All parts of this question concern the function f(:1:) = 2sina: + 30081:. (a) Find the smallest positive constant M that satisfies M Z |f(k)(t)| for every possible combination of an integer k 2 0 and an evaluation point t E (00,100). Hint: A standard trigonometric identity implies that, for a certain angle (15, one has f(x) = V 13 sin (a: + 925) for all real 3:. Answer: M = Recall the standard decomposition at) 2 Tn(m) + En(a:), _ f(n+1)(t) n+1 ' (n + 1)! "3 for some at between 0 and :11. This is valid for every integer n > 0. in which Lagrange's formula says Edna) In both parts below, estimate Edge) using Lagrange's formula with the constant M found in part (a). (Use technology as required.) (b) Find the smallest n for which the polynomial value Tn(0.7) provides an approximation for f(0.7) that is guaranteed to be accurate to within 6 decimal places: Answer: 71 = Hint: To guarantee D correct digits after the decimal point, accounting for rounding, one must have |En(0.7)| g 0.5 x 1013. (C) Suppose n : 7 is prescribed. Find the largest positive number a such that the approximation T7($) for f(:c) is guaranteed to be accurate to within 4 decimal places, for all a: in the symmetric interval (a, a). Answer: a, =