One variable functions

Derivative

Consider a 1 variable function
The derivative of this function can be understood as several things:

The slope of a tangent line at point .
The growth rate of the function at .
The velocity of the function at .
All of these are different ways of expressing the same thing.

The tangent of a function

The tangent line of a function at a point can be calculated as:

Taylor polynomial

The taylor polynomial is a series of sums of the derivatives of a function that can be used to approximate this function.
Some examples of taylor polynomial of degrees 1, 2 and 3 of centered at point are:
This polynomial will progressively approximate the function as the degree of the taylor polynomial grows.

Taylors polynomial: generic form

You can express the taylor polynomial of degree centered at a as:

Example: approximating the sine function near 0

First derivative:
Second derivative:
Third derivative:
Fourth derivative:

Now, let's rewrite the Taylor polynomials for the sine function centered at

3rd-degree Taylor approximation of the sine function centered at :

Simplifying this, we get:

4th-degree Taylor approximation of the sine function centered at :

Simplifying this, we get:

Lagrange error bound

being the value between (the sampling point) and (the point the polynomial is centered around)
#todo you basically make choices that maximize the term aka choose the c value that makes the error bigger and replace all sine and cosines with 1 etc.

Convergence radius

Given a series in the form of where is a general term of a sequence, is a value which is the convergence center around which we'll place the radius and is a variable that changes in value changing whether the series converges or not.
We can calculate the convergence radius with either:

Attention

When calculating the radius and

Cases based on radius values

When the series only converges at
When the series converges in the area
The cases where or need to be studied separately to see if they converge or not
When the series converges at