Ostrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias


 Primrose Gilbert
 1 years ago
 Views:
Transcription
1 Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology, PO Box 448, Melbourne City, MC 800, Victori, Austrli Emil ddress, SS Drgomir: URL: TM Rssis) Deprtment of Mthemtics, Ntionl Technicl University of Athens, Zogrfou Cmpus, 5780 Athens, Greece
2 99 Mthemtics Subject Clssifiction Primry 6D5, 6D0; Secondry 4A55, 4A99 Abstrct Ostrowski type inequlities for univrite nd multivrite rel functions nd their nturl pplictions for numericl qudrtures re presented
3 Contents Prefce v Chpter Generlistions of Ostrowski Inequlity nd Applictions Introduction Generlistions for Functions of Bounded Vrition 3 3 Generlistions for Functions whose Derivtives re in L 5 4 Generlistion for Functions whose Derivtives re in L p 8 5 Generlistions in Terms of L norm 4 Bibliogrphy 53 Chpter Integrl Inequlities for n Times Differentible Mppings 55 Introduction 55 Integrl Identities 56 3 Integrl Inequlities 64 4 The Convergence of Generl Qudrture Formul 7 5 Grüss Type Inequlities 75 6 Some Prticulr Integrl Inequlities 80 7 Applictions for Numericl Integrtion 04 8 Concluding Remrks 8 Bibliogrphy 9 Chpter 3 Three Point Qudrture Rules 3 Introduction 3 Bounds Involving t most First Derivtive 3 33 Bounds for n Time Differentible Functions 86 Bibliogrphy Chpter 4 Product Brnches of Peno Kernels nd Numericl Integrtion 5 4 Introduction 5 4 Fundmentl Results 7 43 Simpson Type Formule 4 44 Perturbed Results 6 45 More Perturbed Results Using Seminorms Concluding Remrks 4 Bibliogrphy 43 Chpter 5 Ostrowski Type Inequlities for Multiple Integrls 45 5 Introduction 45 iii
4 CONTENTS iv 5 An Ostrowski Type Inequlity for Double Integrls Other Ostrowski Type Inequlities Ostrowski s Inequlity for Hölder Type Functions 73 Bibliogrphy 8 Chpter 6 Some Results for Double Integrls Bsed on n Ostrowski Type Inequlity 83 6 Introduction 83 6 The One Dimensionl Ostrowski Inequlity Mpping Whose First Derivtives Belong to L, b) Numericl Results Appliction For Cubture Formule Mpping Whose First Derivtives Belong to L p, b) Appliction For Cubture Formule Mppings Whose First Derivtives Belong to L, b) Integrl Identities Some Integrl Inequlities Applictions to Numericl Integrtion 3 Bibliogrphy 35 Chpter 7 Product Inequlities nd Weighted Qudrture 37 7 Introduction 37 7 Weight Functions Weighted Interior Point Integrl Inequlities Weighted Boundry Point Lobtto) Integrl Inequlities Weighted Three Point Integrl Inequlities 339 Bibliogrphy 35 Chpter 8 Some Inequlities for RiemnnStieltjes Integrl Introduction Some Trpezoid Like Inequlities for RiemnnStieltjes Integrl Inequlities of Ostrowski Type for the RiemnnStieltjes Integrl Some Inequlities of Grüss Type for RiemnnStieltjes Integrl 39 Bibliogrphy 40
5 v Prefce It ws noted in the prefce of the book Inequlities Involving Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, 99, by DS Mitrinović, JE Pečrić nd AM Fink; since the writing of the clssicl book by Hrdy, Littlewood nd Poly 934), the subject of differentil nd integrl inequlities hs grown by bout 800% Ten yers on, we cn confidently ssert tht this growth will increse even more significntly Twenty pges of Chpter XV in the bove mentioned book re devoted to integrl inequlities involving functions with bounded derivtives, or, Ostrowski type inequlities This is now itself specil domin of the Theory of Inequlities with mny powerful results nd lrge number of pplictions in Numericl Integrtion, Probbility Theory nd Sttistics, Informtion Theory nd Integrl Opertor Theory The min im of this present book, jointly written by the members of the Victori University node of RGMIA Reserch Group in Mthemticl Inequlities nd Applictions, is to present selected number of results on Ostrowski type inequlities Results for univrite nd multivrite rel functions nd their nturl pplictions in the error nlysis of numericl qudrture for both simple nd multiple integrls s well s for the RiemnnStieltjes integrl re given In Chpter, uthored by SS Drgomir nd TM Rssis, generlistions of the Ostrowski integrl inequlity for mppings of bounded vrition nd for bsolutely continuous functions vi kernels with n brnches including pplictions for generl qudrture formule, re given Chpter, uthored by A Sofo, builds on the work in Chpter He investigtes generlistions of integrl inequlities for ntimes differentible mppings With the id of the modern theory of inequlities nd by use of generl Peno kernel, explicit bounds for interior point rules re obtined Firstly, he develops integrl equlities which re then used to obtin inequlities for ntimes differentible mppings on the Lebesgue spces L, b, L p, b, < p < nd L, b Secondly, some prticulr inequlities re obtined which include explicit bounds for perturbed trpezoid, midpoint, Simpson s, NewtonCotes, left nd right rectngle rules Finlly, inequlities re lso pplied to vrious composite qudrture rules nd the nlysis llows the determintion of the prtition required for the ccurcy of the result to be within prescribed error tolernce In Chpter 3, uthored by P Cerone nd SS Drgomir, unified tretment of three point qudrture rules is presented in which the clssicl rules of midpoint, trpezoidl nd Simpson type re recptured s prticulr cses Riemnn integrls re pproximted for the derivtive of the integrnd belonging to vriety of norms The Grüss inequlity nd number of vrints re lso presented which provide vriety of inequlities tht re suitble for numericl implementtion Mppings tht re of bounded totl vrition, Lipschitzin nd monotonic re lso investigted with reltion to RiemnnStieltjes integrls Explicit priori bounds re provided llowing the determintion of the prtition required to chieve prescribed error tolernce
6 CONTENTS vi It is demonstrted tht with the bove clsses of functions, the verge of midpoint nd trpezoidl type rule produces the best bounds In Chpter 4, uthored by P Cerone, product brnches of Peno kernels re used to obtin results suitble for numericl integrtion In prticulr, identities nd inequlities re obtined involving evlutions t n interior nd t the end points It is shown how previous work nd rules in numericl integrtion re recptured s prticulr instnces of the current development Explicit priori bounds re provided llowing the determintion of the prtition required for chieving prescribed error tolernce In the min, OstrowskiGrüss type inequlities re used to obtin bounds on the rules in terms of vriety of norms In Chpter 5, uthored by NS Brnett, P Cerone nd SS Drgomir, new results for Ostrowski type inequlities for double nd multiple integrls nd their pplictions for cubture formule re presented This work is then continued in Chpter 6, uthored by G Hnn, where n Ostrowski type inequlity in two dimensions for double integrls on rectngle region is developed The resulting integrl inequlities re evluted for the clss of functions with bounded first derivtive They re employed to pproximte the double integrl by one dimensionl integrls nd function evlutions using different types of norms If the onedimensionl integrls re not known, they themselves cn be pproximted by using suitble rule, to produce cubture rule consisting only of smpling points In ddition, some generlistions of n Ostrowski type inequlity in two dimensions for n  time differentible mppings re given The result is n integrl inequlity with bounded n  time derivtives This is employed to pproximte double integrls using one dimensionl integrls nd function evlutions t the boundry nd interior points In Chpter 7, uthored by John Roumeliotis, weighted qudrture rules re investigted The results re vlid for generl weight functions The robustness of the bounds is explored for specific weight functions nd for vriety of integrnds A comprison of the current development is mde with trditionl qudrture rules nd it is demonstrted tht the current development hs some dvntges In prticulr, this method llows the nodes nd weights of n n point rule to be esily obtined, which my be preferentil if the region of integrtion vries Other explicit error bounds my be obtined in dvnce, thus mking it possible to determine the weight dependent prtition required to chieve certin error tolernce In the lst chpter, SS Drgomir presents recent results in pproximting the RiemnnStieltjes integrl by the use of Trpezoid type, Ostrowski type nd Grüss type inequlities Applictions for certin clsses of weighted integrls re lso given This book is intended for use in the fields of integrl inequlities, pproximtion theory, pplied mthemtics, probbility theory nd sttistics nd numericl nlysis
7 vii The Editors, Melbourne nd Athens, December 000
8
9 CHAPTER Generlistions of Ostrowski Inequlity nd Applictions by SS DRAGOMIR nd TM RASSIAS Abstrct Generliztions of Ostrowski integrl inequlity for mppings of bounded vrition nd for bsolutely continuous functions vi kernels with n brnches plus pplictions for generl qudrture formule re given Introduction The following result is known in the literture s Ostrowski s inequlity see for exmple, p 468) Theorem Let f :, b R be differentible mpping on, b) with the property tht f t) M for ll t, b) Then ) f x) b ) f t) dt b x +b 4 + b ) b ) M for ll x, b The constnt 4 is the best possible in the sense tht it cnnot be replced by smller constnt A simple proof of this fct cn be done by using the identity: ) f x) b where f t) dt + b t if t x p x, t) : t b if x < t b p x, t) f t) dt, x, b, which lso holds for bsolutely continuous functions f :, b R The following Ostrowski type result for bsolutely continuous functions holds see 7, 0 nd 8)
10 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS Theorem Let f :, b R be bsolutely continuous on, b Then, for ll x, b, we hve: f x) 3) f t) dt b 4 + x +b ) b b ) f if f L, b ; where r nd p+) p x b ) p+ + + x +b b f ; The constnts 4, Theorem b x b ) p+ p b ) p f q if f L q, b, r, ) re the usul Lebesgue norms on L r, b, ie, g : ess sup g t) t,b g r : g t) r dt p+) p nd ) r, r, ) p + q, p > ; respectively re shrp in the sense presented in The bove inequlities cn lso be obtined from the Fink result in on choosing n nd performing some pproprite computtions If one drops the condition of bsolute continuity nd ssumes tht f is Hölder continuous, then one my stte the result see 5) Theorem 3 Let f :, b R be of r H Hölder type, ie, 4) f x) f y) H x y r, for ll x, y, b, where r 0, nd H > 0 re fixed Then for ll x, b we hve the inequlity: f x) 5) f t) dt b b ) r+ ) r+ H x x + b ) r r + b b The constnt r+ is lso shrp in the bove sense Note tht if r, ie, f is Lipschitz continuous, then we get the following version of Ostrowski s inequlity for Lipschitzin functions with L insted of H) due to SS Drgomir 3, see lso ) 6) f x) b Here the constnt 4 is lso best f t) dt x +b 4 + b ) b ) L
11 3 SS Drgomir nd TM Rssis Moreover, if one drops the condition of the continuity of the function, nd ssumes tht it is of bounded vrition, then the following result due to Drgomir my be stted see lso 4 or ) Theorem 4 Assume tht f :, b R is of bounded vrition nd denote by b its totl vrition Then 7) f x) b for ll x, b The constnt is the best possible f t) dt + x +b b b f) If we ssume more bout f, ie, f is monotoniclly incresing, then the inequlity 7) my be improved in the following mnner see lso ) Theorem 5 Let f :, b R be monotonic nondecresing Then for ll x, b, we hve the inequlity: 8) f x) f t) dt b { x + b) f x) + b sgn t x) f t) dt {x ) f x) f ) + b x) f b) f x)} b + x +b f b) f ) b } All the inequlities in 8) re shrp nd the constnt is the best possible For recent generlistion of this result see 6 where further extensions were given The min im of the present chpter is to provide number of generlistions for kernels with N brnches of the bove Ostrowski type inequlity Nturl pplictions for qudrture formule re lso given Generlistions for Functions of Bounded Vrition Some Inequlities We strt with the following theorem 4 Theorem 6 Let I k : x 0 < x < < x k < x k b be division of the intervl, b nd α i i 0,, k + ) be k+ points so tht α 0, α i x i, x i i,, k) nd α k+ b If f :, b R is of bounded vrition on, b, then
12 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 4 we hve the inequlity: 9) f x) dx ν h) + mx ν h) b f), k α i+ α i ) f x i ) { α i+ x } i + x i+ b, i 0,, k f) where ν h) : mx {h i i 0,, k }, h i : x i+ x i i 0,, k ) nd b f) is the totl vrition of f on the intervl, b Proof Define the kernel K :, b R given by see lso 4) t α, t, x ) t α, t x, x ) K t) : t α k, t x k, x k ) t α k, t x k, b Integrting by prts in RiemnnStieltjes integrl, we hve successively K t) df t) k xi+ k k K t) df t) x i t α i+ ) f t) xi+ x i xi+ x i xi+ x i t α i+ ) df t) f t) dt k α i+ x i ) f x i ) + x i+ α i+ ) f x i+ ) k α ) f ) + α i+ x i ) f x i ) k + i x i+ α i+ ) f x i+ ) + b α n ) f b) k k α ) f ) + α i+ x i ) f x i ) + x i α i ) f x i ) + b α n ) f b) i f t) dt k α ) f ) + α i+ α i ) f x i ) + b α n ) f b) i k α i+ α i ) f x i+ ) f t) dt, i f t) dt f t) dt f t) dt
13 5 SS Drgomir nd TM Rssis nd then we hve the integrl equlity which is of interest in itself too: 0) f t) dt k α i+ α i ) f x i ) Using the modulus properties, we hve k K t) df t) k However, xi+ t α i+ ) f t) dt sup x i t x i,x i+ xi+ x i xi+ x i K t) df t) K t) df t) K t) df t) : T x i+ t α i+ f) x i+ mx {α i+ x i, x i+ α i+ } f) x i Then T x i+ x i ) + α i+ x i + x i+ x i x i+ f)dt k h i + α i+ x x i + x i+ i+ f) x i mx,,k h i + α i+ x i + x i+ k x i+ f) x i { ν h) + mx α i+ x } i + x i+ b, i 0,, k f) : V x i Now, s then mx nd, consequently, α i+ x i + x i+ h i, { α i+ x i + x i+ The theorem is completely proved V ν h) }, i 0,, k ν h) b f) Now, if we ssume tht the points of the division I k re given, then the best inequlity we cn obtin from Theorem 6 is embodied in the following corollry:
14 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 6 Corollry 7 Let f nd I k be s bove Then we hve the inequlity: f x) dx ) x ) f ) k + x i+ x i ) f x i ) + b x k ) f b) i b ν h) f) Proof We choose in Theorem 6, In this cse we get α 0, α + x, α x + x,, α k x k + x k, α k x k + x k k α i+ α i ) f x i ) α α 0 ) f ) + α α ) f x ) + nd α k+ b + α k α k f x k ) + b α k ) f b) ) + x x + x f ) + + x ) f x ) xk + b + + x ) k + x k f x k ) + b x ) k + b f b) k x ) f ) + x i+ x i ) f x i ) + b x k ) f b) i Now, pplying the inequlity 9), we get ) The following corollry for equidistnt prtitioning lso holds Corollry 8 Let I k : x i : + b ) i i 0,, k) k be n equidistnt prtitioning of, b If f is s bove, then we hve the inequlity: b k f ) + f b) b ) k i) + ib ) f x) dx b ) + f k k k b k b ) f) i
15 7 SS Drgomir nd TM Rssis A Generl Qudrture Formul Let n : x n) 0 < x n) < < x n) n < xn) n b be sequence of division of, b nd consider the sequence of numericl integrtion formule I n f, n, w n ) : n j0 w n) j f x n) j where w n) j j 0,, n) re the qudrture weights nd n j0 wn) j b The following theorem provides sufficient condition for the weights w n) j I n f, n, w n ) pproximtes the integrl f x) dx see lso 4) ) so tht Theorem 9 Let f :, b R be function of bounded vrition on, b If the qudrture weights w n) j stisfy the condition i 3) x n) i w n) j x n) i+ for ll i 0,, n, j0 then we hve the estimte I 4) n f, n, w n ) f x) dx h ν n)) i + mx + ν h n)) b f), j0 w n) j xn) i + x n) b i+, i 0,, n f) where ν h n)) { } : mx h n) i i 0,, n nd h n) i : x n) i+ xn) i In prticulr, 5) lim I n f, n, w n ) νh n) ) 0 uniformly by rpport of the w n Proof Define the sequence of rel numbers Note tht α n) α n) nd observe lso tht α n) i+ i+ : + i n+ + n x n) i j0 j0, x n) i+ f x) dx w n) j, i 0,, n w n) j + b b, Define α n) 0 : nd compute α n) α n) 0,
16 Then GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 8 α n) i+ αn) i + n i j0 α n) n+ αn) n ) α n) i+ αn) i f w n) j i j0 w n) j n b + x n) i ) n j0 w n) i i,, n ), w n) j w n) i f x n) i Applying the inequlity 9), we get the estimte 4) w n) n ) I n f, n, w n ) The uniform convergence by rpport of qudrture weights w n) j lst inequlity is obvious by the Now, consider the equidistnt prtitioning of, b given by E n : x n) i : + i b ) i 0,, n) n nd define the sequence of numericl qudrture formule n I n f, w n ) : w n) i f + in b ) The following corollry which cn be more useful in prctice holds: Corollry 0 Let f be s bove If the qudrture weight w n) j condition: 6) i n b i j0 then we hve: I 7) n f, w n ) f x) dx b n + mx i + b ) n b f) j0 In prticulr, we hve the limit uniformly by rpport of w n w n) j i +, i 0,, n ; n w n) j i + lim I n f, w n ) n b ) n f x) dx, stisfy the b, i 0,, n f)
17 9 SS Drgomir nd TM Rssis 3 Prticulr Inequlities The following proposition holds 4 see lso 4) Proposition Let f :, b R be function of bounded vrition on, b Then we hve the inequlity: 8) f x) dx α ) f ) + b α) f b) b ) + α + b b f) for ll α, b The proof follows by Theorem 6 choosing x 0, x b, α 0, α α, b nd α b Remrk ) If in 8) we put α b, then we get the left rectngle inequlity 9) b f x) dx b ) f ) b ) f); b) If α, then by 8) we get the right rectngle inequlity b 0) f x) dx b ) f b) b ) f); c) It is esy to see tht the best inequlity we cn get from 8) is for α +b obtining the trpezoid inequlity see lso 4) ) f x) dx f ) + f b) b ) b b ) f) Another proposition with mny interesting prticulr cses is the following one 4 see lso 4): Proposition Let f be s bove nd x b, α x α b Then we hve ) f x) dx α ) f ) + α α ) f x ) + b α ) f b) b ) + x + b + α + x + α x + b + α + x α x + b b f) b ) + x + b b b f) b ) f)
18 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 0 Proof Consider the division x 0 x x b nd the numbers α 0, α, x, α x, b nd α 3 b Now, pplying Theorem 6, we get f x) dx α ) f ) + α α ) f x ) + b α ) f b) { mx {x, b x } + mx α + x, α x } + b b f) 4 b ) + x + b + α + x + α x + b + α + x α x + b b f) nd the first inequlity in ) is proved Now, let observe tht α + x x, α x + b b x Consequently, mx { α + x nd the second inequlity in ) is proved The lst inequlity is obvious, α x } + b mx {x, b x } Remrk ) If we choose bove α, α b, then we get the following Ostrowski type inequlity obtined by Drgomir in the recent pper : 3) f x) dx b ) f x ) b ) + x + b b f) for ll x, b We note tht the best inequlity we cn get in 3) is for x +b obtining the midpoint inequlity see lso ) ) + b 4) f x) dx f b ) b b ) f) b) If we choose in ) α 5+b 6, α +5b 6 nd x 5+b 6, +5b 6, then we get f x) dx b f ) + f b) + f x ) 5) 3 b ) + x + b { + mx x + b } 3, + b b x f) 3
19 SS Drgomir nd TM Rssis In prticulr, if we choose in 5), x +b, then we get the following Simpson s inequlity 9 f x) dx b ) f ) + f b) + b 6) + f 3 b 3 b ) f) 4 Prticulr Qudrture Formule Let us consider the prtitioning of the intervl, b given by n : x 0 < x < < x n < x n b nd put h i : x i+ x i i 0,, n ) nd ν h) : mx {h i i 0,, n } The following theorem holds 4: Theorem 3 Let f :, b R be bsolutely continuous on, b nd k Then we hve the composite qudrture formul 7) where 8) A k n, f) : T n, f) + k nd f x) dx A k n, f) + R k n, f), n k k j) xi + jx i+ f k j 9) T n, f) : n f x i ) + f x i+ ) h i is the trpezoid qudrture formul The reminder R k n, f) stisfies the estimte 30) R k n, f) b k ν h) f) h i get Proof Applying Corollry 8 on the intervls x i, x i+ i 0,, n ) we xi+ x i x k h i+ i f) f x) dx k x i f x i ) + f x i+ ) h i + h i k k k j) xi + jx i+ f k j
20 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS Now, using the generlized tringle inequlity, we get: R k n, f) n xi+ f x) dx x i k f x i) + f x i+ ) h i + h k i k j) xi + jx i+ f k k n k f) ν h) n k h i x i+ x i nd the theorem is proved x i+ x i f) ν h) k b f) j The following corollries hold: Corollry 4 Let f be s bove Then we hve the formul: 3) f x) dx T n n, f) + M n n, f) + R n, f) where M n n, f) is the midpoint qudrture formul, n ) xi + x i+ M n n, f) : f h i nd the reminder R n, f) stisfies the inequlity: 3) R n, f) b 4 ν h) f) Corollry 5 Under the bove ssumptions we hve b f x) dx n ) xi + x i+ 33) T n n, f) + f h i 3 3 n ) xi + x i+ + f h i + R 3 n, f) 3 The reminder R 3 n, f) stisfies the bound: 34) R 3 n, f) b 6 ν h) f) The following theorem holds 4 see lso 4): Theorem 6 Let f nd n be s bove nd ξ i x i, x i+ i 0,, n ) Then we hve the qudrture formul: 35) n f x) dx ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) + R ξ, n, f)
21 3 SS Drgomir nd TM Rssis The reminder R ξ, n, f) stisfies the estimtion: 36) R ξ, n, f) { ν h) + mx ξ i x } i + x i+ b, i 0,, n f) ν h) for ll ξ i s bove b f), get Proof Apply Proposition on the intervl x i, x i+ i 0,, n ) to xi+ f x) dx ξ i x i ) f x i ) + x i+ ξ i ) f x i+ ) x i { h i + mx ξ i x } x i + x i+ i+ f) x i Summing over i from 0 to n, using the generlized tringle inequlity nd the properties of the mximum mpping, we get 36) Corollry 7 Let f nd n be s bove Then we hve ) the left rectngle rule 37) n f x) dx f x i ) h i + R l n, f) ; ) the right rectngle rule 38) n f x) dx f x i+ ) h i + R r n, f) ; 3) the trpezoid rule 39) where nd f x) dx T n, f) + R T n, f) b R l n, f) R r n, f) ν h) f) R T n, f) b ν h) f) The following theorem lso holds 4
22 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 4 Theorem 8 Let f nd n be s bove nd ξ i x i, x i+, x i α ) i x i+, then we hve the qudrture formul: α ) i 40) n f x) dx n + α ) i x i ) f x i ) + x i+ α ) i n ) f x i+ ) + R ) α ) i α ) i f ξ i ) ) ξ, α ), α ), n, f ξ i The reminder R ξ, α ), α ), n, f ) stisfies the estimtion R ξ, α ), α ) 4), n, f) { ν h) + mx,,n ξ i x i + x i+ { + mx mx,,n α) i x i + ξ i, mx,,n α) i ξ }} i + x i+ b f) ν h) + mx,n ξ i x i + x i+ b f ν h) f) Proof Apply Proposition on the intervl x i, x i+ to obtin xi+ f x) dx x i ) ) ) α ) i x i f x i ) + α ) i α ) i f ξ i ) + x i+ α ) i f x i+ ) h i + ξ i x i + x i+ { + mx α ) i x i + ξ i, α ) i ξ } x i + x i+ i+ f) x i Summing over i from 0 to n nd using the properties of modulus nd mximum, we get the desired inequlity We shll omit the detils The following corollry is the result of Drgomir from the recent pper Corollry 9 Under the bove ssumptions, we hve the Riemnn s qudrture formul: 4) n f x) dx f ξ i ) h i + R R ξ, n, f)
23 5 SS Drgomir nd TM Rssis The reminder R R ξ, n, f) stisfies the bound 43) R R ξ, n, f) { ν h) + mx ξ i x } i + x i+ b, i 0,, n f) ν h) b f) for ll ξ i x i, x i+ i 0,, n) Finlly, the following corollry which generlizes Simpson s qudrture formul holds Corollry 0 Under the bove ssumptions nd if ξ i xi++5x i 6, xi+5xi+ 6 i 0,, n ), then we hve the formul: 44) f x) dx n f x i ) + f x i+ ) h i + n f ξ 6 3 i ) h i + S f, n, ξ) The reminder S f, n, ξ) stisfies the estimte: 45) S f, n, ξ) { { ν h) + mx ξ i x } i + x i+,,n { + mx mx,,n ξ i x i + x i+ 3, mx,,n x i + x i+ 3 The proof follows by the inequlity 5) nd we omit the detils ξ i }} b Remrk 3 Now, if we choose in 44), ξ i xi+xi+, then we get Simpson s qudrture formul 9 46) f x) dx n f x i ) + f x i+ ) h i 6 + n ) xi + x i+ f h i + S f, n ) 3 where the reminder term S f, n ) stisfies the bound: 47) S f, n ) b 3 ν h) f) f) 3 Generlistions for Functions whose Derivtives re in L 3 Some Inequlities We strt with the following result 5 Theorem Let I k : x 0 < x < < x k < x k b be division of the intervl, b, α i i 0,, k + ) be k + points so tht α 0, α i
24 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 6 x i, x i i,, k) nd α k+ b If f :, b R is bsolutely continuous on, b, then we hve the inequlity: 48) k f x) dx α i+ α i ) f x i ) k k h i + α i+ x ) i + x i+ f 4 k f h i b ) f ν h), where h i : x i+ x i i 0,, k ) nd ν h) : mx {h i i 0,, k } The constnt 4 in the first inequlity nd the constnt in the second nd third inequlity re the best possible Proof Define the mpping K :, b R given by see the proof of Theorem 6) K t) : t α, t, x ) ; t α, t x, x ) ; t α k, t x k, x k ) ; t α k, t x k, b Integrting by prts, we hve successively: K t) f t) dt k xi+ k k K t) f t) dt x i t α i+ ) f t) xi+ x i xi+ x i xi+ x i t α i+ ) f t) dt f t) dt k α i+ x i ) f x i ) + x i+ α i+ ) f x i+ ) f t) dt k k α ) f ) + α i+ x i ) f x i ) + x i+ α i+ ) f x i+ ) + b α n ) f b) i f t) dt
25 7 SS Drgomir nd TM Rssis k k α ) f ) + α i+ x i ) f x i ) + x i α i ) f x i ) + b α n ) f b) i f t) dt k α ) f ) + α i+ α i ) f x i ) + b α n ) f b) i k α i+ α i ) f x i ) f t) dt nd then we hve the integrl equlity: k 49) f t) dt α i+ α i ) f x i ) Using the properties of modulus, we hve K t) f 50) t) dt k xi+ k K t) f t) dt x i k xi+ x i A simple clcultion shows tht 5) xi+ x i t α i+ dt for ll i 0,, k xi+ x i t α i+ f t) dt f αi+ i K t) f t) dt K t) f t) dt k xi+ x i xi+ f t) dt t α i+ dt α i+ t) dt + t α i+ ) dt x i α i+ x i+ α i+ ) + α i+ x i ) 4 h i + α i+ x ) i + x i+ Now, by 49) 5), we get the first inequlity in 48) Assume tht the first inequlity in 48) holds for constnt c > 0, ie, k 5) f x) dx α i+ α i ) f x i ) k k c h i + α i+ x ) i + x i+ f If we choose f :, b R, f x) x, α 0, α b, x 0, x b in 5), we obtin b ) c b ) b ) +, 4
26 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 8 from where we get c 4, nd the shrpness of the constnt 4 is proved The lst two inequlities s well s the shrpness of the constnt we omit the detils re obvious nd Now, if we ssume tht the points of the division I k re given, then the best inequlity we cn get from Theorem is embodied in the following corollry: Corollry Let f, I k be s bove Then we hve the inequlity 53) f x) dx k x ) f ) + x i+ x i ) f x i ) + b x k ) f b) k 4 f h i i The constnt 4 is the best possible one Proof Similr to the proof of Corollry 7 The cse of equidistnt prtitioning is importnt in prctice Corollry 3 Let I k : x i +i b k i 0,, k) be n equidistnt prtitioning of, b If f is s bove, then we hve the inequlity b k f ) + f b) b ) k i) + ib 54) f x) dx b ) + f k k k 4k b ) f The constnt 4 is the best possible one Remrk 4 If k, then we hve the inequlity see for exmple 4) f ) + f b) 55) f x) dx b ) 4 b ) f Choose f :, b R, f x) x +b, which is LLipschitzin with L nd if t ), +b f x) if t +b, b Then f nd f x) dx f ) + f b) b ) i b ), 4 nd the equlity is obtined in 55), showing tht the constnt 4 is shrp
27 9 SS Drgomir nd TM Rssis 3 A Generl Qudrture Formul Let n : x n) 0 < x n) < < x n) n < xn) n b be sequence of divisions of, b nd consider the sequence of numericl integrtion formule where w n) j b I n f, n, w n ) : n j0 w n) j f x n) j j 0,, n) re the qudrture weights nd ssume tht n j0 wn) j The following theorem contins sufficient condition for the weights w n) j so tht I n f, n, w n ) pproximtes the integrl f x) dx with n error expressed in terms of f see lso 5) Theorem 4 Let f :, b R be n bsolutely continuous mpping on, b If the qudrture weights w n) j j 0,, n) stisfy the condition i 56) x n) i w n) j x n) i+ for ll i 0,, n ; j0 then we hve the estimtion I 57) n f, n, w n ) f x) dx n n i h n) i f n h n) i j0 j0 w n) j xn) i ), f b ) ν h n)), where ν h n)) : mx{h n) i : i 0,, n } nd h n) i In prticulr, if f <, then lim I n f, n, w n ) νh n) ) 0 uniformly by the influence of the weights w n + x n) i+ f : x n) i+ xn) i f x) dx Proof Similr to the proof of Theorem 9 nd we omit the detils The cse when the prtitioning is equidistnt is importnt in prctice Consider, then, the prtitioning E n : x n) i : + i b i 0,, n), n nd define the sequence of numericl qudrture formule n I n f, w n ) : w n) i f + i b ) n, w n) j b n j0
28 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS 0 The following result holds: Corollry 5 Let f :, b R be bsolutely continuous on, b If the qudrture weights w n) i stisfy the condition: i n i w n) j i + i 0,, n ); b n j0 then we hve the estimte I n f, w n ) f x) dx f n 4n b i ) + n f b ) In prticulr, if f <, then uniformly by the influence of w n lim I n f, w n ) n j0 w n) j i + f x) dx b n 33 Prticulr Inequlities In this subsection we point out some prticulr inequlities which generlize some clssicl results such s: rectngle inequlity, trpezoid inequlity, Ostrowski s inequlity, midpoint inequlity, Simpson s inequlity nd others in terms of the supnorm of the derivtive 5 see lso 4) Proposition 6 Let f :, b R be bsolutely continuous on, b nd α, b Then we hve the inequlity: 58) f x) dx α )f ) + b α)f b) 4 b ) + α + b ) f b ) f The constnt 4 is the best possible one Proof Follows from Theorem by choosing x 0, x b, α 0, α α, b nd α b Remrk 5 ) If in 58) we put α b, then we get the left rectngle inequlity : 59) f x) dx b ) f ) b ) f
29 SS Drgomir nd TM Rssis b) If α, then by 58) we obtin the right rectngle inequlity 60) f x) dx b ) f ) b ) f 6) c) It is cler tht the best estimtion we cn hve in 58) is for α +b getting the trpezoid inequlity see lso ) f x) dx f ) + f b) b ) 4 b ) f Another prticulr integrl inequlity with mny pplictions is the following one 5: Proposition 7 Let f :, b R be n rbitrry bsolutely continuous mpping on, b nd α x α b Then we hve the inequlity: 6) f x) dx α )f ) + α α )fx ) + b α ) f b) 8 b ) + x + b ) + α + x ) + α x ) + b f Proof Follows by Theorem nd we omit the detils Corollry 8 Let f be s bove nd x, b Then we hve Ostrowski s inequlity: 63) f x) dx b ) fx ) 4 b ) + x + b ) f Remrk 6 If we choose x +b in 63), we obtin the midpoint inequlity 64) ) + b f x) dx b ) f 4 b ) f The following corollry generlizing Simpson s inequlity holds: Corollry 9 Let f be s bove nd x 5+b 6, +5b 6 Then we hve the inequlity f x) dx b f ) + f b) + fx ) 65) b ) + x + b ) f Proof Follows by Proposition 7 using simple computtion nd we omit the detils
30 GENERALISATIONS OF OSTROWSKI INEQUALITY AND APPLICATIONS Remrk 7 Let us observe tht the best estimtion we cn obtin from 65) is tht one for which x +b, obtining the Simpson s inequlity 0 66) f x) dx b 3 f ) + f b) + b + f ) 5 36 b ) f The following corollry lso holds Corollry 30 Let f be s bove nd α +b α b Then we hve the inequlity 67) 8 b ) + f x) dx α )f ) + α α )f α 3 + b ) + α + 3b b ) ) + b α )f b) f The proof is obvious by Proposition 7 by choosing x +b Remrk 8 The best estimtion we cn obtin from 67) is tht one for which α 3+b 4 nd α +3b 4, obtining the inequlity 3 68) f x) dx b f ) + f b) + b + f ) 8 b ) f The following proposition generlizes the threeeights rule of NewtonCotes: Proposition 3 Let f be s bove nd x x b nd α, x, α x, x, α 3 x, b Then we hve the inequlity 69) f x) dx α )f ) + α α )fx ) +α 3 α )fx ) + b α 3 )f b) 4 x ) + x x ) + b x ) + α + x ) + α x ) + x + α 3 x ) + b f The proof is obvious by Theorem The next corollry contins generliztion of the threeeights rule of Newton Cotes in the following wy:
Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationINTERCHANGING TWO LIMITS. Zoran Kadelburg and Milosav M. Marjanović
THE TEACHING OF MATHEMATICS 2005, Vol. VIII, 1, pp. 15 29 INTERCHANGING TWO LIMITS Zorn Kdelburg nd Milosv M. Mrjnović This pper is dedicted to the memory of our illustrious professor of nlysis Slobodn
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationCHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationA Note on Complement of Trapezoidal Fuzzy Numbers Using the αcut Method
Interntionl Journl of Applictions of Fuzzy Sets nd Artificil Intelligence ISSN  Vol.  A Note on Complement of Trpezoidl Fuzzy Numers Using the αcut Method D. Stephen Dingr K. Jivgn PG nd Reserch Deprtment
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationQUADRATURE METHODS. July 19, 2011. Kenneth L. Judd. Hoover Institution
QUADRATURE METHODS Kenneth L. Judd Hoover Institution July 19, 2011 1 Integrtion Most integrls cnnot be evluted nlyticlly Integrls frequently rise in economics Expected utility Discounted utility nd profits
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810193 Aveiro, Portugl
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationBabylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
More informationA new generalized Jacobi Galerkin operational matrix of derivatives: two algorithms for solving fourthorder boundary value problems
AbdElhmeed et l. Advnces in Difference Equtions (2016) 2016:22 DOI 10.1186/s1366201607532 R E S E A R C H Open Access A new generlized Jcobi Glerkin opertionl mtrix of derivtives: two lgorithms for
More information15.6. The mean value and the rootmeansquare value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the rootmensqure vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationAn Undergraduate Curriculum Evaluation with the Analytic Hierarchy Process
An Undergrdute Curriculum Evlution with the Anlytic Hierrchy Process Les Frir Jessic O. Mtson Jck E. Mtson Deprtment of Industril Engineering P.O. Box 870288 University of Albm Tuscloos, AL. 35487 Abstrct
More informationQuasilog concavity conjecture and its applications in statistics
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 R E S E A R C H Open Access Qusilog concvity conjecture nd its pplictions
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationOnline Multicommodity Routing with Time Windows
KonrdZuseZentrum für Informtionstechnik Berlin Tkustrße 7 D14195 BerlinDhlem Germny TOBIAS HARKS 1 STEFAN HEINZ MARC E. PFETSCH TJARK VREDEVELD 2 Online Multicommodity Routing with Time Windows 1 Institute
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More information4 Approximations. 4.1 Background. D. Levy
D. Levy 4 Approximtions 4.1 Bckground In this chpter we re interested in pproximtion problems. Generlly speking, strting from function f(x) we would like to find different function g(x) tht belongs to
More informationOptimal Control of Serial, MultiEchelon Inventory/Production Systems with Periodic Batching
Optiml Control of Seril, MultiEchelon Inventory/Production Systems with Periodic Btching GeertJn vn Houtum Deprtment of Technology Mngement, Technische Universiteit Eindhoven, P.O. Box 513, 56 MB, Eindhoven,
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationThe Riemann Integral. Chapter 1
Chpter The Riemnn Integrl now of some universities in Englnd where the Lebesgue integrl is tught in the first yer of mthemtics degree insted of the Riemnn integrl, but now of no universities in Englnd
More informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationThe Fundamental Theorem of Calculus for Lebesgue Integral
Divulgciones Mtemátics Vol. 8 No. 1 (2000), pp. 75 85 The Fundmentl Theorem of Clculus for Lebesgue Integrl El Teorem Fundmentl del Cálculo pr l Integrl de Lebesgue Diómedes Bárcens (brcens@ciens.ul.ve)
More informationHomogenization of a parabolic equation in perforated domain with Neumann boundary condition
Proc. Indin Acd. Sci. Mth. Sci. Vol. 112, No. 1, Februry 22, pp. 195 27. Printed in Indi Homogeniztion of prbolic eqution in perforted domin with Neumnn boundry condition A K NANDAKUMARAN nd M RAJESH Deprtment
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationORBITAL MANEUVERS USING LOWTHRUST
Proceedings of the 8th WSEAS Interntionl Conference on SIGNAL PROCESSING, ROBOICS nd AUOMAION ORBIAL MANEUVERS USING LOWHRUS VIVIAN MARINS GOMES, ANONIO F. B. A. PRADO, HÉLIO KOII KUGA Ntionl Institute
More informationReal Analysis HW 10 Solutions
Rel Anlysis HW 10 Solutions Problem 47: Show tht funtion f is bsolutely ontinuous on [, b if nd only if for eh ɛ > 0, there is δ > 0 suh tht for every finite disjoint olletion {( k, b k )} n of open intervls
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationEuler Euler Everywhere Using the EulerLagrange Equation to Solve Calculus of Variation Problems
Euler Euler Everywhere Using the EulerLgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationCOMPONENTS: COMBINED LOADING
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
More informationIntroduction to Integration Part 2: The Definite Integral
Mthemtics Lerning Centre Introduction to Integrtion Prt : The Definite Integrl Mr Brnes c 999 Universit of Sdne Contents Introduction. Objectives...... Finding Ares 3 Ares Under Curves 4 3. Wht is the
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationEconomics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999
Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,
More informationDecision Rule Extraction from Trained Neural Networks Using Rough Sets
Decision Rule Extrction from Trined Neurl Networks Using Rough Sets Alin Lzr nd Ishwr K. Sethi Vision nd Neurl Networks Lbortory Deprtment of Computer Science Wyne Stte University Detroit, MI 48 ABSTRACT
More informationReal Analysis and Multivariable Calculus: Graduate Level Problems and Solutions. Igor Yanovsky
Rel Anlysis nd Multivrible Clculus: Grdute Level Problems nd Solutions Igor Ynovsky 1 Rel Anlysis nd Multivrible Clculus Igor Ynovsky, 2005 2 Disclimer: This hndbook is intended to ssist grdute students
More informationThe Definite Integral
Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know
More informationITS HISTORY AND APPLICATIONS
NEČAS CENTER FOR MATHEMATICAL MODELING, Volume 1 HISTORY OF MATHEMATICS, Volume 29 PRODUCT INTEGRATION, ITS HISTORY AND APPLICATIONS Antonín Slvík (I+ A(x)dx)=I+ b A(x)dx+ b x2 A(x 2 )A(x 1 )dx 1 dx 2
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More informationJaERM SoftwareasaSolution Package
JERM SoftwresSolution Pckge Enterprise Risk Mngement ( ERM ) Public listed compnies nd orgnistions providing finncil services re required by Monetry Authority of Singpore ( MAS ) nd/or Singpore Stock
More informationProject 6 Aircraft static stability and control
Project 6 Aircrft sttic stbility nd control The min objective of the project No. 6 is to compute the chrcteristics of the ircrft sttic stbility nd control chrcteristics in the pitch nd roll chnnel. The
More informationNovel Methods of Generating SelfInvertible Matrix for Hill Cipher Algorithm
Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting SelfInvertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering
More informationDerivatives and Rates of Change
Section 2.1 Derivtives nd Rtes of Cnge 2010 Kiryl Tsiscnk Derivtives nd Rtes of Cnge Te Tngent Problem EXAMPLE: Grp te prbol y = x 2 nd te tngent line t te point P(1,1). Solution: We ve: DEFINITION: Te
More informationDlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report
DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationNumerical Methods of Approximating Definite Integrals
6 C H A P T E R Numericl Methods o Approimting Deinite Integrls 6. APPROXIMATING SUMS: L n, R n, T n, AND M n Introduction Not only cn we dierentite ll the bsic unctions we ve encountered, polynomils,
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationDIFFERENTIATING UNDER THE INTEGRAL SIGN
DIFFEENTIATING UNDE THE INTEGAL SIGN KEITH CONAD I hd lerned to do integrls by vrious methods shown in book tht my high school physics techer Mr. Bder hd given me. [It] showed how to differentite prmeters
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationRecognition Scheme Forensic Science Content Within Educational Programmes
Recognition Scheme Forensic Science Content Within Eductionl Progrmmes one Introduction The Chrtered Society of Forensic Sciences (CSoFS) hs been ccrediting the forensic content of full degree courses
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationAA1H Calculus Notes Math1115, Honours 1 1998. John Hutchinson
AA1H Clculus Notes Mth1115, Honours 1 1998 John Hutchinson Author ddress: Deprtment of Mthemtics, School of Mthemticl Sciences, Austrlin Ntionl University Emil ddress: John.Hutchinson@nu.edu.u Contents
More informationSecondDegree Equations as Object of Learning
Pper presented t the EARLI SIG 9 Biennil Workshop on Phenomenogrphy nd Vrition Theory, Kristinstd, Sweden, My 22 24, 2008. Abstrct SecondDegree Equtions s Object of Lerning Constnt Oltenu, Ingemr Holgersson,
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationCURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.
CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e
More informationApplications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
More informationLower Bound for EnvyFree and Truthful Makespan Approximation on Related Machines
Lower Bound for EnvyFree nd Truthful Mespn Approximtion on Relted Mchines Lis Fleischer Zhenghui Wng July 14, 211 Abstrct We study problems of scheduling jobs on relted mchines so s to minimize the mespn
More informationLITTLEWOODTYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE. Peter Borwein and Tamás Erdélyi
LITTLEWOODTYPE PROBLEMS ON SUBARCS OF THE UNIT CIRCLE Peter Borwein nd Tmás Erdélyi Abstrct. The results of this pper show tht mny types of polynomils cnnot be smll on subrcs of the unit circle in the
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More informationSpace Vector Pulse Width Modulation Based Induction Motor with V/F Control
Interntionl Journl of Science nd Reserch (IJSR) Spce Vector Pulse Width Modultion Bsed Induction Motor with V/F Control Vikrmrjn Jmbulingm Electricl nd Electronics Engineering, VIT University, Indi Abstrct:
More information#A12 INTEGERS 13 (2013) THE DISTRIBUTION OF SOLUTIONS TO XY = N (MOD A) WITH AN APPLICATION TO FACTORING INTEGERS
#A1 INTEGERS 13 (013) THE DISTRIBUTION OF SOLUTIONS TO XY = N (MOD A) WITH AN APPLICATION TO FACTORING INTEGERS Michel O. Rubinstein 1 Pure Mthemtics, University of Wterloo, Wterloo, Ontrio, Cnd mrubinst@uwterloo.c
More informationHillsborough Township Public Schools Mathematics Department Computer Programming 1
Essentil Unit 1 Introduction to Progrmming Pcing: 15 dys Common Unit Test Wht re the ethicl implictions for ming in tody s world? There re ethicl responsibilities to consider when writing computer s. Citizenship,
More informationFactoring Trinomials of the Form. x 2 b x c. Example 1 Factoring Trinomials. The product of 4 and 2 is 8. The sum of 3 and 2 is 5.
Section P.6 Fctoring Trinomils 6 P.6 Fctoring Trinomils Wht you should lern: Fctor trinomils of the form 2 c Fctor trinomils of the form 2 c Fctor trinomils y grouping Fctor perfect squre trinomils Select
More informationThe Fundamental Theorem of Calculus
Section 5.4 Te Funmentl Teorem of Clculus Kiryl Tsiscnk Te Funmentl Teorem of Clculus EXAMPLE: If f is function wose grp is sown below n g() = f(t)t, fin te vlues of g(), g(), g(), g(3), g(4), n g(5).
More information6 Energy Methods And The Energy of Waves MATH 22C
6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More information