matematicas visuales visual math
Real Analysis

Sequences and Series
Geometric sequence | matematicas visuales
Geometric sequences graphic representations
Sum of a geometric series of ratio 1/4 | matematicas visuales
One intuitive example of how to sum a geometric series. A geometric series of ratio less than 1 is convergent.
Sum of a geometric series of ratio 1/2 | matematicas visuales
The geometric series of ratio 1/2 is convergent. We can represent this series using a rectangle and cut it in half successively. Here we use a rectangle such us all rectangles are similar.
Gamma, Euler's constant | matematicas visuales
Gamma, the Euler's constant, is defined using a covergent series.

Powers and Polynomials
Powers with natural exponents | matematicas visuales
Power with natural exponents are simple and important functions.
Powers with positive rational exponents | matematicas visuales
We extend the definition of power of positive integer exponent and their inversas if we consider positive rational exponents.
Polynomial of degree 3 with 3 real roots | matematicas visuales
We can see how the graph of a polynomial of degree three with three real roots changes when we change a root.
Polynomial approximations | matematicas visuales
The powers of natural exponent are the base of the polynomials. We can consider the polynomic function that passes through a series of points of the plane. This is an interpolation problem that is solved here using the Lagrange interpolating polynomial.

Integral
Riemann integral | matematicas visuales
The integral concept is associate to the concept of area. We began considering the area limited by the graph of a function and the x-axis between two vertical lines.
The error approximating the integral | matematicas visuales
When approximating an integral using rectangles we commit an error. In some cases, for example, in monotonic functions, we can limit the magnitude of the error.
Integral of powers | matematicas visuales
Archimedes' Method to calculate the area of a parabolic segment | matematicas visuales
Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.
Kepler: The best proportions for a wine barrel | matematicas visuales
Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

Taylor's Polynomials
Taylor polynomials: Exponential function | matematicas visuales
By increasing the degree, Taylor polynomial approximates the exponential function more and more.
Mercator and Euler: Logarithm Function | matematicas visuales
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
Taylor polynomials: Sine function | matematicas visuales
By increasing the degree, Taylor polynomial approximates the sine function more and more.
Taylor polynomials: Square root | matematicas visuales
The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials: Rational function 1 | matematicas visuales
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials: Rational function 2 | matematicas visuales
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials: Rational function with two real singularities | matematicas visuales
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: Rational function without real singularities | matematicas visuales
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.

Exponentials and Logarithms
Una propiedad de la integral de la hipérbola (Spanish) | matematicas visuales
Una propiedad de la integral de la hipérbola | 'Mostración'(Spanish) | matematicas visuales
El logaritmo de un producto (Spanish) | matematicas visuales
Definición de logaritmo como una integral (Spanish) | matematicas visuales
Mercator and Euler: Logarithm Function | matematicas visuales
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
Approximation of number e (Spanish) | matematicas visuales
Two definitions of number e (Spanish) | matematicas visuales
The exponential as the inverse of the logarithm (Spanish) | matematicas visuales
Hipérbolas, logaritmos y exponenciales (Spanish) | matematicas visuales
Exponencial functions (Spanish) | matematicas visuales
Radioactiv desintegration (Spanish) | matematicas visuales