Chem Math 252. Differentiation & Integration

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1 Chem 3 - Mth 5 Chpter 4 Differetitio & Itegrtio Differetitio & Itegrtio Experimetl dt t discrete poits Need to kow the rte of chge of the depedet vrile with respect to the idepedet vrile Need to kow re uder curve Need to itegrte lytic fuctio tht is too complicted to do lyticlly C do iterpoltio/curvefittig to get lytic fuctio

2 Lier Differetitio ( +Δ ) ( ) f x x f x f ( x) = limδ x Δx ( +Δ ) ( ) f x x f x f ( x ) = Δx Δx smll (Eq ) ( +Δ ) ( Δ ) f x x f x x f ( x ) = Δx Δx smll (Eq ) x ( ) ( ) f x = e f x = e Lier Differetitio x Δx f '(.5) Eq () % error f '(.5) Eq () % error ( x ) f = ( x ) f = ( +Δ ) ( ) f x x f x (Eq ) ( x ) k Δx ( +Δ ) ( Δ ) f x x f x x (Eq ) Smller spcig ot ecessrily etter f = Δx ( ) ( ) f x f x k+ k x x k+ k Exct vlue

3 3 poit Differetitio Lier differetitio igores ctul poit ( ) ( ) ( ) ( ) f x = p f x h + p f x + p f x + h Mke exct for f ( x) ( ) = f x = x x ( ) = ( ) f x x x 3 Mple Sheet ( + ) + ( ) ( ) ( ) h f x h h h f x h f x h f ( x ) = hh h h ( x ) f = ( + ) ( + ) ( ) f x h f x h h Multi-poit Differetitio Formule oly derived for equl spcig No equl spcig solve equtios umericlly Mple f ( x ) = f ( x h) 8f ( x h) 8f ( x h) f ( x h) h

4 Multi-poit Differetitio Coefficiet Demomitor -4h -3h -h -h h h 3h 4h Exct to st derivtive h - Qudrtic h Qurtic 6h th order 84h th order Multi-poit Differetitio Coefficiet Demomitor -4h -3h -h -h h h 3h 4h Exct to d derivtive h Qudrtic h Qurtic 54h th order 54h th order 4

5 Multi-poit Differetitio Coefficiet Demomitor -4h -3h -h -h h h 3h 4h Exct to 3 rd derivtive h Qurtic 48h th order 4h th order si x x f ( x) = x x = Techique Exmple x =.4π = h =. f ( x) % error f ( x) % error f ( x) % error 3-poit poit poit poit Exct vlue

6 Numericl Itegrtio Midpoit Formul Uses vlue of fuctio d slope t midpoit of itervl + + ( ) = ( ) + ( ) f x dx w f w f Determie w & w ( ) = = f x w + f ( x) = x = w + w w = w = ( ) = ( ) ( ) f x dx f + 6

7 Composite Midpoit Formul suitervls (equl spcig) h = ( ) 3 5 ( ) = ( + ) + ( + ) + ( + ) + ( + ) + ( + ) + + ( + ) f xdx h f h f h f h f h f h f h i = h f + h i= Trpezoidl Itegrtio Approximte f(x) y lier fuctio over itervl [,] ( ) f ( ) ( ) f f ( x) = f ( ) + x ( ) f ( ) f f ( x) dx = f ( ) dx + ( x ) dx ( ) f ( ) ( ( ) ( )) f = f ( )( ) + ( )( ) ( ( ) ( ))( ( ) ) f ( ) f ( ) ( ) = f + f f + = + 7

8 Trpezoidl Itegrtio Alterte derivtio Lier comitio of edpoits tht give est estimte of itegrl ( ) = ( ) + ( ) f xdx wf w f Determie w & w ( ) = = + f x w w f ( x) = x = w + w w = w = Composite Trpezoidl Itegrtio suitervls (equl spcig) h = h f xdx f f h f h f h f h f h f ( ) = ( ( ) + ( + ) + ( + ) + ( + ) + ( + ) + ( + 3 ) + + ( )) ( ( ) ( ) ( ) ( 3 ) ( )) = h f + f + h + f + h + f + h + + f 8

9 Simpso s Rule Comies Trpezoidl d Midpoit Also referred to s 3 - poit + ( ) = ( ) + ( ) + ( ) f xdx wf w f w f 3 Pckge Determie w w & w 3 f ( x) = = w+ w + w3 ( ) f x = x = w + w + w 3 ( ) w = w3 = w = f ( x) = x = w + w + w 3 f ( x) dx = f ( ) f ( ) + f ( ) 6 h = f ( ) + 4 f ( + h ) + f ( + h ) 3 Composite Simpso s Rule suitervls (equl spcig) h = h f x dx f f h f h f h f h f h f h 6 ( ) = ( ( ) + 4 ( + ) + ( + ) + ( + ) + 4 ( + 3 ) + ( + 4 ) + ( + )) h = f f + h + f i h f ih i= ( ) ( ) ( [ ] ) ( ) 9

10 Newto-Cotes Formul Geerliztio to use more th 3 poits Trpezoidl exct up to lier ( st order NC) Simpso s exct up to qudrtic (y defiitio ut turs out to e exct for up to cuic) ( d order NC) Equivlet to itegrtio of Lgrgi iterpoltio fuctios 3 rd order NC Use 4 poits d fuctios up to cuic Higher orders c give lrger errors Newto-Cotes Formul rd 3 order + 3h 3h f ( xdx ) = f( ) + 3f( + h) + 3f( + h) + f( + 3h) 8 Pckge th 4 order + 4h h f ( x) dx= 7f ( ) + 3f ( + h) + f ( + h) + 3f ( + 3h) + 7f ( + 4h) 45

11 Gussi Qudrtures So fr evluted fuctio t fixed poits & optimized coefficiets Optimize loctios lso φ ( zdz ) wφ( z) = i= i i φ( x) dx = φ( ) Optimize w i & z i x ( + ) z = z dz -poit Gussi Qudrtures φ ( zdz ) = wφ( z) Need two equtios Mke exct for φ(z) =, & φ(z) = z ( z) For φ = φ = w ( zdz ) = wφ( z) For φ ( z) ( zdz ) = wφ( z) φ = z z zdz = = = w z z = φ ( zdz ) = φ( )

12 -poit ( zdz ) = w ( z) + w ( z) φ φ φ Gussi Qudrtures Need four equtios Mke exct for φ(z) =, φ(z) = z, φ(z) = z, φ(z) = z 3 Does ot give uique solutio Mke symmetric out Gussi Qudrture.mws ( zdz ) = w ( z) + ( z) φ φ φ Gussi Qudrtures ( z) For φ = ( zdz ) = w ( z) + ( z) φ φ φ [ ] dz = w = w w = For φ = ( z) = z ( zdz ) = ( z) + ( z) φ φ φ zdz = z + z For φ 3 ( z) ( zdz ) = ( z) + ( z) zdz= z z = = z 3 3 z = 3 = z φ φ φ

13 φ( zdz ) = wiφ( zi) i= Gussi Qudrtures Roots (z i ) Weight Fctors (w i ) Two-Poit Formul ± Three-Poit Formul ± Four-Poit Formul ± ± Gussi Qudrtures φ( zdz ) = wiφ( zi) i= Roots (z i ) Weight Fctors (w i ) ± ± ± ± ± Five-Poit Formul Six-Poit Formul

14 φ( zdz ) = wiφ( zi) i= Gussi Qudrtures Roots (z i ) ± ± ± ± ± ± ± ± ± ± ± ± Te-Poit Formul Fiftee-Poit Formul Weight Fctors (w i ) Other forms e 8 z e ( ) φ z dz z ( ) φ z dz Gussi Qudrtures 4

15 Gussi Qudrtures - Exmple π e x dx Simpso s Rule Use itervls Gussi Qudrture 3 d 5 poit Simpsos Gussi Qudrture Exmple.mws 5

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