mws_gen_dif_txt_continuous - Chapter 02.02 Differentiation...

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02.01.1 Chapter 02.02 Differentiation of Continuous Functions After reading this chapter, you should be able to: 1. derive formulas for approximating the first derivative of a function, 2. derive formulas for approximating derivatives from Taylor series, 3. derive finite difference approximations for higher order derivatives, and 4. use the developed formulas in examples to find derivatives of a function. The derivative of a function at x is defined as ( ) ( ) ( ) x x f x x f x f x + = 0 lim To be able to find a derivative numerically, one could make x finite to give, ( ) ( ) ( ) x x f x x f x f + . Knowing the value of x at which you want to find the derivative of ( ) x f , we choose a value of x to find the value of ( ) x f . To estimate the value of ( ) x f , three such approximations are suggested as follows. Forward Difference Approximation of the First Derivative From differential calculus, 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 + The above is the forward divided difference approximation of the first derivative. It is called forward because you are taking a point ahead of x . To find the value of ( ) x f at i x x = , we may choose another point x ahead as 1 + = i x x . This gives ( ) ( ) ( ) x x f x f x f i i i + 1
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