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mws_gen_dif_ppt_continuous

# mws_gen_dif_ppt_continuous - FORWARD DIFFERENCE METHOD PPT

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10/14/11 http://numericalmethods.eng.usf.edu 1 Differentiation-Continuous Functions Major: All Engineering Majors Authors: Autar Kaw, Sri Harsha Garapati http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

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Differentiation – Continuous Functions http://numericalmethods.eng.usf.edu
http://numericalmethods.eng.usf.edu 3 Forward Difference Approximation ( 29 ( 29 ( 29 x x f x x f x x f Δ Δ 0 Δ lim - + = For a finite ' Δ ' x ( 29 ( 29 ( 29 x x f x x f x f - +

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http://numericalmethods.eng.usf.edu 4 x x+Δx f(x) Figure 1  Graphical Representation of forward difference approximation of first derivative. Graphical Representation Of  Forward Difference  Approximation
http://numericalmethods.eng.usf.edu 5 Example 1 The velocity of a rocket is given by ( 29 30 0 , 8 . 9 2100 10 14 10 14 ln 2000 4 4 - - × × = t t t t ν where  ' ' ν is given in m/s and  ' ' t is given in seconds.  a) Use forward difference approximation of the first derivative of        to  calculate the acceleration at            . Use a step size of            . b) Find the exact value of the acceleration of the rocket. c) Calculate the absolute relative true error for part (b). ( 29 t ν s t 16 = s t 2 Δ =

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http://numericalmethods.eng.usf.edu 6 Example 1 Cont. Solution ( 29 ( 29 ( 29 t t t t a i i i - + ν ν 1 16 = i t 2 Δ = t 18 2 16 1 = + = + = + t t t i i ( 29 ( 29 ( 29 2 16 18 16 ν ν - a
http://numericalmethods.eng.usf.edu 7 Example 1 Cont. ( 29 ( 29 ( 29 18 8 . 9 18 2100 10 14 10 14 ln 2000 18 4 4 - - × × = ν m/s 02 . 453 = ( 29 ( 29 ( 29 16 8 . 9 16 2100 10 14 10 14 ln 2000 16 4 4 - - × × = ν m/s 07 . 392 = Hence ( 29 ( 29 ( 29 2 16 18 16 ν ν - a

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http://numericalmethods.eng.usf.edu 8 Example 1 Cont. 2 07 . 392 02 . 453 - 2 m/s 474 . 30 The exact value of  ( 29 16 a can be calculated by differentiating  ( 29 t t t 8 . 9 2100 10 14 10 14 ln 2000 4 4 - - × × = ν as ( 29 ( 29 [ ] t ν dt d t a = b)
http://numericalmethods.eng.usf.edu 9 Example 1 Cont. Knowing that ( 29 [ ] t t dt d 1 ln = and  2 1 1 t t dt d - = ( 29 8 . 9 2100 10 14 10 14 10 14 2100 10 14 2000 4 4 4 4 - - × × × - × = t dt d t t a ( 29 ( 29 ( 29 8 . 9 2100 2100 10 14 10 14 1 10 14 2100 10 14 2000 2 4 4 4 4 - - - × × - × - × = t t t t 3 200 4 . 29 4040 + - - - =

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