Assignment_2

Assignment_2 - Time t (sec) 0 0.5 1.5 Position x (m) 0 1...

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ME 303 ADVANCED ENGINEERING MATHEMATICS Assignment 2 Question 1. Note a feature of all finite difference formulas in handout 2.2: the coefficients of f always sum to zero. For example, in the 4 th order central x f f f f dx df i i i i x i Δ + + = + + 12 8 8 2 1 1 2 , we have (-1+8-8+1)=0. This is a useful check in formula derivations. Why do these coefficients always sum to zero? Question 2. Derive the formulas and verify in each case that the f coefficients sum to zero: a) the 1 st order, backward difference formula for i x dx f d 2 2 b) the 3 rd order, forward difference formula for i x dx df Question 3. Consider three unequally spaced points i x , x x x i i Δ = 1 , x x x i i Δ + = + 2 1 a) Derive the central difference formula for i x dx f d 2 2 of the highest possible accuracy using these three points. b) Experimental data for position versus time of an object are given below. Use (a) to estimate the object’s acceleration at 5 . 0 = t (sec).
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Unformatted text preview: Time t (sec) 0 0.5 1.5 Position x (m) 0 1 1.7 Question 4. a) Consider x x f 2 sin ) ( = . Find the exact value of 1 = x dx df b) Sketch ) ( x f to verify that 15 . = Δ x is a reasonable grid spacing here. With 15 . = Δ x , estimate 1 = x dx df using first, second and third order forward differences. (c) Try a first order forward difference with 05 . = Δ x and . 005 . = Δ x Question 5. a) Sketch the function x e y x f y cos 3 ) , ( 2 − = for 2 / π ≤ ≤ x and 3 ≤ ≤ y b) Find the exact values of x f ∂ ∂ , y f ∂ ∂ , and 2 2 x f ∂ ∂ at ( 1 , 1 = = y x ) c) From (a), verify that 1 . = Δ = Δ y x is a reasonable grid spacing for this function. Estimate x f ∂ ∂ , y f ∂ ∂ , and 2 2 x f ∂ ∂ at ( 1 , 1 = = y x ) using second order, central difference formulas....
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This note was uploaded on 09/16/2011 for the course ME 303 taught by Professor Serhiyyarusevych during the Spring '10 term at Waterloo.

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