mws_gen_dif_ppt_continuous

mws_gen_dif_ppt_continuous - Differentiation-Continuous...

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1/10/2010 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
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http://numericalmethods.eng.usf.edu 3 Forward Difference Approximation ( ) ( ) ( ) x x f x x f x x f Δ Δ 0 Δ lim + = For a finite ' Δ ' x ( ) ( ) ( ) x x f x x f x f +
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http://numericalmethods.eng.usf.edu 4 x x+ Δ x f(x) Figure 1 Graphical Representation of forward difference approximation of first derivative. Graphical Representation Of Forward Difference Approximation
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http://numericalmethods.eng.usf.edu 5 Example 1 The velocity of a rocket is given by ( ) 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). ( ) t ν s t 16 = s t 2 Δ =
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http://numericalmethods.eng.usf.edu 6 Example 1 Cont. Solution ( ) ( ) ( ) t t t t a i i i + ν 1 16 = i t 2 Δ = t 18 2 16 1 = + = + = + t t t i i ( ) ( ) ( ) 2 16 18 16 a
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http://numericalmethods.eng.usf.edu 7 Example 1 Cont. ( ) ( ) ( ) 18 8 . 9 18 2100 10 14 10 14 ln 2000 18 4 4 × × = ν m/s 02 . 453 = ( ) ( ) ( ) 16 8 . 9 16 2100 10 14 10 14 ln 2000 16 4 4 × × = m/s 07 . 392 = Hence ( ) ( ) ( ) 2 16 18 16 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 ( ) 16 a can be calculated by differentiating ( ) t t t 8 . 9 2100 10 14 10 14 ln 2000 4 4 × × = ν as ( ) ( ) [ ] t ν dt d t a = b)
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http://numericalmethods.eng.usf.edu 9 Example 1 Cont. Knowing that ( ) [ ] t t dt d 1 ln = and 2 1 1 t t dt d = ( ) 8 . 9 2100 10 14 10 14 10 14 2100 10 14 2000 4 4 4 4 × × × × = t dt d t t a ( ) ( ) ( ) 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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http://numericalmethods.eng.usf.edu 10 Example 1 Cont. ( ) ( ) ( ) 16 3 200 16 4 . 29 4040 16 + = a 2 m/s 674 . 29 = The absolute relative true error is 100 Value True Value e Approximat - Value True x t = 100 674 . 29 474 . 30 674 . 29 x = % 6967 . 2 =
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http://numericalmethods.eng.usf.edu 11 Backward Difference Approximation of the First Derivative We know ( ) ( ) ( ) x x f x x f x x f
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This note was uploaded on 06/12/2011 for the course EML 3041 taught by Professor Kaw,a during the Spring '08 term at University of South Florida.

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mws_gen_dif_ppt_continuous - Differentiation-Continuous...

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