Mean Value Theorem. (Source: math.libretexts.org Figure \(\PageIndex{5}\): The Mean Value Theorem says that for a function that meets its conditions, at some point the tangent line has the same slope as the secant line between the ends. For this function, there are two values \(c_1\) and \(c_2\) such that the tangent line to \(f\) at \(c_1\) and \(c_2\) has the same slope as the secant line. (Source:math.libretexts.org mWe know that \(f(x)=\sqrt{x}\) is continuous over \([0,9]\) and differentiable over \((0,9).\) Therefore, \(f\) satisfies the hypotheses of the Mean Value Theorem, and there must exist at least one value \(c∈(0,9)\) such that \(f′(c)\) is equal to the slope of the line connecting \((0,f(0))\) and \((9,f(9))\) (Figure). To determine which value(s) of \(c\) are guaranteed, first calculate the derivative of \(f\). The derivative \(f′(x)=\frac{1}{(2\sqrt{x})}\). The slope of the line connecting \((0,f(0))\) and \((9,f(9))\) is given by (Source:ath.libretexts.org)))

One application that helps illustrate the Mean Value Theorem involves velocity. For example, suppose we drive a car for 1 h down a straight road with an average velocity of 45 mph. Let \(s(t)\) and \(v(t)\) denote the position and velocity of the car, respectively, for \(0≤t≤1\) h. Assuming that the position function \(s(t)\) is differentiable, we can apply the Mean Value Theorem to conclude that, at some time \(c∈(0,1)\), the speed of the car was exactly (Source: math.libretexts.org)

Let’s now look at three corollaries of the Mean Value Theorem. These results have important consequences, which we use in upcoming sections. (Source: math.libretexts.org c. The instantaneous velocity is given by the derivative of the position function. Therefore, we need to find a time \(t\) such that \(v(t)=s′(t)=v_{avg}=−40\) ft/sec. Since \(s(t)\) is continuous over the interval \([0,5/2]\) and differentiable over the interval \((0,5/2),\) by the Mean Value Theorem, there is guaranteed to be a point \(c∈(0,5/2)\) such that (Source:math.libretexts.org mAt this point, we know the derivative of any constant function is zero. The Mean Value Theorem allows us to conclude that the converse is also true. In particular, if \(f′(x)=0\) for all \(x\) in some interval \(I\), then \(f(x)\) is constant over that interval. This result may seem intuitively obvious, but it has important implications that are not obvious, and we discuss them shortly. (Source:ath.libretexts.org)))

If \(f\) is continuous over \([a,b]\) and differentiable over \((a,b)\), then there exists a point \(c∈(a,b)\) such that \[f'(c)=\frac{f(b)−f(a)}{b−a}. \nonumber\] This is the Mean Value Theorem. (Source: math.libretexts.org This fact is important because it means that for a given function \(f\), if there exists a function \(F\) such that \(F′(x)=f(x)\); then, the only other functions that have a derivative equal to \(f\) are \(F(x)+C\) for some constant \(C\). We discuss this result in more detail later in the chapter. (Source:math.libretexts.org))

Rolle’s theorem, in analysis, special case of the mean-value theorem of differential calculus. Rolle’s theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b. In other words, if a continuous curve passes through the same y-value (such as the x-axis) twice and has a unique tangent line (derivative) at every point of the interval, then somewhere between the endpoints it has a tangent parallel to the x-axis. The theorem was proved in 1691 by the French mathematician (Source: www.britannica.com)

Mean Value Theorem is an important theorem in calculus. The first form of the mean value theorem was proposed in the 14th century by Parmeshwara, a mathematician from Kerela, India. Further, a simpler version of this was proposed by Rolle in the 17th century: Rolle's Theorem, which was proved only for polynomials and was not a part of the calculus. Finally, the present version of the Mean Value Theorem was proposed by Augustin Louis Cauchy in the year 1823. (Source: www.cuemath.com Bhaskara II. Other than being useful in proving the mean-value theorem, Rolle’s theorem is seldom used, since it establishes only the existence of a solution and not its value. (Source:www.britannica.com))

The mean value theorem states that for a curve passing through two given points there is one point on the curve where the tangent is parallel to the secant passing through the two given points. Rolle's theorem has been derived from this mean value theorem. (Source: www.cuemath.com www.cuemath.com))The mean value theorem states that for a curve f(x) passing through two given points (a, f(a)), (b, f(b)), there is at least one point (c, f(c)) on the curve where the tangent is parallel to the secant passing through the two given points. The mean value theorem is defined herein calculus for a function f(x): [a, b] → R, such that it is continuous and differentiable across an interval. (Source:

Here we have proved that the tangent at c is parallel to the secant passing through the points (a, f(a)), (b, f(b)). This mean value theorem is used to prove a statement across a closed interval. Further, Rolle's theorem is derived from this mean value theorem. (Source: www.cuemath.com)

 

 

 

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