Taylor polynomial

The -th order Taylor polynomial is the first terms of a Taylor series, such as linear approximation and quadratic approximation

It is an approximation of near

Linear Taylor Polynomial

The linear approximation (first-order Taylor polynomial) of around is:

This represents the tangent plane at the point .

Second-order Taylor Polynomial

quadratic approximation

The quadratic approximation to a univariate function at the point is given by:

It has the same function value, first derivative and second derivative with at :

For multivariate function, the quadratic approximation (second-order Taylor polynomial) of around is:

where

• is the gradient of the function
• is the Hessian matrix

This includes both the linear and quadratic terms and provides a more accurate approximation by considering the curvature of near .

Note that the quadratic term (order two) of this quadratic approximation will be quadratic form

Example

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