All parts of this question concern the function f(x) = 6 sin x + 2 cos...

  

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All parts of this question concern the function f(x) = 6 sin x + 2 cos x. (a) Find the smallest positive constant M that satisfies M > f(k) (t) for every possible combination of an integer k ≥ 0 and an evaluation point t€ (-∞0, +∞0). Hint: A standard trigonometric identity implies that, for a certain angle , one has f(x) = √40 sin (x + o) for all real x. f(n+1) (t) (n + 1)! Recall the standard decomposition f(x) = Tn(x) + En(x), in which Lagrange's formula says En(x) = valid for every integer n ≥ 0. In both parts below, estimate En(x) using Lagrange's formula with the constant M found in part (a). (Use technology as required.) for some t between 0 and x. This is (b) Find the smallest n for which the polynomial value T₁ (0.3) provides an approximation for f(0.3) that is guaranteed to be accurate to within 11 decimal places: Hint: To guarantee D correct digits after the decimal point, accounting for rounding, one must have En(0.3)| ≤ 0.5 × 10-. (c) Suppose n = 9 is prescribed. Find the largest positive number a such that the approximation T9 (x) for f(x) is guaranteed to be accurate to within 9 decimal places, for all x in the symmetric interval (-a, a). All parts of this question concern the function f(x) = 6 sin x + 2 cos x. (a) Find the smallest positive constant M that satisfies M > f(k) (t) for every possible combination of an integer k ≥ 0 and an evaluation point t€ (-∞0, +∞0). Hint: A standard trigonometric identity implies that, for a certain angle , one has f(x) = √40 sin (x + o) for all real x. f(n+1) (t) (n + 1)! Recall the standard decomposition f(x) = Tn(x) + En(x), in which Lagrange's formula says En(x) = valid for every integer n ≥ 0. In both parts below, estimate En(x) using Lagrange's formula with the constant M found in part (a). (Use technology as required.) for some t between 0 and x. This is (b) Find the smallest n for which the polynomial value T₁ (0.3) provides an approximation for f(0.3) that is guaranteed to be accurate to within 11 decimal places: Hint: To guarantee D correct digits after the decimal point, accounting for rounding, one must have En(0.3)| ≤ 0.5 × 10-. (c) Suppose n = 9 is prescribed. Find the largest positive number a such that the approximation T9 (x) for f(x) is guaranteed to be accurate to within 9 decimal places, for all x in the symmetric interval (-a, a).

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a To find the smallest positive constant M such that M fx fx for all a and all k 0 we ... View the full answer