hw7 - 4. Find b,c,d so that the following is a natural...

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Homework #7 1. Consider data points ( x 0 - h,f ( x 0 - h )) , ( x 0 ,f ( x 0 )) , ( x 0 + h,f ( x 0 + h )). (a) Find the interpolating polynomial of least degree p ( x ) passing through these points in Newton form. (b) Write down an approximation for f 0 ( x 0 ) by simplifying the expression for p 0 ( x 0 ). (c) Write down an approximation for R x 0 + h x 0 - h f ( x ) dx by simplifying the expression for R x 0 + h x 0 - h p ( x ) dx . 2. (a) Draw a graph of the piecewise linear interpolating polynomial for the data given in the following table: x 0 1 3 4 6 f ( x ) 1 0 1 2 1 (b) Write down the equation for the linear piece in the interval [4 , 6]. (c) Suppose we create the piecewise quadratic interpolating polynomial by using the quadratic passing through the first three points in x [0 , 3] and the quadratic passing through the last three points in x [3 , 6]. Find the values of this piecewise polynomial at x = 2 and x = 5. 3. Consider the data ( x 1 ,y 1 ) , ( x 2 ,y 2 ) ,..., ( x n ,y n ) . Suppose we want a spline of degree 4 polynomials in each interval [ x i ,x i +1 ] for i = 1 , 2 ,...,n - 1. How many derivatives of the spline can we make continuous?
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Unformatted text preview: 4. Find b,c,d so that the following is a natural cubic spline S ( x ) = ( 1 + 2 x-x 3 , if 0 x < 1 , 2 + b ( x-1) + c ( x-1) 2 + d ( x-1) 3 , if 1 x < 2 . 5. (Matlab) (a) Plot the piecewise linear polynomial of problem #2(a) over [0 , 6] and turn it in. (b) Plot the piecewise quadratic polynomial of problem #2(c) over [0 , 6] and turn it in. (c) Plot the interpolating polynomial of least degree for the data in problem #2 and turn it in. 6. (Math 274) Consider data points taken from the sine function at n evenly spaced points x i : 0 = x 1 < x 2 < ... < x n = 2 . Let p ( x ) be the piecewise linear polynomial interpolating these points. Use bounds to nd n that will ensure that | f ( x )-p ( x ) | 10-7 . 1...
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This note was uploaded on 10/17/2010 for the course MATH 174 taught by Professor Staff during the Spring '08 term at UCSD.

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