Taylor Series Approximation of a Function

Under certain conditions we may approximate an analytic function about a specified point on the function by an infinite series. The most useful series for our purposes is the Taylor series.

The Taylor series of a function about the point is given as

The equation above allows us to approximate the value of the function as an infinite series for any point sufficiently close to while only knowing the value of the function and its’ derivatives at . As an example, consider the equation. This is an inverted “W”-shaped function with roots at 0,. We are interested in a region centered about , so we may begin by evaluating the above equation with and for increasing values of .

For ,

For ,

For ,

etc.

For our 4th order function, any values of will result in the derivative being equal to . Therefore, we have a finite number of terms in the full Taylor series expansion up to a maximum of . The following graph shows the original function and the Taylor series approximations for . In this case, the series obtained when is algebraically equivalent to our original function.

graph

The Taylor series may also be defined for multivariate functions. This allows us to expand the usefulness of the series to functions of multiple variables, and as we shall see later, allow us to predict the function values for small deviations about a nominal point, as well as ascribe sensitivity of the function to the independent variables.

The Taylor series of a function about the point is given as

We’ll use this later as we begin to discuss error propagation.

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