The rational function

is, as in the example of the square root, a special case of Newton's Binomial Theorem. We can calculate its Taylor's series at x = 0 easily.

This function has a singularity at the point x =- 1. The approximation is good in ranges from -1 to +1. Again we find a useful approach that is centered on the origin.

At the point x = +1 their ordinates are alternately equal to 1 and 0, while that of the original curve has the value 1/2.

We can compare this behavior with that of the rational function 2.

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Taylor polynomials (6): Rational function with two real singularities

This function has two real singularities at -1 and 1. Taylor polynomials approximate the function in an interval centered at the center of the series. Its radius is the distance to the nearest singularity.

Taylor polynomials (7): Rational function without real singularities

This is a continuos function and has no real singularities. However, the Taylor series approximates the function only in an interval. To understand this behavior we should consider a complex function.

Mercator and Euler: Logarithm Function

Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.

Taylor polynomials: Rational function with two complex singularities

We will see how Taylor polynomials approximate the function inside its circle of convergence.

Taylor polynomials (1): Exponential function

By increasing the degree, Taylor polynomial approximates the exponential function more and more.

Taylor polynomials (2): Sine function

By increasing the degree, Taylor polynomial approximates the sine function more and more.

Polynomial functions and derivative (1): Linear functions

The derivative of a lineal function is a constant function.

Polynomial functions and derivative (2): Quadratic functions

The derivative of a quadratic function is a linear function, it is to say, a straight line.

Polynomial functions and derivative (3): Cubic functions

The derivative of a cubic function is a quadratic function, a parabola.

Polynomial functions and derivative (4): Lagrange polynomials (General polynomial functions)

Lagrange polynomials are polynomials that pases through n given points. We use Lagrange polynomials to explore a general polynomial function and its derivative.