A simple pendulum.An idealized simple pendulum is given by a mass m hanging from

Question

A simple pendulum.An idealized simple pendulum is given by a mass m hanging from a massless string of length L. Its motion is described by the angle ?(t), where t is time. The distance travelled by the pendulum is the arclengths (t) =L?(t). According to Newton A simple pendulum.An idealized simple pendulum is given by a mass m hanging from a massless string of length L. Its motion is described by the angle ?(t), where t is time. The distance travelled by the pendulum is the arclengths (t) =L?(t). According to Newton

Explanation / Answer

At x = 0, sin(0) = 0, therefore the error at this point is 0.
Let the error function be f(x) = |sin(x) - x|
Since the taylor approximation for sin(x) is x - x^3 / 3! + x^5/5!, sin(x) is smaller than x. Therefore,
f(x) = -sin(x) + x if x>0
f(x) = sin(x) - x if x<0
The derivative of f(x) is:
f'(x) = -cos(x) + 1 if x>0
f'(x) = cos(x) - 1 if x<0

The error increases faster as x moves from 0 to pi/10. Therefore the error is the biggest when x = pi/10 or -pi/10.

sin (-pi/10) = -0.309 and -pi/10 = -0.314
Error = |-0.309 - (-0.314)| = 0.005
sin (pi/10) = 0.309 and pi/10 = 0.314
Error = 0.005

The upper bound for the error = 0.005 where ?pi/10 ? x ? pi/10

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