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EXAMPLE 2 (a) What is the maximum error possible in using the approximation below when -0.2 < x < 0.2? Use this approximation to find sin(10°) correct to six decimal places. sin(x) = x -_+. 3! 5! (b) For what values of x is this approximation accurate to 5e-05?

EXAMPLE 2 (a) What is the maximum error possible in using the approximation below when -0.2 < x < 0.2? Use this approximation to find sin(10°) correct to six decimal places. sin(x) = x -_+. 3! 5! (b) For what values of x is this approximation accurate to 5e-05?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
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EXAMPLE 2 approximation below when -0.2 < x < 0.2? Use this approximation to find sin(10°) correct to six decimal places. (a) What is the maximum error possible in using the sin(x) = x -. + 3! 5! (b) For what values of x is this approximation accurate to 5e-05? SOLUTION (a) Notice that the Maclaurin series sin(x) = x · 3! 5! 7! is alternating for all nonzero values of x, and the successive terms decrease size because | x | < 1, so we can use the Alternating Series Estimation Theorem. The error in approximating sin(x) by the first three terms of its Maclaurin series is at most | x 17 7! If -0.2 < x < 0.2, then | x | < 0.2, so the error is smaller than (0.2)7 * 0.000000C V 5040 To find sin(10°) we first convert to radian measure. 10n sin(10°) = sin(_) = sin( 18 180 1 1 + ( 18 18 3! 18 5! Thus, correct to six decimal places, sin(10°) - 0.1736481 (b) The error will be smaller than 5e-05 if | x |7 < 5e-05 5040 Solving this inequality for x, we get | x |7 < 0.252 | x | < (0.252) - 0.8212696 V So the given approximation is accurate to within 5e-05 when | x | < 0.8212696
Transcribed Image Text:EXAMPLE 2 approximation below when -0.2 < x < 0.2? Use this approximation to find sin(10°) correct to six decimal places. (a) What is the maximum error possible in using the sin(x) = x -. + 3! 5! (b) For what values of x is this approximation accurate to 5e-05? SOLUTION (a) Notice that the Maclaurin series sin(x) = x · 3! 5! 7! is alternating for all nonzero values of x, and the successive terms decrease size because | x | < 1, so we can use the Alternating Series Estimation Theorem. The error in approximating sin(x) by the first three terms of its Maclaurin series is at most | x 17 7! If -0.2 < x < 0.2, then | x | < 0.2, so the error is smaller than (0.2)7 * 0.000000C V 5040 To find sin(10°) we first convert to radian measure. 10n sin(10°) = sin(_) = sin( 18 180 1 1 + ( 18 18 3! 18 5! Thus, correct to six decimal places, sin(10°) - 0.1736481 (b) The error will be smaller than 5e-05 if | x |7 < 5e-05 5040 Solving this inequality for x, we get | x |7 < 0.252 | x | < (0.252) - 0.8212696 V So the given approximation is accurate to within 5e-05 when | x | < 0.8212696
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