Construct a sequence of interpolating values y_n to f(1 + sqrt 10), where f(x) =
ID: 2973349 • Letter: C
Question
Construct a sequence of interpolating values y_n to f(1 + sqrt 10), where f(x) = (1 + x^2)^-1 for -5 is less than or equal to x is less than or equal to 5 as follows. For each n = 1, 2, ..., 10, let h = 10 / n and y_n = P_n(1 + sqrt 10), where P_n (x) is the interpolating polynomial for f(x) at the nodes x_0^(n), x_1^(n), ..., x_n^(n) and xj^(n) = -5 +jh, for each j = 0, 1, 2, ... , n. Does the sequence {y_n} appear to converge to f(1 + sqrt 10)? Inverse interpolation: Suppose f is an element of C^1 [a, b], f(x) does not equal 0 on [a, b] and f has one zero p in [a, b]. Let x_0, ... , x_n be n + 1 distinct numbers in [a, b] with f(xg) = yg, for each k = 0, 1, ..., n. To approximate p, construct the interpolating polynomial of degree n on the nodes y_0, ... , yg for f^-1. Since yg = f(xg) and 0 = f(p), it follows that f^-1(yg) = xg and p = f^-1(0).Explanation / Answer
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