201-Lect 7 - MECH201 Engineering Analysis (2009) p. 3 1...

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Unformatted text preview: MECH201 Engineering Analysis (2009) p. 3 1 REVISION 1.1 Polynomial Approximation and Interpolations Given a set of points ( x i , y i ), where i = 0, 1, 2, 3, n , a polynomial of n degree can be fitted. The polynomial is in the form: ( ) = = n j j j x A x y so that x = x i , y = y i. This give rises to n+1 linear equation with A j being the unknown. However the above equation can be converted into a set of linear equations in A j in which A j can be solved. For every point ( x i , y i ) we have a linear equation. Thus this result in a set of linear equations: n j i j i j x A y = = An iteration process will solve for the values of A j . . Once A j are known the polynomial is define and we may use the polynomial to represent a derivative or an integration. =- = n j j j x jA dx dy 1 1 and b a n j j j b a x j A ydx = + + = 1 1 Problem For n=2 and the points are equally spaced at interval x , let x = x 1- x and x 2 = x 1 + x . Show that: ( ) ( ) 2 1 2 1 2 1 2 1 2 2 2 x x x y y y x x x y y y y- +- +- - + = 1.2 Approximation We may reverse the above approach and use the know value of f(x A ) to estimate the value of f(x B ) . That is: ( ) ( ) ( ) tan B A C f x f x x = + or ( ) ( ) B A C df f x f x x dx = + MECH201 Engineering Analysis (2009) p. 4 In other words, ( ) A f x is an estimation of ( ) B f x and the error of the estimation is C df x dx . Note that the location of C, that is the value of x C cannot be specified at this stage except that x C lies between x A and x B . If x is small, the error may be close to the value of A df x dx . On the other hand we may simply say that the error is of order x [O( x )]. 1.3 The Taylor Series The Taylor Series of expansion represents the value of a function f(x) at x B by the function and all its derivatives at point x A . That is ( ) ( ) ( ) [ ] ( ) [ ] ( ) [ ] ( ) 2 3 ' 1 1 '' ''' ... 2! 3! 1 .... .. ! B A B A A B A B A A A n n B A A f x f x x x f x x f x x f x x f n = +- +- +- + +- + Examples Write down the Taylor series expansion for the following functions: 1. sin(x) around x = 0 2. 1 sin( ) x around x = /4 3. (1+x) 3 around x = 0 and x = 1 4. x e around x = 0 5. ln(x) around x = 0 ? and x = 1 6. ln(1+x) around x = 0....
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201-Lect 7 - MECH201 Engineering Analysis (2009) p. 3 1...

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