مبرهنة القيمة المتوسطة
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حسبان تفاضلي
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حسبان تكاملي
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المتسلسلات
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حسبان المتجهات
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حسبان متعدد المتغيرات
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حسبان متخصص
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في الرياضييات، تقول مبرهنة القيمة المتوسطة Mean value theorem : أنه من أجل قطاع من منحن ، هناك نقطة على هذا المنحنيقد يكون فيها تدرج (ميل) المنحني مساويا للتدرج الوسطي للقطاع ككل . تستخدم هذه المبرهنة لإثبات مبرهنات تؤدي لإستنتاجات عامة (شاملة) حول التابع على مجال محدد ما بدءا من فرضيات محلية حول مشتقات النقاط لهذا المجال .
More precisely, if
وهي واحدة من أبرز النتائج في التحليل الحقيقي real analysis.
الصيغة الشكلية
Let
The mean value theorem is a generalization of Rolle's theorem, which assumes , so that the right-hand side above is zero.
The mean value theorem is still valid in a slightly more general setting. One only needs to assume that
exists as a finite number or equals
Note that the theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define for all real . Then
while for any real .
These formal statements are also known as Lagrange's Mean Value Theorem.
الإثبات
The expression
Define
By Rolle's theorem, since is differentiable and , there is some in for which , and it follows from the equality that,
A simple application
Assume that f is a continuous, real-valued function, defined on an arbitrary interval I of the real line. If the derivative of f at every interior point of the interval I exists and is zero, then f is constant in the interior.
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
This implies that f(a) = f(b). Thus, f is constant on the interior of I and thus is constant on I by continuity. (See below for a multivariable version of this result.)
Remarks:
- Only continuity of f, not differentiability, is needed at the endpoints of the interval I. No hypothesis of continuity needs to be stated if I is an open interval, since the existence of a derivative at a point implies the continuity at this point. (See the section continuity and differentiability of the article derivative.)
- The differentiability of f can be relaxed to one-sided differentiability, a proof given in the article on semi-differentiability.
مبرهنة كوشي للقيمة المتوسطة
Cauchy's mean value theorem, also known as the extended mean value theorem, is a generalization of the mean value theorem. It states: If functions f and g are both continuous on the closed interval [a, b], and differentiable on the open interval (a, b), then there exists some c ∈ (a, b), such that
Of course, if g(a) ≠ g(b) and if g′(c) ≠ 0, this is equivalent to:
Geometrically, this means that there is some tangent to the graph of the curve
which is parallel to the line defined by the points (f(a), g(a)) and (f(b), g(b)). However Cauchy's theorem does not claim the existence of such a tangent in all cases where (f(a), g(a)) and (f(b), g(b)) are distinct points, since it might be satisfied only for some value c with f′(c) = g′(c) = 0, in other words a value for which the mentioned curve is stationary; in such points no tangent to the curve is likely to be defined at all. An example of this situation is the curve given by
which on the interval [−1, 1] goes from the point (−1, 0) to (1, 0), yet never has a horizontal tangent; however it has a stationary point (in fact a cusp) at t = 0.
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 g(t) = t.
إثبات مبرهنة كوشي للقيمة المتوسطة
The proof of Cauchy's mean value theorem is based on the same idea as the proof of the mean value theorem.
- Suppose g(a) ≠ g(b). Define h(x) = f(x) − rg(x), where r is fixed in such a way that h(a) = h(b), namely
- Since f and g are continuous on [a, b] and differentiable on (a, b), the same is true for h. All in all, h satisfies the conditions of Rolle's theorem: consequently, there is some c in (a, b) for which h′(c) = 0. Now using the definition of h we have:
- Therefore:
- which implies the result.
- If g(a) = g(b), then, applying Rolle's theorem to g, it follows that there exists c in (a, b) for which g′(c) = 0. Using this choice of c, Cauchy's mean value theorem (trivially) holds.
التعميم للمحددات
Assume that and are differentiable functions on that are continuous on . Define
There exists such that .
Notice that
and if we place , we get Cauchy's mean value theorem. If we place and we get Lagrange's mean value theorem.
The proof of the generalization is quite simple: each of and are determinants with two identical rows, hence . The Rolle's theorem implies that there exists such that .
مبرهنة القيمة المتوسطة في عدة متغيرات
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.
Let be an open convex subset of , and let be a differentiable function. Fix points , and define . Since is a differentiable function in one variable, the mean value theorem gives:
for some between 0 and 1. But since and , computing explicitly we have:
where
In particular, when the partial derivatives of
As an application of the above, we prove that is constant if is open and connected and every partial derivative of is 0. Pick some point , and let . We want to show for every . For that, let . Then E is closed and nonempty. It is open too: for every ,
for every in some neighborhood of . (Here, it is crucial that and are sufficiently close to each other.) Since is connected, we conclude .
The above arguments are made in a coordinate-free manner; hence, they generalize to the case when is a subset of a Banach space.
مبرهنة القيمة المتوسطة للدوال ذات القيم المتجهية
There is no exact analog of the mean value theorem for vector-valued functions.
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:
Theorem. For a continuous vector-valued function differentiable on , there exists such that .
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 → Rm is a differentiable function (where U ⊂ Rn is open) and if x + th, x, h ∈ Rn, t ∈ [0, 1] is the line segment in question (lying inside U), then one can apply the above parametrization procedure to each of the component functions fi (i = 1, ..., m) of f (in the above notation set y = x + h). In doing so one finds points x + tih on the line segment satisfying
But generally there will not be a single point x + t*h on the line segment satisfying
for all i simultaneously. For example, define:
Then , but and are never simultaneously zero as ranges over .
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
On the other hand, we have, by the fundamental theorem of calculus followed by a change of variables,
Thus, the value f′(x + t*h) at the particular point t* has been replaced by the mean value
This last version can be generalized to vector valued functions:
-
Lemma 1. Let U ⊂ Rn be open, f : U → Rm continuously differentiable, and x ∈ U, h ∈ Rn vectors such that the line segment x + th, 0 ≤ t ≤ 1 remains in U. Then we have:
- where Df denotes the Jacobian matrix of f and the integral of a matrix is to be understood componentwise.
Proof. Let f1, ..., fm denote the components of f and define:
Then we have
The claim follows since Df is the matrix consisting of the components
-
Lemma 2. Let v : [a, b] → Rm be a continuous function defined on the interval [a, b] ⊂ R. Then we have
Proof. Let u in Rm denote the value of the integral
Now we have (using the Cauchy–Schwarz inequality):
Now cancelling the norm of u from both ends gives us the desired inequality.
-
Mean Value Inequality. If the norm of Df(x + th) is bounded by some constant M for t in [0, 1], then
Proof. From Lemma 1 and 2 it follows that
Mean value theorems for definite integrals
First mean value theorem for definite integrals
Let f : [a, b] → R be a continuous function. Then there exists c in [a, b] such that
Since the mean value of f on [a, b] is defined as
we can interpret the conclusion as f achieves its mean value at some c in (a, b).
In general, if f : [a, b] → R is continuous and g is an integrable function that does not change sign on [a, b], then there exists c in (a, b) such that
Proof of the first mean value theorem for definite integrals
Suppose f : [a, b] → R is continuous and g is a nonnegative integrable function on [a, b]. By the extreme value theorem, there exists m and M such that for each x in [a, b], and . Since g is nonnegative,
Now let
If , we're done since
means
so for any c in (a, b),
If I ≠ 0, then
By the intermediate value theorem, f attains every value of the interval [m, M], so for some c in [a, b]
that is,
Finally, if g is negative on [a, b], then
and we still get the same result as above.
QED
Second mean value theorem for definite integrals
There are various slightly different theorems called the second mean value theorem for definite integrals. A commonly found version is as follows:
- If G : [a, b] → R is a positive monotonically decreasing function and φ : [a, b] → R is an integrable function, then there exists a number x in (a, b] such that
Here stands for , the existence of which follows from the conditions. Note that it is essential that the interval (a, b] contains b. A variant not having this requirement is:
- If G : [a, b] → R is a monotonic (not necessarily decreasing and positive) function and φ : [a, b] → R is an integrable function, then there exists a number x in (a, b) such that
Mean value theorem for integration fails for vector-valued functions
If the function returns a multi-dimensional vector, then the MVT for integration is not true, even if the domain of is also multi-dimensional.
For example, consider the following 2-dimensional function defined on an -dimensional cube:
Then, by symmetry it is easy to see that the mean value of over its domain is (0,0):
However, there is no point in which , because everywhere.
A probabilistic analogue of the mean value theorem
Let X and Y be non-negative random variables such that E[X] < E[Y] < ∞ and
Let g be a measurable and differentiable function such that E[g(X)], E[g(Y)] < ∞, and let its derivative g′ be measurable and Riemann-integrable on the interval [x, y] for all y ≥ x ≥ 0. Then, E[g′(Z)] is finite and
Generalization in complex analysis
As noted above, the theorem does not hold for differentiable complex-valued functions. Instead, a generalization of the theorem is stated such:
Let f : Ω → C be a holomorphic function on the open convex set Ω, and let a and b be distinct points in Ω. Then there exist points u, v on Lab (the line segment from a to b) such that
Where Re() is the Real part and Im() is the Imaginary part of a complex-valued function.
انظر أيضاً
- Newmark-beta method
- Mean value theorem (divided differences)
- Racetrack principle
- Stolarsky mean
المراجع
- ^ Weisstein, Eric. "Mean-Value Theorem". MathWorld. Wolfram Research. Retrieved 24 March 2011.
- ^ (in الإنجليزية). Krishna Prakashan Media.
- ^ W., Weisstein, Eric. "Extended Mean-Value Theorem". mathworld.wolfram.com (in الإنجليزية). Retrieved 2018-10-08.
- ^ "Cauchy's Mean Value Theorem". Math24 (in الإنجليزية). Retrieved 2018-10-08.
- ^ Rudin, Walter (1976). . New York: McGraw-Hill. p. 113. ISBN .
- ^ "Mathwords: Mean Value Theorem for Integrals". www.mathwords.com.
- ^ Michael Comenetz (2002). Calculus: The Elements. World Scientific. p. 159. ISBN .
- ^ Hobson, 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.
- ^ 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.
- ^ "Complex Mean-Value Theorem". PlanetMath. PlanetMath.
وصلات خارجية
- نطقب:SpringerEOM
- PlanetMath: Mean-Value Theorem
- Eric W. Weisstein, Mean value theorem at MathWorld.
- Eric W. Weisstein, Cauchy's Mean-Value Theorem at MathWorld.
- "Mean Value Theorem: Intuition behind the Mean Value Theorem" at the Khan Academy