Math Model

In our differential equation, we have approximated by . When is the magnitude of the error caused by this approximation less than, say, 10% of the magnitude of the actual function?

The actual function is , so 10% of that is . The absolute error between and is . Thus, the absolute error is less than 10% of the magnitude of the actual function when

For what values of is close to ? To help you, a graph shows in blue and in green.

plots for sin
	error

What if you wanted a 5% error? Estimate the valid range of theta for this case. We have addressed the issue of how close is to , but not the issue of how close the solution of the linearized pendulum equation is to the solution of the nonlinear pendulum equation. We will return to this more difficult and important issue later in the module.


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