By Samuel Daniel Conte
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5) does indeed describe a polynomial of degree < n. , the coefficients a0, . . , an in the Lagrange form are simply the values of the polynomial p(x) at the points x0 , . . , xn . 7) is a polynomial of degree < n which interpolates f(x) at x0, . . , xn. This establishes the following theorem. 1 Given a real-valued function f(x) and n + 1 distinct points x0, . . , xn, there exists exactly one polynomial of degree < n which interpolates f(x) at x0, . . , xn. 7) is called the Lagrange formula for the interpolating polynomial.
2-3, the polynomial of degree < k which interpolates a function f(x) at x0, . . , xk is f(x) itself if f(x) is a polynomial of degree < k. This fact may be used to check the accuracy of the computed interpolating polynomial. 4 to carry out such a check as follows: For n = 4, 8, 12, . . , 32, find the polynomial pn(x) of degree < n which interpolates the function at 0,1,2, . . ,n. Then estimate where the yi's are a suitably large number of points in [0, n] . 3-6 Prove that the first derivative p'2(x) of the parabola interpolating f(x) at x0 < xl < x2 is equal to the straight line which takes on the value f[xi-1, xi] at the point (xi-1 + xi) /2, for i = 1, 2.
Xk; it is called the kth divided difference of f(x) at the points x0, . . 11) The first divided difference, at any rate, is a ratio of differences. 2-1 Prove that (x - xn). 2-l as 22-2 Calculate the limit of the formula for while all other points remain fixed. 2-3 Prove that the polynomial of degree < n which interpolates f(x) at n + 1 distinct points is f(x) itself in case f(x) is a polynomial of degree < n. 2-4 Prove that the kth divided difference p[x0, . . , xk] of a polynomial p(x) of degree < k is independent of the interpolation points x0, xl, .