3.01.Splines - Approximation Authors: Bill Davis, Horacio...

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Unformatted text preview: Approximation Authors: Bill Davis, Horacio Porta and Jerry Uhl 1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 3.01 Splines BASICS B.1) Some remarkable plots explained by order of contact Sometimes functions whose formulas are strikingly different have plots that are strikingly similar. Here are a few for your plotting pleasure. B.1.a.i) Here are plots of f @ x D = Cos @ 2 x D , and g @ x D = 1- 4 x 2 on the same axes for - 0.5 x 0.5. Clear @ x, f, g D ; f @ x _ D = Cos @ 2 x D ; g @ x _ D = 1- 4 x 2 ; Plot @8 f @ x D , g @ x D< , 8 x,- 0.5, 0.5 < , AxesLabel 8 "x" , "" < , PlotStyle 88 GrayLevel @ 0.4 D , Thickness @ 0.01 D< , 8 Red, Thickness @ 0.01 D<<D- 0.4- 0.2 0.2 0.4 x 0.2 0.4 0.6 0.8 1.0 Describe what you see. Answer: Very similar plots near x = 0. As you advance x from the left of 0 to the right of 0, you can make the transition from one curve to the other with ease. B.1.a.ii) Here are plots of f @ x D = Sin @ x D 2 , and g @ x D = x 2 Cos @ x D 2 3 on the same axes for - 1 x 1. Clear @ x, f, g D ; f @ x _ D = Sin @ x D 2 ; g @ x _ D = x 2 Cos @ x D 2 3 ; Plot @8 f @ x D , g @ x D< , 8 x,- 1, 1 < , AxesLabel 8 "x" , "" < , PlotStyle 88 GrayLevel @ 0.4 D , Thickness @ 0.01 D< , 8 Red, Thickness @ 0.01 D<<D- 1.0- 0.5 0.5 1.0 x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Describe what you see. Answer: Very similar plots near x = 0. As you advance x from the left of 0 to the right of 0, you can make the transition from one curve to the other with ease. B.1.a.iii) Here are plots of f @ x D = 1 + Sin @ x D and g @ x D = 60 + 60x + 3x 2- 7x 3 60 + 3x 2 on the same axes for - 3 x 3. f @ x _ D = 1 + Sin @ x D ; g @ x _ D = 60 + 60 x + 3 x 2- 7 x 3 60 + 3 x 2 ; Plot @8 f @ x D , g @ x D< , 8 x,- 3, 3 < , AxesLabel 8 "x" , "" < , PlotStyle 88 GrayLevel @ 0.4 D , Thickness @ 0.01 D< , 8 Red, Thickness @ 0.01 D<<D- 3- 2- 1 1 2 3 x 0.5 1.0 1.5 2.0 Describe what you see. Answer: Very similar plots near x = 0. As you advance x from the left of 0 to the right of 0, you can make the transition from one curve to the other with ease. B.1.a.iv) Here are plots of f @ x D = Cos @ p x D , and g @ x D = - 1 + p 2 2 H x- 1 L 2 on the same axes for 0 x 2: Clear @ x, f, g D ; f @ x _ D = Cos @ p x D ; g @ x _ D = - 1 + 1 2 p 2 H x- 1 L 2 ; Plot @8 f @ x D , g @ x D< , 8 x, 0, 2 < , AxesLabel 8 "x" , "" < , PlotStyle 88 GrayLevel @ 0.4 D , Thickness @ 0.01 D< , 8 Red, Thickness @ 0.01 D<<D 0.5 1.0 1.5 2.0 x- 1 1 2 3 4 Describe what you see. Answer: Very similar plots near x = 1. As you advance x from the left of 1 to the right of 1, you can make the transition from one curve to the other with ease....
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This note was uploaded on 10/11/2011 for the course MATH 231 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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3.01.Splines - Approximation Authors: Bill Davis, Horacio...

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