All parts of this question concern the function f(x) = 7 sin x + 3 cos x. (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 te (-∞0, +∞0). Hint: A standard trigonometric identity implies that, for a certain angle , one has f(x) = √58 sin (x + p) for all real x. Answer: M = sqrt(58) B Answer: n = 10 Recall the standard decomposition f(x) = T₁(x) + E₂(x), in which Lagrange's formula says E,(™) 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.) f(n+1) (t) (n + 1)! = Answer: a = -x+1 for some t between 0 and x. This is (b) Find the smallest n for which the polynomial value T, (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 |E, (0.3)| ≤ 0.5 × 10-D. (c) Suppose n = 6 is prescribed. Find the largest positive number a such that the approximation T6(x) for f(x) is guaranteed to be accurate to within 5 decimal places, for all in the symmetric interval (-a, a).
All parts of this question concern the function f(x) = 7 sin x + 3 cos x. (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 te (-∞0, +∞0). Hint: A standard trigonometric identity implies that, for a certain angle , one has f(x) = √58 sin (x + p) for all real x. Answer: M = sqrt(58) B Answer: n = 10 Recall the standard decomposition f(x) = T₁(x) + E₂(x), in which Lagrange's formula says E,(™) 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.) f(n+1) (t) (n + 1)! = Answer: a = -x+1 for some t between 0 and x. This is (b) Find the smallest n for which the polynomial value T, (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 |E, (0.3)| ≤ 0.5 × 10-D. (c) Suppose n = 6 is prescribed. Find the largest positive number a such that the approximation T6(x) for f(x) is guaranteed to be accurate to within 5 decimal places, for all in the symmetric interval (-a, a).
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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Just need part c), please show all work.
Transcribed Image Text:All parts of this question concern the function f(x) = 7 sin x + 3 cos x.
(a) Find the smallest positive constant M that satisfies M ≥ |ƒ(*) (t)| for every possible combination of an integer k > 0 and an evaluation point
te (-∞0, +∞0).
Hint: A standard trigonometric identity implies that, for a certain angle , one has f(x) = √58 sin (x + ) for all real x.
Answer: M = sqrt(58)
=
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.)
f(n+1) (t),
(n + 1)!
Answer: n = 10
(b) Find the smallest n for which the polynomial value T, (0.3) provides an approximation for f(0.3) that is guaranteed to be accurate to within 11 decimal
places:
-x+¹ for some t between 0 and x. This is
Answer: a =
Hint: To guarantee D correct digits after the decimal point, accounting for rounding, one must have |E₂(0.3)| ≤ 0.5 × 10-⁰.
(c) Suppose n = 6 is prescribed. Find the largest positive number a such that the approximation T6(x) for f(x) is guaranteed to be accurate to within 5
decimal places, for all in the symmetric interval (-a, a).
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