Dedicated to Professor D. D. Stancu on his 75th birthday Mathematical Subject Classification: 25A24, 65D30

Transcription

1 Generl Mthemtics Vol. 10, No. 1 2 (2002), About some properties of intermedite point in certin men-vlue formuls Dumitru Acu, Petrică Dicu Dedicted to Professor D. D. Stncu on his 75th birthdy. Abstrct In this pper we study property of the intermedite point (see [10]) from the qudrture formul of the trpeze, Simpson, s formul nd the men-vlue formul of N. Ciorănescu Mthemticl Subject Clssifiction: 25A24, 65D30 1. In the specilized literture there re lot of men-vlue formuls (the men-vlue theorem for derivtes, the men-vlue theorems for Riemnn integrls, the qudrture formuls, the cubture formuls, etc.). In generl they contin one or more intermedite points. This points hve different properties which result from the properties of the clsses of functions, to which the men-vlue theorems refer to. 51

2 52 Dumitru Acu, Petrică Dicu In [10] B. Jcobson demonstrted tht for the men-vlue formul f(t)dt = (b )f(c), where f is continuous function on the rel closed intervl [, b], we hve: c b b = 1 2 In [13] E. C. Pop finds the following results: 1.1 Let f, g : [, b] R, two continuous functions with f derivble in, g non-negtive nd f ()g() 0. From the first men-vlue theorem for Riemnn integrl we hve: for ny x (, b] there is c x [, x] such tht: f(t)g(t)dt = f(c x ) g(t)dt Then: c x x x = Let f, g : [, b] R, f continuous, g monotone nd derivble with f()g () 0. From the second men-vlue for Riemnn integrl, we hve: for ny x (, b] there is c x [, x] such tht: f(t)g(t)dt = g() c x f(t)dt + g(x) f(t)dt. c x Then: c x x x = Let f, g : [, b] R, two continuous functions on [, b], derivble on (, b), derivble two times in nd g (x) 0 for ny x [, b], f ()g ()

3 About some properties of intermedite point f ()g (). By using Cuchy, s theorem, for ny x [, b] we hve g() g(x) nd there is c x (, x) so tht: Then f(x) f() g(x) g() = f (c x ) g (c x ). c x x x = 1 2. In [5] S. Anit gives the following result: Let x 0 I, I intervl nd f : I R function derivble of (n + 2) times on I nd f (n+2) (x 0 ) 0. According to Tylor, s theorem we hve: (1) f(x) = f(x 0 ) + x x 0 1! f (x 0 ) (x x 0) n f (n) (x 0 )+ n! + (x x 0) n+1 f (n+1) (c x ). (n + 1)! If for ny x I {x 0 } we fix c x x 0 for which reltion (1) holds, then: c x x 0 = 1 x x 0 x x 0 n + 2. V. Rdu demonstrted in [14] tht if f, g : I R R (n + 1) times derivble on I nd, x I, then there is c x between nd x such tht: (2) n f (k) ()(x ) k f(x) k! k=0 n g (k) ()(x ) k g(x) k! k=0 = f (n+1) (c x ) g (n+1) (c x ). In [4] D. Acu demonstrted tht, for the intermedite point in formul (2), we hve: c x x x = 1 n + 2.

4 54 Dumitru Acu, Petrică Dicu In this pper we study the property shown bove for the intermedite point from the qudrture formul of the trpeze, the qudrture formul of Simpson nd the men-vlue formul of N. Ciorănescu [6]. 2. First, let us consider the qudrture formul of the trpeze. If f : [, b] R, f C 2 [, b], then for ny x (, b] there is c x (, x) such tht: (3) f(t)dt = 1 2 (x )3 (x ) [f() + f(x)] f (c x ). 12 (see [7], [9], [11], [12]). In the bove condition we hve the following theorem: Theorem 1. If f C 3 [, b] nd f () 0, then for the intermedite point c x tht ppers in formul (3), it follows: c x x x = 1 2. Proof. Let F, G : [, b] R defined s follows: F (x) = f(t)dt 1 2 (x )3 (x )[f() + f(x)] + f (), G(x) = (x ) Since F nd G re two times derivble on [, b] nd G (x) 0, G (x) 0 for ny x (, b], we hve: F (x) = 1 2 f (x) 1 2 f() 1 2 (x )f (x) (x )2 f () F (x) = 1 2 (x )f (x) (x )f () = 1 2 (x )[f (x) f ()]. G (x) = 4(x ) 3, G (x) = 12(x ) 2.

5 About some properties of intermedite point By using succesive l, Hospitl rule, we obtin: F (x) 1 x G (x) = 2 (x )[f (x) f ()] x 12(x ) 2 = 1 24 f () Hence: (4) On the other hnd: F (x) x G(x) = 1 24 f (). (5) (x ) 3 F (x) f (x )3 (c x ) + f () G(x) = (x ) 4 = 1 f (c x ) f () = 12 x Hence: = 1 f (c x ) f () cx 2 c x x. F (x) x G(x) = 1 12 f c x () x x becuse c x when x. From reltions (4) nd (5) we obtin: c x x x = 1 2 which is in fct the conclusion of Theorem Now we will consider Simpson, s formul. If f : [, b] R, f C 4 [, b], then ny x (, b], there is c x (, b) such tht: (6) f(t)dt = 1 ( ) + x [f() 6 (x ) + 4f 2 (see [7], [9], [11], [12]). Our min result is contined in the following: ] (x )5 + f(x) 2880 f (IV ) (c x ).

6 56 Dumitru Acu, Petrică Dicu Theorem 2. If f C 5 [, b] nd f (5) () 0, then for the intermedite point c x which ppers in formul (6) we hve: c x x x = 1 2 Proof. Let us consider F, G : [, b] R defined by: F (x) = f(t)dt 1 ( ) + x [f() 6 (x ) + 4f 2 G(x) = (x ) 6. (x )5 + f(x) ] f (IV ) (), Since F nd G re fifth times derivble on [, b], G (k) (x) 0 for k = 1, 5, x (, b] we hve: F (5) (x) = 1 6 f (IV ) (x) 5 24 f (IV ) ( ) + x f (IV ) () 1 [ ( ) ] 1 + x 6 (x ) 8 f (5) + f (5) (x), 2 G (5) (x) = 6!(x ). We hve: 1 [ F (5) (x) f (IV ) (x) f (IV ) () ] x G (5) (x) = 6 x 6!(x ) + x 2 6!(x ) 1 6 6! = 1 6! [ 1 8 f (5) ( ) + x 2 [ ( 5 + x f (IV ) x 2 ] + f (5) (x) = ( ) f (5) () = ! 48 f (5) (). ) ] f (IV ) ()

7 About some properties of intermedite point (7) Hence: On the other hnd: F (x) x G(x) = 1 5! 48 f (5) (). (x ) 5 F (x) G(x) = 2880 f (IV ) (x )5 (c x ) f (IV ) () (x ) 6 = = 1 f (IV ) (c x ) f (IV ) () cx 2880 c x x. By pplying the it x, we hve c x nd we obtin: (8) F (x) x G(x) = f (5) c x () x x. From the reltions (7) nd (8), it follow c x x x = 1 2 wht is exctly the ssertion of Theorem The lst prt of the pper is dedicted to the study of the intermedite, s point property from the men-vlue formul of N. Ciorănescu [6]. Being given: function f : [, b] R, f C m [, b], sequence of orthogonl polynomils (p n ) n 0 (degree of p n is equl n) on [, b], in respect to weight function w : [, b] (0, ), N. Ciorănescu demonstrted tht the following formul is vlid: (9) f(x)p m (x)w(x)dx = f (m) (c b ) m! x m p m (x)w(x)dx. Theorem 3. If f C (m+1) [, b] nd f (m+1) () 0, then the intermedite point of the men-vlue formul (9) stisfies the reltion: b c b b = 1 m + 2

8 58 Dumitru Acu, Petrică Dicu Proof. To obtin this result we use the so-clled method of prmeters introduced by D. D. Stncu 46 yers go in [15] nd [16], in order to construct qudrture formule of high degree of exctness nd by using the Hildebrnd, s V - method we will obtin the result from the theorem. For more informtions bout the use of this effective method it cn be consulted [1], [2], [3], [16]. First we hve two results. Lemm 1. In orthogonlity conditions of the (p m )(x) on [, b] R ssocited with the non-negtive weight function w on [, b] we construct the polynoms (V k ) k=0,m with Hildebrnd, s V - method (see [1], [2], [3], [8]) s follows V 0 (x) = w(x)p m (x), V j (x) = V j 1 (t)dt, j = 1, m. Then: V j () = 0, V j (b) = 0, for ny j = 1, m. Proof. We hve V j () = 0, for ny j = 1, m. V 1 (b) = V 0 (t)dt = w(t)p m (t)dt = 0 (from the orthogonlity of the polynoms (p m ) m 0 ). Using the integrtion by prts, we obtin: V 2 (b) = V 1 (t)dt = tv 1 (t) b tv 1(t)dt = tv 0 (t)dt = tp m (t)w(t)dt = 0.

9 About some properties of intermedite point In the sme wy, it follows V 3 (b) = V 2 (t)dt = tv 2 (t) b tv 1 (t)dt = = tv 2 (t) b t2 2 V 1(t) b + 1 t 2 V 0 (t)dt = V m (b) = V m 1 (t)dt = tv m 1 (t) b tv m 2 (t)dt = = t2 2 V m 2(t) b Hence V j (b) = 0, for ny j = 1, m. t 3 V m 3 (t)dt =... = ( 1)m 1 t m 1 V 0 (t)dt = 0. (m 1)! Lemm 2. We hve the following equlities: i) f(x)p m (x)w(x)dx = ( 1) m b f (m) (x)v m (x)dx. ii) x m V (m) m (x)dx = ( 1) m m! V m (x)dx. Proof. Using the integrtion by prts nd tking into ccount Lemm 1, we hve: i) f(x)p m (x)w(x)dx = f (x)v 1 (x)dx = f(x)v 0 (x)dx = f(x)v 1(x)dx = f(x)v 1 (x) b f (x)v 2 (x)dx =... = ( 1) m f (m) (x)v m (x)dx.

10 60 Dumitru Acu, Petrică Dicu ii) x m V m (m) (x)dx = x m (V (m 1) m ) (x)dx = x m V m (m 1) (x) b m x m 1 V m (m 1) (x)dx =... = ( 1) m m! V m (x)dx. We now come bck to the proof of the Theorem 3. Using the Lemm 2 results in reltions (9) we obtin successively f(x)p m (x)w(x)dx = ( 1) m f (m) (x)v m (x)dx = = f (m) (c b ) m! Therefore: x m p m (x)w(x)dx = f (m) (c b ) m! = f (m) (c b ) b ( 1) m m! V m (x)dx. m! x m V (m) m (x)dx = (10) f (m) (x)v m (x)dx = f (m) (c b ) V m (x)dx. Now, we consider the functions F, G : [, b] R defined by: F (b) = f (m) (x)v m (x)dx f (m) () V m (x)dx, G(b) = (b ) m+2. We hve: F () = 0, F (b) = f (m) (b)v m (b) f (m) ()V m (b) = 0, F (b) = f (m+1) (b)v m (b) + f (m) (b)v m(b) f (m) ()V m(b) =

11 About some properties of intermedite point = f (m+1) (b)v m (b) + f (m) (b)v m 1 (b) f (m) ()V m 1 (b) = 0.. F (k) (b) = f (m+k 1) (b)v m (b) + C 1 k 1f (m+k 2) (b)v m 1 (b) f (m) (b)v m k+1 (b) f (m) ()V m k+1 (b) = 0. F (m) (b) = f (2m 1) (b)v m (b) + C 1 m 1f (2m 2) (b)v m 1 (b) f (m) (b)v 1 (b) f (m) ()V 1 (b) = 0. F (m+1) (b) = f (2m) (b)v m (b) + C 1 mf (2m 1) (b)v m 1 (b) C m mf (m) (b)v 0 (b) (11) f (m) ()V 0 (b) = [f (m) (b) f (m) ()]V 0 (b). Anlog: G (b) = (m + 2)(b ) m+1,..., G (m+1) (b) = (m + 2)!(b ). Therefore, using the l, Hospitl rule, we obtin F (m+1) (b) b G (m+1) (b) = b According to the l, Hospitl rules it follows It is obvious tht we hve: F (b) G(b) = f (m) (c b ) [f (m) (b) f (m) ()] V 0 (b) = f (m+1) () (m + 2)!(b ) (m + 2)! V 0(). F (x) b G(x) = f (m+1) () (m + 2)! V 0() f (m) (x)v m (x)dx f (m) () V m (x)dx (b ) m+2 = V m (x)dx f (m) () V m (x)dx = (b ) m+2 =

12 62 Dumitru Acu, Petrică Dicu = V m (x)dx (b ) m+1 f (m) (c b ) f (m) () c b cb b. By pssing to it with b, we hve c b, we will obtin: (12) becuse: nd: F (b) b G(b) = V 0() (m + 1)! f (m+1) () b c b b, f (m) (c b ) f (m) () b b = f (m+1) (), b V m (x)dx m+1 = (b ) b = b V m (b) (m + 1)(b ) m = b V 1 (b) (m + 1)!(b ) = b From reltions (11) nd (12) we hve: c b b b = 1 m + 2 wht is exctly the ssertion of Theorem 3. V m 1 (b) m 1 =... = m(m + 1)(b ) V 0 (b) (m + 1)! = V 0() (m + 1)!. Remrk 1. For w(x) = 1 nd m = 0 form Theorem 3 we obtin B. Jcobson, s result. References [1] D. Acu, V-optiml qudrture formuls of Guss - Christoffel type, Clcolo, vol. 34, No. 1-4, 1997,

13 About some properties of intermedite point [2] D. Acu, Extreml problems in the numericl integrtion of the functions, Doctor thesis (Cluj - Npoc), 1980 (in Romnin). [3] D. Acu, On the D. D. Stncu method of prmeters, Studi Univ. Bbeş - Bolyi, Mthemtic, vol. XLII, nr. 1, Mrch, 1997, 1-7. [4] D. Acu, About the intermedite point in the men-vlue theorems, The nnul session of scientificl comunictions of the Fculty of Sciences in Sibiu, My, 2001 (in Romnin). [5] S. Anit, Problem C: 275, Gzet Mtemtică, nr. 9, 1987 (in Romnin). [6] N. Ciorănescu, L générlistion de l premiére formule de l moyenne, L, einsegment Mth., Genéve, 37 (1938), [7] A. Ghizzetti nd A. Ossicini, Qudrture Formule, Birkhäuser Verlg Bsel und Stuttgrt, [8] F. B. Hildebrnd, Introduction to numericl nlysis, New York, Mc Grw - Hill, [9] D. V. Ionescu, Numericl qudrtures, Buchrest, Technicl Editure, 1957 (in Romnin). [10] B. Jcobson, On the men-vlue theorem for integrls, The Americn Mthemticl Monthly, vol. 89, 1982, [11] A. Lupş, C. Mnole, Numericl nlysis chpters, Publishing House of the Lucin Blg University of Sibiu, Sibiu, 1994 (in Romnin).

14 64 Dumitru Acu, Petrică Dicu [12] A. Lupş, Numericl Methods, Constnt Publishing House, Sibiu, 1994 (in Romnin). [13] E. C. Pop, An intermedite point property in some the men - vlue theorems, Astr Mtemtică, vol. 1, nr. 4, 1990, 3-7 (in Romnin). [14] V. Rdu, Elementry mthemtics lessons, Spiridon Publishing House, 1996 (in Romnin). [15] D. D. Stncu, A method for construction of the qudrture formuls of high degree exctness, Comunic. Acd. R.P.R. (Buchrest), 4 (8), 1957, [16] D. D. Stncu, Sur quelques formules générles de qudrture du type Guss - Christoffel, Mtemtic (Cluj), 1 (24), 1959, Lucin Blg University of Sibiu Deprtment of Mthemtics Str. Dr. I. Rţiu, nr Sibiu, Romni E-mil ddress: depmth@ulbsibiu.ro