Question

1) Given a pair of values for x and f (x), f (x) is unknown, as follows: x | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 | 1.1 | 1.3 f(x) | 0.003 | 0.067 | 0.148 | 0.248 | 0.370 | 0.518 | 0.697 (a) What are the exact degrees of the polynomial through the seven data points? (b) What is the best degree of polynomial to interpolate the seven data points? (c) With the best degree stated in answer (b), find the value of the function at x = 0.58 using the interpolated polynomial: (ii) Lagrange (iii) Newton (iv) Newton-Gregory forward (v) Newton-Gregory backward Choose points that minimize interpolation errors. (d) Calculate the estimated error in calculating f (0.8) with polynomials (ii), (iii), (iv). (e) With the inverse interpolation technique, determine the value of x so that f (x) = 0.200. 2. (a) Show for cubic interpolation that: | E3(x) | = | f(x) - p3(x) | ? h^4/27 max | f^(4)(c) | , x0 ? c ? x3 which in this case p3(x) is a 3rd degree interpolated polynomial which is interpolate x0 = 0, x1 = h, x2 = 2h, dan x3 = 3h . (b) Find h (distance between points) for points that are equidistant from the function f(x) = ?x between x = 1 to x = 2 so that the interpolation of the cubic polynomial in the list of values has an error of less than 0.000001. How many data points with h of that size?

          1) Given a pair of values for x and f (x), f (x) is unknown, as follows:

x | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 | 1.1 | 1.3
f(x) | 0.003 | 0.067 | 0.148 | 0.248 | 0.370 | 0.518 | 0.697

(a) What are the exact degrees of the polynomial through the seven data points?
(b) What is the best degree of polynomial to interpolate the seven data points?
(c) With the best degree stated in answer (b), find the value of the function at x = 0.58 using the interpolated polynomial:
(ii) Lagrange
(iii) Newton
(iv) Newton-Gregory forward
(v) Newton-Gregory backward
Choose points that minimize interpolation errors.
(d) Calculate the estimated error in calculating f (0.8) with polynomials (ii), (iii), (iv).
(e) With the inverse interpolation technique, determine the value of x so that f (x) = 0.200.
2. (a) Show for cubic interpolation that:
| E3(x) | = | f(x) - p3(x) | ? h^4/27 max | f^(4)(c) | , x0 ? c ? x3
which in this case p3(x) is a 3rd degree interpolated polynomial which is interpolate x0 = 0, x1 = h, x2 = 2h, dan x3 = 3h .
(b) Find h (distance between points) for points that are equidistant from the function f(x) = ?x between x = 1 to x = 2 so that the interpolation of the cubic polynomial in the list of values has an error of less than 0.000001. How many data points with h of that size?

1) Given a pair of values for x and f (x), f (x) is unknown, as follows:

x | 0.1 | 0.3 | 0.5 | 0.7 | 0.9 | 1.1 | 1.3
f(x) | 0.003 | 0.067 | 0.148 | 0.248 | 0.370 | 0.518 | 0.697

(a) What are the exact degrees of the polynomial through the seven data points?
(b) What is the best degree of polynomial to interpolate the seven data points?
(c) With the best degree stated in answer (b), find the value of the function at x = 0.58 using the interpolated polynomial:
(ii) Lagrange
(iii) Newton
(iv) Newton-Gregory forward
(v) Newton-Gregory backward
Choose points that minimize interpolation errors.
(d) Calculate the estimated error in calculating f (0.8) with polynomials (ii), (iii), (iv).
(e) With the inverse interpolation technique, determine the value of x so that f (x) = 0.200.
2. (a) Show for cubic interpolation that:
| E3(x) | = | f(x) - p3(x) | ? h^4/27 max | f^(4)(c) | , x0 ? c ? x3
which in this case p3(x) is a 3rd degree interpolated polynomial which is interpolate x0 = 0, x1 = h, x2 = 2h, dan x3 = 3h .
(b) Find h (distance between points) for points that are equidistant from the function f(x) = ?x between x = 1 to x = 2 so that the interpolation of the cubic polynomial in the list of values has an error of less than 0.000001. How many data points with h of that size?

Added by Alejandra M.

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