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What Do Functions Do for Tiny X?

Looking at small values of functions. Motivating the existence of the Taylor expansion.

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Work in groups to try to create the best approximations to these physical quantities.

Building Approximations for Sin(x)

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

In general, any function whose derivatives of all orders are finite can be written as a power series

$$f(x) = f(0) + x f'(0) + \frac{x^2}{2!}f''(0) + \dots + \frac{x^n}{n!}f^{(n)}(0)+\dots$$
This expansion is an infinite series (not a polynomial). Truncating this series at a given point provides us with a polynomial approximation to f(x).

The question of how big the errors are in this approximation is a difficult one to answer, and more details will be discovered at university in Numerical Analysis courses.