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All parts of this question concern the function f(x) = 6 sin x + 7 cosx. (a) Find the smallest positive constant M that satisfies M2f) (t) for every possible combination of an integer k20 and an evaluation point t€ (-∞, +∞o). Hint: A standard trigonometric identity implies that, for a certain angle d. one has f(x) = √85 sin (x+) for all real x. Answer: M = f(n+1) (t) (n+1)! In both parts below, estimate En(a) using Lagrange's formula with the constant M found in part (a). (Use technology as required.) Recall the standard decomposition f(x) = T₂(x) + En(x), in which Lagrange's formula says En(x) = (b) Find the smallest n for which the polynomial value T₁, (0.4) provides an approximation for f(0.4) that is guaranteed to be accurate to within 10 decimal places: Answer: n == -x+1 for some t between 0 and x. This valid for every integer n 20. Hint: To guarantee D correct digits after the decimal point, accounting for rounding, one must have |E, (0.4)| ≤ 0.5 × 10-D. (c) Suppose n = 7 is prescribed. Find the largest positive number a such that the approximation T7(x) for f(x) is guaranteed to be accurate to within 8 decimal places, for all in the symmetric interval (-a, a). Answer: a =

All parts of this question concern the function f(x) = 6 sin x + 7 cosx. (a) Find the smallest positive constant M that satisfies M2f) (t) for every possible combination of an integer k20 and an evaluation point t€ (-∞, +∞o). Hint: A standard trigonometric identity implies that, for a certain angle d. one has f(x) = √85 sin (x+) for all real x. Answer: M = f(n+1) (t) (n+1)! In both parts below, estimate En(a) using Lagrange's formula with the constant M found in part (a). (Use technology as required.) Recall the standard decomposition f(x) = T₂(x) + En(x), in which Lagrange's formula says En(x) = (b) Find the smallest n for which the polynomial value T₁, (0.4) provides an approximation for f(0.4) that is guaranteed to be accurate to within 10 decimal places: Answer: n == -x+1 for some t between 0 and x. This valid for every integer n 20. Hint: To guarantee D correct digits after the decimal point, accounting for rounding, one must have |E, (0.4)| ≤ 0.5 × 10-D. (c) Suppose n = 7 is prescribed. Find the largest positive number a such that the approximation T7(x) for f(x) is guaranteed to be accurate to within 8 decimal places, for all in the symmetric interval (-a, a). Answer: a =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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All parts of this question concern the function f(x) = 6 sin x + 7 cosa. (a) Find the smallest positive constant M that satisfies M > f(*) (t) for every possible combination of an integer k ≥ 0 and an evaluation point t € (-∞, +∞). Hint: A standard trigonometric identity implies that, for a certain angle , one has f(x) = √85 sin (x + o) for all real x. Answer: M = f(n+1) (t) (n + 1)! In both parts below, estimate En(x) using Lagrange's formula with the constant M found in part (a). (Use technology as required.) EB Recall the standard decomposition f(x) = T₂(x) + En(x), in which Lagrange's formula says En(x) = Answer: n = (b) Find the smallest n for which the polynomial value T, (0.4) provides an approximation for f(0.4) that is guaranteed to be accurate to within 10 decimal places: -"+1 for some t between 0 and x. This is valid for every integer n ≥ 0. Answer: a = Hint: To guarantee D correct digits after the decimal point, accounting for rounding, one must have |E, (0.4)| ≤ 0.5 × 10-D (c) Suppose n = 7 is prescribed. Find the largest positive number a such that the approximation T7(x) for f(x) is guaranteed to be accurate to within 8 decimal places, for all a in the symmetric interval (-a, a).
Transcribed Image Text:All parts of this question concern the function f(x) = 6 sin x + 7 cosa. (a) Find the smallest positive constant M that satisfies M > f(*) (t) for every possible combination of an integer k ≥ 0 and an evaluation point t € (-∞, +∞). Hint: A standard trigonometric identity implies that, for a certain angle , one has f(x) = √85 sin (x + o) for all real x. Answer: M = f(n+1) (t) (n + 1)! In both parts below, estimate En(x) using Lagrange's formula with the constant M found in part (a). (Use technology as required.) EB Recall the standard decomposition f(x) = T₂(x) + En(x), in which Lagrange's formula says En(x) = Answer: n = (b) Find the smallest n for which the polynomial value T, (0.4) provides an approximation for f(0.4) that is guaranteed to be accurate to within 10 decimal places: -"+1 for some t between 0 and x. This is valid for every integer n ≥ 0. Answer: a = Hint: To guarantee D correct digits after the decimal point, accounting for rounding, one must have |E, (0.4)| ≤ 0.5 × 10-D (c) Suppose n = 7 is prescribed. Find the largest positive number a such that the approximation T7(x) for f(x) is guaranteed to be accurate to within 8 decimal places, for all a in the symmetric interval (-a, a).
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