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# (Solved): (a) Given a function f that satisfies f(-1) = 3, f(0) = -4, f(1) = 5 and f(2)=-6. Find the unique La ...

(a) Given a function f that satisfies f(-1) = 3, f(0) = -4, f(1) = 5 and f(2)=-6. Find the unique Lagrange interpolating polynomial of degree 3 that agrees with all given function values. Show your steps. (b) Given a closed and bounded interval [c, d], and suppose this interval consists of (n + 1) regularly spaced nodes (including the two endpoints, c and d). Denote the nodes as {X1, X2, X3,..., Xn+1}. n+1 Prove that for all x € [c, d], [(x − x₁)(x − x₂) ... (x − Xn+1)| ≤n! ⋅ (d = c) (c) Let f(x) = sinx, and 4 nodes are given (including the two endpoints), at x = 0,7, and An interpolating polynomial of degree 3, P3 (x) is used to approximate f(x) on [0,1]. Show that when P3(x) is used to "approximate" f(x) at x = 2, the maximum possible error is less than 0.003. Show your steps. 12

(a) Given a function that satisfies and . Find the unique Lagrange interpolating polynomial of degree 3 that agrees with all given function values. Show your steps. (b) Given a closed and bounded interval , and suppose this interval consists of regularly spaced nodes (including the two endpoints, and ). Denote the nodes as . Prove that for all (c) Let , and 4 nodes are given (including the two endpoints), at and . An interpolating polynomial of degree is used to approximate on . Show that when is used to "approximate" at , the maximum possible error is less than 0.003 . Show your steps.

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