Evaluating the Derivatives of Two Types of Functions

Transcription

1 Available Olie at Iteatioal Joual of Compute Sciece ad Mobile Computig A Mothly Joual of Compute Sciece ad Ifomatio Techology ISSN X IJCSMC Vol. 2 Issue. 7 July 2013 pg RESEARCH ARTICLE Evaluatig the Deivatives of Two Types of Fuctios Chii-Huei Yu Depatmet of Maagemet ad Ifomatio Na Jeo Istitute of Techology Taia City Taiwa chiihuei@mail.jtc.edu.tw Abstact This pape uses the mathematical softwae Maple fo the auxiliay tool to study the diffeetial poblem of two types of fuctios. We ca obtai the closed foms of ay ode deivatives of these two types of fuctios by usig Eule's fomula DeMoive's fomula fiite geometic seies ad Leibiz diffeetial ule ad hece geatly educe the difficulty of calculatig thei highe ode deivative values. I additio we povide two examples to do calculatio pactically. The eseach methods adopted i this study ivolved fidig solutios though maual calculatios ad veifyig these solutios by usig Maple. This type of eseach method ot oly allows the discovey of calculatio eos but also helps modify the oigial diectios of thikig fom maual ad Maple calculatios. Theefoe Maple povides isights ad guidace egadig poblem-solvig methods. Keywods deivatives closed foms Eule's fomula DeMoive's fomula fiite geometic seies Leibiz diffeetial ule Maple I. INTRODUCTION The compute algeba system (CAS) has bee widely employed i mathematical ad scietific studies. The apid computatios ad the visually appealig gaphical iteface of the pogam ede ceative eseach possible. Maple possesses sigificace amog mathematical calculatio systems ad ca be cosideed a leadig tool i the CAS field. The supeioity of Maple lies i its simple istuctios ad ease of use which eable begies to lea the opeatig techiques i a shot peiod. I additio though the umeical ad symbolic computatios pefomed by Maple the logic of thikig ca be coveted ito a seies of istuctios. The computatio esults of Maple ca be used to modify ou pevious thikig diectios theeby fomig diect ad costuctive feedback that ca aid i impovig udestadig of poblems ad cultivatig eseach iteests. Iquiig though a olie suppot system povided by Maple o bowsig the Maple website ( ca facilitate futhe udestadig of Maple ad might povide uexpected isights. As fo the istuctios ad opeatios of Maple we ca efe to [1]-[7]. ( ) I calculus ad egieeig mathematics cuicula fidig f ( c) ( the -th ode deivative value of ( ) fuctio f ( at x c ) i geeal ecessay goes though two pocedues: Fistly evaluatig ( the - ( ) th ode deivative of f ( ) ad secodly substitutig x c to. Whe evaluatig the highe ode deivative values of a fuctio (i.e. is lage) these two pocedues will make us face with iceasigly complex calculatios. Theefoe to obtai the aswes though maual calculatios is ot a easy thig. I this pape we maily study the evaluatio of deivatives of the followig two types of fuctios 2013 IJCSMC All Rights Reseved 108

2 Chii-Huei Yu Iteatioal Joual of Compute Sciece ad Mobile Computig Vol.2 Issue. 7 July pg ) 2) 1 x cos x x cos( + 1) x + x cos x 2 (1) ( + 1) ( + 2) x si x x si( + 1) x + x si x 2 whee is a positive itege is a eal umbe. We ca obtai the closed foms of ay ode deivatives of these two types of fuctios by usig Eule's fomula DeMoive's fomula fiite geometic seies ad Leibiz diffeetial ule; these ae the majo esults i this pape (i.e. Theoems 1 ad 2) ad hece geatly educe the difficulty of detemiig highe ode deivatives values of these two types of fuctios. As fo the elated study of the diffeetial poblems ca efe to [8]-[16]. O the othe had we popose two fuctios to detemie the closed foms of thei ay deivatives ad calculate some of thei highe ode deivative values pactically. The eseach methods adopted i this study ivolved fidig solutios though maual calculatios ad veifyig these solutios by usig Maple. This type of eseach method ot oly allows the discovey of calculatio eos but also helps modify the oigial diectios of thikig fom maual ad Maple calculatios. Fo this easo Maple povides isights ad guidace egadig poblem-solvig methods. (2) II. MAIN RESULTS Fistly we itoduce a otatio ad some fomulas used i this study. Notatio. Suppose t is ay eal umbe ad m is ay positive itege. Defie ( t) m t ( t 1) ( t m + 1) ad ( t ) 0 1. Fomulas. (i) Eule's fomula. e ix cos x + i si x whee x is ay eal umbe. (ii) DeMoive's fomula. (cos x + i si cos x + i si x whee is ay itege x is ay eal umbe. (iii) Fiite geometic seies z + z z z 1 z whee is ay positive itege z is a complex umbe ad z 1. (iv) Leibiz diffeetial ule ([17]):Let be a positive itege ad ae fuctios such that thei m -th ode deivatives g ( exist fo all fuctio m The the -th ode deivative of the poduct ( fg ) ( m ) ( m ) ( g ( m m whee m!. m!( m)! The followig is the fist esult i this study we detemie the closed foms of ay ode deivatives of fuctio (1). Theoem 1. Suppose m ae positive iteges is a eal umbes. Let the domai of the fuctio 1) 2) 1 x cos x x cos( + 1) x + x cos x 2 the m -th ode deivative of f ( be { 01 2 cos 2 x R x exist x x x + x 0} 2013 IJCSMC All Rights Reseved 109. The

3 Chii-Huei Yu Iteatioal Joual of Compute Sciece ad Mobile Computig Vol.2 Issue. 7 July pg m m m j k j ( m j) k) j k x cos k 1 j (3) fo all x satisfy 2 x exist x 0 ad 1 2 x 0. Poof. Takig z x e ix ito the fiite geometic seies we obtai ix + 1 ( x e ) ix x e ix ix 2 ix 1+ x e + ( x e ) + + ( x e ) 1) i( + 1) x 1 x e ix 1 x e ix 2 i2x ix 1 + x e + x e + + x e (by DeMoive's fomula) 1) 1) {[1 x cos( + 1) x] ix si( + 1) x}[(1 x cos + ix si x] 2 2x (by Eule's fomula) Usig the equal of the eal pats of both sides of (4) we have x k e ikx k 0 (4) 1) 2) x cos x x cos( + 1) x + x cos x 2 2x Theefoe we obtai ay m -th ode deivative of f ( k x cos kx (5) k 0 m ( m ) m k ( j) ( m j) x ) (cos kx ) (by Leibiz diffeetial ule) k 0 j m m m j k j ( m j) k) j k x cos k 1 j fo all x satisfy 2 x exist x x 0 Next we evaluate the closed foms of ay ode deivatives of fuctio (2). Theoem 2. If the assumptios ae the same as Theoem 1 ad suppose the domai of the fuctio 1) 2) x si x x si( + 1) x + x si x 2 m -th ode deivative of g ( is { 01 2 cos 2 x R x exist x x x + x 0}. The the g m ( m m m j k j ( m j) k) j k x si k 1 j (6) fo all x satisfy 2 x exist x 0 ad 1 2 x 0 Poof. By the equal of the imagiay pats of both sides of (4) we have 1) 2) x si x x si( + 1) x + x si x 2 2x Thus we obtai ay m -th ode deivative of g ( k x si kx (7) k IJCSMC All Rights Reseved 110

4 Chii-Huei Yu Iteatioal Joual of Compute Sciece ad Mobile Computig Vol.2 Issue. 7 July pg g m m m k ( j) ( (si ) ( m j x kx ) (by Leibiz diffeetial ule) k 0 j m m m j k j ( m j) k) j k x si k 1 j fo all x satisfy 2 x exist x x 0 III. EXAMPLES I the followig we povide two fuctios to detemie the closed foms of thei ay ode deivatives ad some of thei highe ode deivative values pactically. O the othe had we use Maple to calculate the appoximatios of these highe ode deivative values ad thei closed foms fo veifyig ou aswes. Example 1. If the domai of the fuctio 2 / 3 2 8/ 3 1 x cos x x cos3x + x cos 2x (8) 2 / 3 4 / 3 2 / 3 4 / 3 is { R x 0 0} x (the case of 2 / 3 2 i Theoem 1) By Theoem 1 we ca obtai ay m -th ode deivative of f ( 2 m m m j 2k / 3 j ( m j) 2k / 3) j k x cos k 1 j (9) 2 / 3 4 / 3 fo all x satisfy x x 0. Thus we obtai the 13-th ode deivative value of f ( at 5 x k / 3 j ( 13) j 5 5k (13 j) f 2k / 3) j k cos + 4 k 1 j j (10) Next we use Maple to veify the coectess of (10). >f:x->(1-x^(2/3)*cos(-x^2*cos(3*+x^(8/3)*cos(2*)/(1-2*x^(2/3)*cos(+x^(4/3)); >evalf((d@@13)(f)(5*pi/4)20); >evalf(sum(13!/(j!*(13-j)!)*poduct(2/3-pp0..(j-1))*(5*pi/4)^(2/3-j)*cos(5*pi/4+(13-j)*pi/2)j0..13)+ sum(13!/(j!*(13-j)!)*poduct(4/3-pp0..(j-1))*2^(13-j)*(5*pi/4)^(4/3-j)*cos(10*pi/4+(13-j)*pi/2)j0..13)20); Example 2. If the domai of the fuctio 4 2 is { R x > 01 2 x 1 / 1 / 0} 1/ 4 5 / 4 x si x x si 4x + x si 3x (11) 1/ 4 1/ 2 x (the case of 1 / 4 3 i Theoem 2) Usig Theoem 2 we obtai ay m -th ode deivative of g ( 2013 IJCSMC All Rights Reseved 111

5 Chii-Huei Yu Iteatioal Joual of Compute Sciece ad Mobile Computig Vol.2 Issue. 7 July pg g m ( 3 m m m j k / 4 j ( m j) k / 4) j k x si k 1 j (12) 1 / 4 1 / 2 fo all x satisfy x > 0 ad 1 2 x 0. Hece we ca evaluate the 10-th ode deivative value of at 2 x k / 4 j ( 10) j 2 2k (10 j) g k / 4) j k si + 3 k 1 j j (13) Usig Maple to veify the coectess of (13) as follows: >g:x->(x^(1/4)*si(-x*si(4*+x^(5/4)*si(3*)/(1-2*x^(1/4)*cos(+x^(1/2)); >evalf((d@@10)(g)(2*pi/3)20); >evalf(sum(10!/(j!*(10-j)!)*poduct(1/4-pp0..(j-1))*(2*pi/3)^(1/4-j)*si(2*pi/3+(10-j)*pi/2)j0..10)+ sum(10!/(j!*(10-j)!)*poduct(1/2-pp0..(j-1))*2^(10-j)*(2*pi/3)^(1/2-j)*si(4*pi/3+(10-j)*pi/2)j0..10)+ sum(10!/(j!*(10-j)!)*poduct(3/4-pp0..(j-1))*3^(10-j)*(2*pi/3)^(3/4-j)*si(2*pi+(10-j)*pi/2)j0..10)20); IV. CONCLUSIONS As metioed the Eule's fomula the DeMoive's fomula the fiite geometic seies ad the Leibiz diffeetial ule play sigificat oles i the theoetical ifeeces of this study. I fact the applicatios of these fomulas ae extesive ad ca be used to easily solve may difficult poblems; we edeavo to coduct futhe studies o elated applicatios. I additio Maple also plays a vital assistive ole i poblem-solvig. I the futue we will exted the eseach topic to othe calculus ad egieeig mathematics poblems ad solve these poblems by usig Maple. These esults will be used as teachig mateials fo Maple o educatio ad eseach to ehace the cootatios of calculus ad egieeig mathematics. REFERENCES [1] D. Richads Advaced Mathematical Methods with Maple New Yok: Cambidge Uivesity Pess [2] F. Gava The Maple Book Lodo: Chapma & Hall/CRC [3] J. S. Robetso Egieeig Mathematics with Maple New Yok: McGaw-Hill [4] C. Tocci ad S. G. Adams Applied Maple fo Egiees ad Scietists Bosto: Atech House [5] C. T. J. Dodso ad E. A. Gozalez Expeimets i Mathematics Usig Maple New Yok: Spige- Velag [6] R. J. Stoeke ad J. F. Kaashoek Discoveig Mathematics with Maple : A Iteactive Exploatio fo Mathematicias Egiees ad Ecoometicias Basel: Bikhause Velag [7] M. L. Abell ad J. P. Baselto Maple by Example 3d ed. New Yok: Elsevie Academic Pess [8] C. -H. Yu A study o the diffeetial poblems usig Maple Iteatioal Joual of Compute Sciece ad Mobile Computig vol. 2 issue. 7 pp [9] C. -H. Yu Applicatio of Maple o solvig some diffeetial poblems Poceedigs of IIE Asia Cofeece 2013 Natioal Taiwa Uivesity of Sciece ad Techology Taiwa o [10] C.-H. Yu A study o some diffeetial poblems with Maple Poceedigs of 6th IEEE/Iteatioal Cofeece o Advaced Ifocomm Techology Natioal Uited Uivesity Taiwa o [11] C. -H. Yu The diffeetial poblem of fou types of fuctios Joual of Kag-Nig vol. 14 i pess. [12] C. -H. Yu A study o the diffeetial poblem of some tigoometic fuctios Joual of Je-Teh vol. 10 i pess IJCSMC All Rights Reseved 112

6 Chii-Huei Yu Iteatioal Joual of Compute Sciece ad Mobile Computig Vol.2 Issue. 7 July pg [13] C. -H. Yu Applicatio of Maple o the diffeetial poblem of hypebolic fuctios Poceedigs of Iteatioal Cofeece o Safety & Secuity Maagemet ad Egieeig Techology 2012 WuFeg Uivesity Taiwa pp [14] C. -H. Yu Applicatio of Maple: takig the diffeetial poblem of atioal fuctios as a example Poceedigs of 2012 Optoelectoics Commuicatio Egieeig Wokshop Natioal Kaohsiug Uivesity of Applied Scieces Taiwa pp [15] C. -H. Yu The diffeetial poblem of two types of expoetial fuctios Joual of Na Jeo vol. 16 D1-1~11. [16] C. -H. Yu Applicatio of Maple: takig the evaluatio of highe ode deivative values of some type of atioal fuctios as a example Poceedigs of 2012 Digital Life Techology Semia Natioal Yuli Uivesity of Sciece ad Techology Taiwa pp [17] T. M. Apostol Mathematical Aalysis 2d ed. Bosto: Addiso-Wesley p IJCSMC All Rights Reseved 113