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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 Δ x , let x = x 1- Δ x and x 2 = x 1 + Δ 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- Δ +- +- Δ- + = 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 θ = + Δ ⋅ or ( ) ( ) B A C df f x f x x dx = + Δ ⋅ 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 Δ ⋅ . 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 Δ x is small, the error may be close to the value of A df x dx Δ ⋅ . On the other hand we may simply say that the error is of order Δ x [O( Δ 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 = π /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....
View Full Document

{[ snackBarMessage ]}

### Page1 / 8

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online