mws_gen_dif_txt_discrete

mws_gen_dif_txt_discrete - Chapter 02.03 Differentiation of...

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02.03.1 Chapter 02.03 Differentiation of Discrete Functions After reading this chapter, you should be able to: 1. find approximate values of the first derivative of functions that are given at discrete data points, and 2. use Lagrange polynomial interpolation to find derivatives of discrete functions. To find the derivatives of functions that are given at discrete points, several methods are available. Although these methods are mainly used when the data is spaced unequally, they can be used for data that is spaced equally as well. Forward Difference Approximation of the First Derivative We know     x x f x x f x f x 0 lim For a finite x ,   x x f x x f x f
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02.03.2 Chapter 02.03 Figure 1 Graphical representation of forward difference approximation of first derivative. So given 1 n data points        n n y x y x y x y x , , , , , , , , 2 2 1 1 0 0 , the value of ) ( x f for 1 i i x x x , 1 ,..., 0 n i , is given by   i i i i i x x x f x f x f 1 1 Example 1 The upward velocity of a rocket is given as a function of time in Table 1. Table 1 Velocity as a function of time. (s) t ) m/s ( ) ( t v 0 0 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 Using forward divided difference, find the acceleration of the rocket at s 16 t . Solution To find the acceleration at s 16 t , we need to choose the two values of velocity closest to s 16 t , that also bracket s 16 t to evaluate it. The two points are s 15 t and s 20 t ) ( x f x x x x
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Differentiation of Discrete Functions 02.03.3    t t t v t a i i i 1 15 i t 20 1 i t 5 15 20 1 i i t t t   
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mws_gen_dif_txt_discrete - Chapter 02.03 Differentiation of...

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