Is s x x a first degree spline why or why not 17 find

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Engineering Fundamentals: An Introduction to Engineering
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Chapter 15 / Exercise 15.23
Engineering Fundamentals: An Introduction to Engineering
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16. Is S ( x ) = | x | a first degree spline ? Why or why not ? 17. Find the natural quadratic spline interpolant for the following data
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Engineering Fundamentals: An Introduction to Engineering
The document you are viewing contains questions related to this textbook.
Chapter 15 / Exercise 15.23
Engineering Fundamentals: An Introduction to Engineering
Moaveni
Expert Verified
Polynomial and Splines Interpolation 181 (a) i x y 0 - 1 2 1 0 1 2 1 / 2 0 3 1 1 4 2 2 5 5 / 2 3 (b) i x y 0 1 2 1 2 1 2 5 / 2 0 3 3 1 4 4 3 18. Are these functions quadratic splines ? Explain why or why not ?Plot their graphs.
19. Determine a quadratic spline that interpolates the data f (0) = 0 , f (1) =
20. Do there exist a,b, c and d so that the function
is a natural cubic spline function ?
182 N. Nassif and D. Fayyad 21. Do there exist coe ffi cients for which the function S ( x ) = 8 < : x + 1 ; - 2 x  - 1 ax 3 + bx 2 + cx + d ; - 1 x 1 x - 1 ; 1 x 2 is a natural cubic spline function ? Why or why not ? 22. Determine the natural cubic spline that interpolates the function f ( x ) = x 6 over the interval [0 , 2] using nodes 0 , 1 and 2 . 23. Find the natural cubic spline interpolant for this table i x y 0 1 0 1 2 1 2 3 0 3 4 1 4 5 0 24. Find the natural cubic spline interpolant for this table i x y 0 1 0 1 2 1 2 3 0 3 4 1 4 5 0 6 1 25. Consider the following set of data generated using the function f ( x ) = xcos x - 2 x 2 + 3 x - 1 i x y 0 0 . 1 - 0 . 620499 1 0 . 2 - 0 . 283986 2 0 . 3 +0 . 006600 3 0 . 4 +0 . 248424 (a) Construct the natural cubic spline for the data above (b) Use the cubic spline constructed above to approximate f (0 . 25 and f 0 (0 . 25), and calculate the absolute error.
Polynomial and Splines Interpolation 183 26. Give an example of a cubic spline with nodes 0 , 1 , 2 , and 3 that is quadratic in [0 , 1], cubic in [1 , 2], and quadratic in [2 , 3]. 27. Give an example of a cubic spline with nodes 0 , 1 , 2 , and 3 that is linear in [0 , 1], but of degree 3 in the other two intervals. 28. List all the ways in which the following functions fail to be natural cubic splines: (a) S ( x ) = 8 < : x + 1 ; - 2 x  - 1 x 3 - 2 x + 1 ; - 1 x 1 x - 1 ; 1 x 2 (b) S ( x ) = x 3 + x - 1 ; - 1 x 0 x 3 - x - 1 ; 0 x 1 29. Determine the coe ffi cients so that the function S ( x ) = x 2 + x 3 ; 0 x 1 a + bx + cx 2 + dx 3 ; 1 x 2 is a cubic spline that has the property S 000 1 ( x ) = 12 30. Use the data points (0 , 1) , (1 , e ) , (2 , e 2 ) , (3 , e 3 ) to form a natural cubic spline that approximates f ( x ) = e x . 31. Use the cubic spline obtained in the preceding exercise, to approximate R 3 0 e x dx . What is the relative error in this approximation ? 32. Construct a natural cubic spline to approximate f ( x ) = e - x based on the nodes x = 0 , 0 . 25 , 0 . 75 and 1. Integrate the spline over the interval [0 , 1] and compare the results to R 1 0 e - x dx . Use the deriva- tives of the spline to approximate f 0 (0 . 5) and f ”(0 . 5). Compare the approximations to the actual values.

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