Remainder Estimation Theorem

Question

Find the largest interval containing x=0 that the Remainder Estimation Theorem allows over which f(x)=sin(3x) can be approximated by p(x)=3x−[(3x)^3]/6 to three decimal-place accuracy throughout the interval.

Enter Interval in Interval Notation: ____

Answer 1

f (x) = sin (3x), p (x) -3x (3x)³

Using Taylor’s Remainder Estimation Theorem around 0 with n = 3:

| R₃ (x) | = | f⁴ (ξ) = x⁴ |=| 81 sin 3ξ x⁴ | ≤ 81⁄24 |x|⁴ .

4! 24

For three–decimal-place accuracy we need |R₃(x) | ≤ 0.0005, so

81⁄24 |x|⁴ ≤ 0.0005 ⇒ |x| ≤ ( 0.0005 • 24) ^1⁄4 ≈ 0.1103.

81

Largest symmetric interval about 0 (rounded safely to three decimals):

Enter Interval in Interval Notation: [-0.110, 0.110]