The Lagrange Mean Value Theorem Of Functions of n Variables

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1 陕西师范大学学士学位论文 The Lagrage Mea Value Theorem Of Fuctios of Variables 作者单位数学与信息科学学院指导老师曹怀信作者姓名李碧专业班级数学与应用数学专业 4 级班

2 The Lagrage Mea Value Theorem of a Fuctio of Variables LI i lass, Grade 4, ollege of Mathematics ad Iformatio Sciece dvisor: Professor O Huai-i bstract: The aim of the paper is to show the summary ad proof of the Lagrage mea value theorem of a fuctio of variables Firstly, we review the mea value theorem of a fuctio of oe variable ad the properties of cotiuous fuctios of two variables ccordig to the geometric cosequece of Lagrage mea value theorem of fuctios of oe variable, a Lagrage mea value theorem of fuctios of two variables is give We use the method of costructive auiliary fuctio to prove the theorem Moreover, the theorem is illustrated with oe eample Fially, the theroems about fuctios of variables are geeralized to fuctios of variables Key words: fuctio of variables; Lagrage mea value theorem; cotiuity; differetiability 元函数的拉格朗日中值定理李碧数学与信息科学学院 4 级班指导教师曹怀信教授摘要 : 本文论述并证明了元函数的拉格朗日中值定理首先回顾了一元函数微分中值定理及二元连续函数相关性质, 再从一元函数的拉格朗日中值定理的几何意义中分析设想了二元函数的拉格朗日中值定理, 利用类似一元函数的证明方法构造辅助函数证明了定理的正确性, 并进行了举例验证最后推广得到了元函数的拉格朗日中值定理关键词 : 元函数 ; 拉格朗日中值定理 ; 连续性 ; 可微性 - -

3 alculus was iveted i the seveteeth cetury to provide a tool for solvig problems ivolvig motio[] The derivative is oe of the fudametal cocepts of calculus Mea value theorems are very importat ad basic theorems i calculus, which make it possible to take derivatives as icrease or decrease, critical poits, ad etreme etc[] The Lagrage mea value theorem asserts that if a fuctio R is cotiuous ad differetiable i a, b, the there is a poit i the ope iterval a, b with the property that the taget lie fo the graph at, f is parallel to the lie passig through a, f a ad b, f b [3] I the study of may pheome oe ecouters fuctios of two or more idepedet variables So it is importat to geeralize the theorems to fuctios of variables, see [4] ad [5] However, they did t reveal the geometrical cosequece of the Lagrage mea value theorem I this paper, it will be give Lemmas Mea Value Theorem of fuctios of oe variable Fuctios of oe variable are the basis of fuctios of several variables Let s review the theorems of fuctios of oe variable Lemma Let I be a ope iterval cotaiig the poit ad suppose that the fuctio f : I R be differetiable at If the poit is either a maimizer or a miimizer of the fuctio f : I R, the f ' = Proof Observe that by the very defiitio of derivative, f f f f lim = lim = f, <, > First suppose that is a maimizer The ad hece O the other had, f f f f f = for i I with < lim, < f f for i I with >, - -

4 ad hece f f f = lim, > Thus ' f = I the case whe is a miimizer, the same proof applies, with iequalities reversed Lemma Rolle s Theorem Suppose that the fuctio R is cotiuous ad R is differetiable ssume, moreover, that f a = f b = The there is a poit i the ope iterval ab, at which ' f = Proof Sice R is cotiuous, accordig to the Etreme Value Theorem, it attais both a miimum value ad a maimum value o[ ab, ] Sice f a = f b =,if both the maimizers ad miimizers occur at the edpoits, the the fuctio R is idetically equal to, so f = at every poit i abotherwise,, the fuctio has either a maimizer or miimizer at some poit i the ope iterval I = ab, ad hece, by the precedig lemm at this poit ' f = Lemma 3Lagrage Mea Value Theorem Suppose that the fuctio R is cotiuous ad R is differetiable The there is a poit i the ope iterval ab, at whichf = slope f b f a b a Proof The lie passig through the poits f a ad, b f b has f b f a, so the equatio of the lie passig through these poits is b a f b f a y = g = fa + a b a Defie the auiliary fuctio d : b R by d = f g for all i [ ab, ] The clearly da = db = Moreover, from the additio theorems for cotiuous ad differetiable fuctios, it follows that d : b R is cotiuous is differetiable We apply Rolle s Theorem to select a poit i the ope iterval ab, at which d = ut - 3 -

5 d = f so that f = f b f a b a f b f a b a for a< < b, The Properties Of otiuous Fuctios Of Two Variables Theorem Suppose that f is cotiuous o the bouded closed regio D R, the f attais both a miimum ad a maimum value o D Proof Let s show f be bouded firstly ssume that f is ubouded Let be a atural umber The there is a poit P=, y D at which f P > hoose such a poit ad label it with the property that P This defies a sequece { } P i D f P > for every atural umber We ca employ the olzao-weierstrass Theorem to fid a subsequece { } k to a poit coverges to P D Sice f is cotiuous at P of { } P that coverges P, the image sequece { f P } k f P This is cotradicts the uboudedess of the sequece { } So f is bouded Defie c = sup f D Let be a atural umber The the umber f P k c is a smaller tha c, ad is therefore ot a upper boud for the set f D Thus there is a poit Q=, y D at which f Q > c hoose such a poit ad label it Q From this choice ad from the fact that c is a upper boud for f D, we see that c < f Q < c for ever atural umber Hece the sequece f Q coverges to c The olzao-weierstrass Theoremasserts that there is a subsequece { Q k } of { } the image sequece { } Q that coverges to a poit Q D Sice f is cotiuous at Q, f coverges to Q Q k f ut { } f is a subsequece Q k of f Q, so c = f Q The poit Q is a maimizer of the fucuio f To complete the proof, we observe that the fuctio f is also cotiuous osequetly, usig what we have just prove, we may select a poit i D at which f attais a maimum value, ad at this poit the fuctio f attais a miimum value - 4 -

6 Theorem If f is partial differetiable at P =, y ad get the etreme value at P, the f P = f P The proof is omitted y = The Lagrage Mea Value Theorem of Fuctios of Two Variables Itroductio From the geometric cosequece of Lagrage Mea Value Theorem of fuctios of oe variable we kow that: Let f is differetiable If the curve l : y = f itersects the lie l : + y + = at poits M ad N The there eists a poit P, f betwee poits M ad N such that f ' that is to say the taget lie to the curve l at poit P parallels to the lie l =, Furthermore, for a fuctio f of two variables, let f be cotiuous o the domai D ad differetiable oit D If the plae π : + y + z + D = ad the curved surface Ω : z = f, y have itersectio poits, the whether there eists a poit P=, y, f, y such that the taget plae to the surface Ω at the poit P parallels to the plae Eamples π That is to say f, y =, f y, y = Suppose that the surface Ω : z = f, y ad the plae have itersectio poits Eample Let f, y = y, the f, y = y f y, y = π : + y + z + D = So there eists the poit P =,,, with the property that the taget plae to the surface Ω is parallel to the plae π Eample Let = + y, the f, y + y - 5 -

7 f, y = f y, y = + y + y Hece, f, y =, f y, y = if ad oly if =, that is =, there eists a poit P with the property that the taget plae to the surface Ω is parallel to the plae π It follows from the above eamples that it demad more coditios to determie the eistece of poit P with the property The Lagrage Mea Value Theorem of fuctios of two variables Teorem Suppose that the fuctio f of two variables satisfies the followig coditios : f is cotiuous o E ; f is differetiable o it E ; 3 There eists a bouded regio E such that E E ad arbitrary poit P=, y E is a root of the equatio + y + z + D = The there eists a poit P it E with the property that f P =, fy P = That is to say the taget plae fo the surface Ω : z = f, y at poit P is parallel to the plae π : + y + z + D = Proof Defie the auiliary fuctio F by F, y = + y+ f, y + D The obviously, P=, y E, we have FP= Moreover, from the additio theorems for cotiuous ad differetiable fuctios, it follows that F is cotiuous o E ad differetiable o it E ccordig to the Theorem,the fuctio attais both a miimum value ad a maimum value o E Sice F, y = for, y E, if both the maimizers ad the miimizers occur at poits i E, the the fuctio F is idetically equal to o E, so F P = F P = at every poit P i it E ad hece, by Theorem we get y F P = F P = Thus, there must eist a poit P it E such that y - 6 -

8 F P = F P =, ie y + f P = + fy P = So, f P =, fy P = Hece the taget plae to the surface Ω : z = f, y at poit P is parallel to the plae π : + y + z + D = Eample 3 Let f y, = + y π : + y z+ = 3 the domai E =, y + y = We ca verify that they satisfy the three coditios of the theorem So it is sure that there eists a poit P it E such that the taget plae to the surface Ω : z = y at poit P is parallel to the plae π : + y z+ = It is easy to get P =,, 3 The Lagrage Mea Value Theorem of Fuctios of Variables We ca geeralize the limit, cotiuity ad differetiatio of fuctios of two variables to obtai the similar defiitios ad properties Defiitio fuctio of variables f o D R is cotiuous at P D if for ε > there eists δ > satisfyig f P f P < ε P U P ; δ D f is cotiuous o D if f is cotiuous at every poit of D Defiitio Let the fuctio y = f,,, be defied o the eighborhood of P =,,, U P For P + Δ, + Δ,, + Δ U if P Δy = f P f P = Δ + Δ + + Δ + ο ρ where i i =,, are costats oly determied by the poit P ad - 7 -

9 ρ = Δ ο ρ is ifiitesimal of higher order tha ρ The f is differetiable at P D pplyig the aalogy proof method, we ca get the followig theorem: i= Theorem 3 Suppose that f is cotiuous o the bouded closed regio R, the f attais both a miimum ad a maimum value o D i Theorem 3 If f is partial differetiable at P =,,, ad get the etreme value at P the f P = f P = = f P = Theorem 33 Suppose that the fuctio f of variables satisfies the followig coditios: f is cotiuous o E R ; f is differetiable o it E ; 3 There eists a bouded regio E such that E E ad arbitrary poit P=,,, E is a root of the equatio = f,,, + The there eists a poit P it E with the property that i f i Pi = i =,,, The proofs are similar to the two variable case Referece [] Liu Li-hua Itro alculus[m] eijig: 国防工业 Press, 4 [] Liu Ji-ia, Qiu Ji-qig, Ha Xiao-big alculus of Oe Variable[M] eijig: Higher Educatio Press, 3 [3] Partrick M Fitpatrick dvaced alculus[m] eijig: hia Machie Press, 3 [4] Xiog Ya Lagrage Mea Value Theorem of fuctios of variables[j] Joural of Xijiag Employee Uiversity, 995, 3: -8 [5] 张飞军外微分初步 [M] 西安 : 陕西师范大学出版社,6 [6] 华东师范大学数学系数学分析 [M] 北京 : 高等教育出版, - 8 -

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges