matematicas visuales visual math
Real Analysis

Geometric sequence | matematicasVisuales Sum of a geometric series of ratio 1/4 | matematicasVisuales Sum of a geometric series of ratio 1/2 | matematicasVisuales Convergence of Series: Integral test | matematicasVisuales Gamma, Euler's constant | matematicasVisuales Polynomial Functions (1): Linear functions | matematicasVisuales Powers with natural exponents (and positive rational exponents) | matematicasVisuales
Polynomial Functions (2): Quadratic functions | matematicasVisuales Polynomial Functions (3): Cubic functions | matematicasVisuales Polynomial Functions (4): Lagrange interpolating polynomial | matematicasVisuales Rational Functions (1): Linear rational functions | matematicasVisuales Rational Functions (2): degree 2 denominator | matematicasVisuales Rational Functions (3): Oblique Asymptote | matematicasVisuales Rational Functions (4): Asymptotic behavior | matematicasVisuales
Polynomial functions and derivative (1): Linear functions | matematicasVisuales Polynomial functions and derivative (2): Quadratic functions | matematicasVisuales Polynomial functions and derivative (3): Cubic functions | matematicasVisuales Polynomial functions and derivative (4): Lagrange polynomials (General polynomial functions) | matematicasVisuales Polynomial functions and derivative (5): Antidifferentiation | matematicasVisuales Definite integral | matematicasVisuales Indefinite integral | matematicasVisuales
Monotonic functions are integrable | matematicasVisuales Integral of powers with natural exponent | matematicasVisuales Archimedes' Method to calculate the area of a parabolic segment | matematicasVisuales Kepler: The best proportions for a wine barrel | matematicasVisuales Polynomial functions and integral (1): Linear functions | matematicasVisuales Polynomial functions and integral (2): Quadratic functions | matematicasVisuales Polynomial functions and integral (3): Lagrange polynomials (General polynomial functions) | matematicasVisuales
The Fundamental Theorem of Calculus (1) | matematicasVisuales The Fundamental Theorem of Calculus (2) | matematicasVisuales Piecewise Linear Functions. Only one piece | matematicasVisuales Piecewise Constant Functions | matematicasVisuales Continuous Piecewise Linear Functions | matematicasVisuales Non continuous Piecewise Linear Functions | matematicasVisuales Exponentials and Logarithms (1): Exponential Functions | matematicasVisuales
Exponentials and Logarithms (2): Logarithm definition as an integral | matematicasVisuales Exponentials and Logarithms (3): One property of the integral of the rectangular hyperbola | matematicasVisuales Exponentials and Logarithms (4): the logarithm of a product | matematicasVisuales Exponentials and Logarithms (5): Approximation of number e | matematicasVisuales Exponentials and Logarithms (6): Two definitions of number e | matematicasVisuales Exponentials and Logarithms (7): The exponential as the inverse of the logarithm | matematicasVisuales Exponentials and Logarithms (8): Hyperbolas, logarithms and exponencials | matematicasVisuales
Mercator and Euler: Logarithm Function | matematicasVisuales Exponentials and Logarithms (9): Radioactive decay (Spanish) | matematicasVisuales Taylor polynomials (1): Exponential function | matematicasVisuales Mercator and Euler: Logarithm Function | matematicasVisuales Taylor polynomials (2): Sine function | matematicasVisuales Taylor polynomials (3): Square root | matematicasVisuales Taylor polynomials (4): Rational function 1 | matematicasVisuales
Taylor polynomials (5): Rational function 2 | matematicasVisuales Taylor polynomials (6): Rational function with two real singularities | matematicasVisuales Taylor polynomials (7): Rational function without real singularities | matematicasVisuales


Sequences and Series
Geometric sequence | matematicasVisuales
Geometric sequences graphic representations. Sum of terms of a geometric sequence and geometric series.
Sum of a geometric series of ratio 1/4 | matematicasVisuales
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 | matematicasVisuales
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.
Convergence of Series: Integral test | matematicasVisuales
Using a decreasing positive function you can define series. The integral test is a tool to decide if a series converges o diverges. If a series converges, the integral test provide us lower and upper bounds.
Gamma, Euler's constant | matematicasVisuales
Gamma, the Euler's constant, is defined using a covergent series.

Powers and Polynomials
Polynomial Functions (1): Linear functions | matematicasVisuales
Two points determine a stright line. As a function we call it a linear function. We can see the slope of a line and how we can get the equation of a line through two points. We study also the x-intercept and the y-intercept of a linear equation.
Powers with natural exponents (and positive rational exponents) | matematicasVisuales
Power with natural exponents are simple and important functions. Their inverse functions are power with rational exponents (a radical or a nth root)
Polynomial Functions (2): Quadratic functions | matematicasVisuales
Polynomials of degree 2 are quadratic functions. Their graphs are parabolas. To find the x-intercepts we have to solve a quadratic equation. The vertex of a parabola is a maximum of minimum of the function.
Polynomial Functions (3): Cubic functions | matematicasVisuales
Polynomials of degree 3 are cubic functions. A real cubic function always crosses the x-axis at least once.
Polynomial Functions (4): Lagrange interpolating polynomial | matematicasVisuales
We can consider the polynomial 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.

Rational Functions
Rational Functions (1): Linear rational functions | matematicasVisuales
Rational functions can be writen as the quotient of two polynomials. Linear rational functions are the simplest of this kind of functions.
Rational Functions (2): degree 2 denominator | matematicasVisuales
When the denominator of a rational function has degree 2 the function can have two, one or none real singularities.
Rational Functions (3): Oblique Asymptote | matematicasVisuales
For large absolute values of x, some rational functions behave like an oblique straight line, we call this line an oblique or slant asymptote.
Rational Functions (4): Asymptotic behavior | matematicasVisuales
You can add a polynomial to a proper rational function. The end behavior of this rational function is very similar to the polynomial.

Polynomial functions and derivative
Polynomial functions and derivative (1): Linear functions | matematicasVisuales
The derivative of a lineal function is a constant function.
Polynomial functions and derivative (2): Quadratic functions | matematicasVisuales
The derivative of a quadratic function is a linear function, it is to say, a straight line.
Polynomial functions and derivative (3): Cubic functions | matematicasVisuales
The derivative of a cubic function is a quadratic function, a parabola.
Polynomial functions and derivative (4): Lagrange polynomials (General polynomial functions) | matematicasVisuales
Lagrange polynomials are polynomials that pases through n given points. We use Lagrange polynomials to explore a general polynomial function and its derivative.
Polynomial functions and derivative (5): Antidifferentiation | matematicasVisuales
If the derivative of F(x) is f(x), then we say that an indefinite integral of f(x) with respect to x is F(x). We also say that F is an antiderivative or a primitive function of f.

Integral
Definite integral | matematicasVisuales
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.
Indefinite integral | matematicasVisuales
If we consider the lower limit of integration a as fixed and if we can calculate the integral for different values of the upper limit of integration b then we can define a new function: an indefinite integral of f.
Monotonic functions are integrable | matematicasVisuales
Monotonic functions in a closed interval are integrable. In these cases we can bound the error we make when approximating the integral using rectangles.
Integral of powers with natural exponent | matematicasVisuales
The integral of power functions was know by Cavalieri from n=1 to n=9. Fermat was able to solve this problem using geometric progressions.
Archimedes' Method to calculate the area of a parabolic segment | matematicasVisuales
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 | matematicasVisuales
Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.

Polynomial functions and integral
Polynomial functions and integral (1): Linear functions | matematicasVisuales
It is easy to calculate the area under a straight line. This is the first example of integration that allows us to understand the idea and to introduce several basic concepts: integral as area, limits of integration, positive and negative areas.
Polynomial functions and integral (2): Quadratic functions | matematicasVisuales
To calculate the area under a parabola is more difficult than to calculate the area under a linear function. We show how to approximate this area using rectangles and that the integral function of a polynomial of degree 2 is a polynomial of degree 3.
Polynomial functions and integral (3): Lagrange polynomials (General polynomial functions) | matematicasVisuales
We can see some basic concepts about integration applied to a general polynomial function. Integral functions of polynomial functions are polynomial functions with one degree more than the original function.

The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus (1) | matematicasVisuales
The Fundamental Theorem of Calculus tell us that every continuous function has an antiderivative and shows how to construct one using the integral.
The Fundamental Theorem of Calculus (2) | matematicasVisuales
The Second Fundamental Theorem of Calculus is a powerful tool for evaluating definite integral (if we know an antiderivative of the function).

Piecewise Functions
Piecewise Linear Functions. Only one piece | matematicasVisuales
As an introduction to Piecewise Linear Functions we study linear functions restricted to an open interval: their graphs are like segments.
Piecewise Constant Functions | matematicasVisuales
A piecewise function is a function that is defined by several subfunctions. If each piece is a constant function then the piecewise function is called Piecewise constant function or Step function.
Continuous Piecewise Linear Functions | matematicasVisuales
A continuous piecewise linear function is defined by several segments or rays connected, without jumps between them.
Non continuous Piecewise Linear Functions | matematicasVisuales
Graphs of these functions are made of disconnected line segments. There are points where a small change in x produces a sudden jump in the value of the function.

Exponentials and Logarithms
Exponentials and Logarithms (1): Exponential Functions | matematicasVisuales
We can study several properties of exponential functions, their derivatives and an introduction to the number e.
Exponentials and Logarithms (2): Logarithm definition as an integral | matematicasVisuales
Using the integral of the equilateral hyperbola we can define a new function that is the natural logarithm function.
Exponentials and Logarithms (3): One property of the integral of the rectangular hyperbola | matematicasVisuales
The natural logaritm can be defined using the integral of the rectangular hiperbola. In this page we are going to see an important property of this integral. Using this property you can justify that the logarithm of a product is the sum of the logarithms.
Exponentials and Logarithms (4): the logarithm of a product | matematicasVisuales
The main property of a logarithm function is that the logarithm of a product is the sum of the logarithms of the individual factors.
Exponentials and Logarithms (5): Approximation of number e | matematicasVisuales
The logarithm of the number e is equal to 1. Using this definition of the number e we can approximate its value.
Exponentials and Logarithms (6): Two definitions of number e | matematicasVisuales
Constant e is the number whose natural logarithm is 1. It can be defined as a limit of a sequence related with the compound interest. Both definitions for e are equivalent.
Exponentials and Logarithms (7): The exponential as the inverse of the logarithm | matematicasVisuales
After the definition of the natural logarithm function as an integral you can define the exponential function as the inverse function of the logarithm.
Exponentials and Logarithms (8): Hyperbolas, logarithms and exponencials | matematicasVisuales
Different hyperbolas allow us to define different logarithms functions and their inversas, exponentials functions.
Mercator and Euler: Logarithm Function | matematicasVisuales
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
Exponentials and Logarithms (9): Radioactive decay (Spanish) | matematicasVisuales

Taylor's Polynomials
Taylor polynomials (1): Exponential function | matematicasVisuales
By increasing the degree, Taylor polynomial approximates the exponential function more and more.
Mercator and Euler: Logarithm Function | matematicasVisuales
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
Taylor polynomials (2): Sine function | matematicasVisuales
By increasing the degree, Taylor polynomial approximates the sine function more and more.
Taylor polynomials (3): Square root | matematicasVisuales
The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (4): Rational function 1 | matematicasVisuales
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (5): Rational function 2 | matematicasVisuales
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
Taylor polynomials (6): Rational function with two real singularities | matematicasVisuales
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 | matematicasVisuales
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.