Univariate Functions §
For a univariate function , we can use its 🍧 Derivatives to represent it as a polynomial:
This is only an approximation of the actual function in the neighborhood around . With , we have the Taylor series.
Multivariate Functions §
With multivariate functions , we’ll use the ❄️ Gradient. The gradient provides a linear approximation around as
where is the gradient of evaluated at .
Generalizing to higher derivatives, for a function that is smooth at , the Taylor polynomial is
where is the th total derivative of evaluated at and . Similarly, the Taylor series is
Note that isn’t defined for vectors. This notation is instead a -fold outer product, . Thus,