NUMERICAL-ANALYSIS Course

Taylor: the origin

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It is based on the Taylor's theorem:

\[ y(x_0 + h) \approx y(x_0) + h\times y\prime (x_0) \]

That we could rewrite

\[ y(x_0) \approx y(x_0) + (x - x_0) \times y\prime (x_0) \]

with $h = x_0 - a$.

The approximation is pretty good when $h$ is really small ( $|h| << 1$ ).

For example, $\sqrt{1,44} = \sqrt{1 + 0,44} \approx \sqrt 1 + \frac{1}{2\sqrt 1} \times 0,44 \approx 1,22$ which is an approximation of $\sqrt{1,44} = 1,2$ .