What is the Taylor theorem used for?

Taylor series

A function that is infinitely differentiable forms a Taylor series. Taylor series are used to approximate the value of a function at a point. For example, most pocket calculators use Taylor series to calculate the sine and other trigonometric functions, since an accurate calculation would be too computationally intensive.

The Taylor series is in principle a tool in mathematics with which one can turn complicated functions into simpler ones.

definition

A function f (x) corresponds to a Taylor series with an infinite number of terms. The position a is the Development office. The development point is the point in whose environment we are interested in the behavior of the function.

Each term in the Taylor series corresponds to a Taylor polynomial. A Taylor series with n terms is also called a Taylor series nth degree. The higher the degree of a Taylor series, the more precisely it agrees with the output function.

A Taylor series at the point a = 0 is also sometimes used MacLaurin series called.

n! is the factorial of n.

Approximate the sine with the Taylor series

Many calculators use Taylor series internally to calculate trigonometric functions such as sine. In the figure on the right we see the sine function as well as the Taylor series of degrees 3, 5 and 7. To write the sine function as a Taylor series, let's first look at the derivation of the sine:

As we can see, after four derivatives we get the sine function again - our output function. The sine function can therefore be differentiated as often as desired and its derivatives are repeated after four runs.
We want to calculate the sine at 0. We know that a Taylor series at position 0 is structured according to the following scheme:

Now we insert the derivative of the sine function:

We also know from the sine and cosine functions that sin (0) is always 0 and cos (0) is always 1. This simplifies our series very much:

After simplifying the series, we get:

The sine function can therefore also be expressed by the following Taylor series:

Taylor series and tangent equation

Often in high school, the tangent equation of a function should be calculated at one point. A first degree Taylor series corresponds to the tangent equation. One also speaks of the "linearization of the function f at the point a".

This is particularly interesting because older CAS calculators often do not have an independent function to determine the tangent equation, but mostly a Taylor function.