Differentiation of Continuous Functions Autar Kaw

Transcription

1 Derentton o Contnuous Functons Autr Kw Ater redng ths chpter, you should be ble to:. derve ormuls or pproxmtng the rst dervtve o uncton,. derve ormuls or pproxmtng dervtves rom Tylor seres, 3. derve nte derence pproxmtons or hgher order dervtves, nd. use the developed ormuls n exmples to nd dervtves o uncton. The dervtve o uncton t x s dened s ( x) lm 0 ( x) ( x) To be ble to nd dervtve numerclly, one could mke x nte to gve, ( x) ( x) ( x). Knowng the vlue o x t whch you wnt to nd the dervtve o ( x), we choose vlue o x to nd the vlue o ( x). To estmte the vlue o ( x), three such pproxmtons re suggested s ollows. Forwrd Derence Approxmton o the Frst Dervtve From derentl clculus, we know ( x) lm For nte x, ( x) 0 ( x) ( x) ( x) ( x) Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 8

2 The bove s the orwrd dvded derence pproxmton o the rst dervtve. It s x t clled orwrd becuse you re tkng pont hed o x. To nd the vlue o x x, we my choose nother pont x hed s x x. Ths gves where x x x x x (x) x x x Fgure Grphcl representton o orwrd derence pproxmton o rst dervtve. Exmple The velocty o rocket s gven by Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 8

3 t ( t) ln 9.8t, 0 t 30 where s gven n m/s nd t s gven n seconds. At t 6s, ) use the orwrd derence pproxmton o the rst dervtve o ( t) to clculte the ccelerton. Use step sze o t s. b) nd the exct vlue o the ccelerton o the rocket. c) clculte the bsolute reltve true error or prt (b). Soluton () ( t ) ( t ) ( t ) t t 6 t t t ( 6) t 6 8 ( 8) ( 6) ( 8) 000 ln ( 8) 53.0 m/s ( 6) 000 ln ( 6) m/s Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 3 o 8

4 Hence ( 6) ( 8) ( 6) m/s (b) The exct vlue o ( 6) cn be clculted by derenttng s t ( t ) ln 9. 8 t d dt Knowng tht ( t) [ ( t) ] d [ ( t) ] d ln nd dt t dt t t 0 00t d dt 0 00t ( t) 9. 8 ( 6) 0 00t t 00 9.t 00 3t m/s ( ) ( 00) 9. 8 Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 8

5 (c) The bsolute reltve true error s t True Vlue Approxmte Vlue 00 True Vlue % Bckwrd Derence Approxmton o the Frst Dervtve We know ( x) lm For nte x, ( x) 0 ( x) ( x) ( x) ( x) I x s chosen s negtve number, ( x) ( x) ( x) ( x) ( x ) Ths s bckwrd derence pproxmton s you re tkng pont bckwrd rom x. To nd the vlue o ( x) t x x, we my choose nother pont x behnd s x x. Ths gves ( x) x x Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 5 o 8

6 where x x (x) x x x Fgure Grphcl representton o bckwrd derence pproxmton o rst dervtve. Exmple The velocty o rocket s gven by t ( t) ln 9.8t,0 t 30 () Use the bckwrd derence pproxmton o the rst dervtve o ( t) to clculte the ccelerton t t 6s. Use step sze o t s. (b) Fnd the bsolute reltve true error or prt (). Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 6 o 8

7 Soluton ( t) ( t ) ( t ) t t 6 t t t ( 6) t 6 ( 6) ( ) ( 6) 000 ln ( 6) m/s ( ) 000 ln ( ) 33. m/s ( 6) ( 6) ( ) m/s (b) The exct vlue o the ccelerton t ( 6 ) 9.67 m/s t 6s rom Exmple s Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 7 o 8

8 The bsolute reltve true error or the nswer n prt () s t % Forwrd Derence Approxmton rom Tylor Seres Tylor s theorem sys tht you know the vlue o uncton (x) t pont x nd ll ts dervtves t tht pont, provded the dervtves re contnuous between x nd x, then ( x) ( x)( x x )! ( x) ( ) x x K Substtutng or convenence x x x ( x)! ( x) x K x ( x) ( x ) K O! The O( x) term shows tht the error n the pproxmton s o the order o x. Cn you now derve rom the Tylor seres the ormul or the bckwrd dvded derence pproxmton o the rst dervtve? As you cn see, both orwrd nd bckwrd dvded derence pproxmtons o O x. Cn we get better the rst dervtve re ccurte on the order o pproxmtons? Yes, nother method to pproxmte the rst dervtve s clled the centrl derence pproxmton o the rst dervtve. From the Tylor seres Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 8 o 8

9 x 3 x x K ()! 3! nd x 3 x x K ()! 3! Subtrctng Equton () rom Equton () ( x) 3 K x 3! x ( x) ( ) x K O 3! hence showng tht we hve obtned more ccurte ormul s the error s o the order o O( x). (x) x x x x Fgure 3 Grphcl representton o centrl derence pproxmton o rst dervtve. Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 9 o 8

10 Exmple 3 The velocty o rocket s gven by ( t) 000ln 9.8,0 30 t t. 0 00t () Use the centrl derence pproxmton o the rst dervtve o ( t) to clculte the ccelerton t t 6 s. Use step sze o t s. (b) Fnd the bsolute reltve true error or prt (). Soluton ( t ) ( t ) ( t ) t t 6 t t t t 6 8 t t t 6 ( 6) ( 8) ( ) ( 8) ( ) ( 8) 000 ln ( 8) 53.0 m/s Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 0 o 8

11 ( ) 000 ln ( ) 33. m/s ( 6) ( 8) ( ) m/s (b) The exct vlue o the ccelerton t ( 6 ) 9.67 m/s t 6 s rom Exmple s The bsolute reltve true error or the nswer n prt () s t % The results rom the three derence pproxmtons re gven n Tble. Tble Summry o ( 6) usng derent derence pproxmtons Type o derence pproxmton Forwrd Bckwrd Centrl ( 6) ( m/s ) t % Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 8

12 Clerly, the centrl derence scheme s gvng more ccurte results becuse the order o ccurcy s proportonl to the squre o the step sze. In rel le, one would not know the exct vlue o the dervtve so how would one know how ccurtely they hve ound the vlue o the dervtve? A smple wy would be to strt wth step sze nd keep on hlvng the step sze untl the bsolute reltve pproxmte error s wthn pre-speced tolernce. Tke the exmple o ndng v ( t) or t ( t ) ln 9. 8 t t t 6 usng the bckwrd derence scheme. Gven n Tble re the vlues obtned usng the bckwrd derence pproxmton method nd the correspondng bsolute reltve pproxmte errors. Tble Frst dervtve pproxmtons nd reltve errors or derent t vlues o bckwrd derence scheme. t v ( t) % From the bove tble, one cn see tht the bsolute reltve pproxmte error decreses s the step sze s reduced. At t 0. 5, the bsolute reltve pproxmte error s %, menng tht t lest sgncnt dgts re correct n the nswer. Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 8

13 Fnte Derence Approxmton o Hgher Dervtves One cn lso use the Tylor seres to pproxmte hgher order dervtve. For x, the Tylor seres s exmple, to pproxmte where x where x 3 K (3)! 3! x x 3 K ()! 3! x x Subtrctng tmes Equton () rom Equton (3) gves ( x) 3 K ( x ) ( x ) ( x) ( x ) ( x ) ( x) O K (5) Exmple The velocty o rocket s gven by t ( t) ln 9.8t,0 t 30 Use the orwrd derence pproxmton o the second dervtve o ( t) to clculte the jerk t t 6 s. Use step sze o t s. Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 3 o 8

14 Soluton j ( t ) ( t ) ( t ) ( t) ( t) t 6 t t t t 6 8 t 0 ( ) t t 6 j ( 6) ( 0) ( 8) ( 6) 0 ( 0) 000 ln m/s ( 0) 9.8 ( 0) ( 8) 000 ln ( 8) 53.0 m/s ( 6) 000 ln ( 6) m/s j ( 6) ( 53.0) Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge o 8

15 m/s The exct vlue o j ( 6) cn be clculted by derenttng twce s t ( t ) ln 9. 8 t d ( t) [ ( t) ] nd dt d j dt Knowng tht ( t) [ ( t) ] d [ ln ( t) ] nd dt t d dt t t 0 00t d dt 0 00t ( t) t t ( ) ( 00) t 00 3t Smlrly t cn be shown tht d j dt ( t) [ ( t) ] Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 5 o 8

16 j ( 6) 8000 ( 00 3t) 8000 [ 00 3(6)] m/s The bsolute reltve true error s 3 t % The ormul gven by Equton (5) s orwrd derence pproxmton o the second O x. Cn we get ormul tht hs dervtve nd hs n error o the order o better ccurcy? Yes, we cn derve the centrl derence pproxmton o the second dervtve. The Tylor seres s where where 3 x ( x)... x (6)! 3!! x x x 3 ( x) K x (7)! 3!! x x Addng Equtons (6) nd (7), gves x x x x x... Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 6 o 8

17 ( x ) ( x) ( x ) ( ) x ( x ) ( x) ( x ) O... Exmple 5 The velocty o rocket s gven by ( t) 000ln 9.8, 0 30 t t, 0 00t () Use the centrl derence pproxmton o the second dervtve o ( t) to clculte the jerk t Soluton t 6 s. Use step sze o t s. The second dervtve o velocty wth respect to tme s clled jerk. The second order pproxmton o jerk then s j ( t ) ( t ) ( t) ( t ) ( t) t 6 t t t t 6 8 t t t 6 j ( 6) ( 8) ( 6) ( ) Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 7 o 8

18 ( 8) 000 ln ( 8) 53.0 m/s ( 6) 000 ln ( 6) m/s ( ) 000 ln ( ) j ( 6) 33. m/s ( 8) ( 6) ( ) ( 39.07) m/s The bsolute reltve true error s t % Source URL: Sylor URL: Attrbuted to: Unversty o South Flord: Holstc Numercl Methods Insttute Pge 8 o 8