Sequences and Series
|
 |
Geometric sequences graphic representations
|
 |
One intuitive example of how to sum a geometric series. A geometric series of ratio less than 1 is convergent.
|
 |
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, the Euler's constant, is defined using a covergent series.
|
Powers and Polynomials
|
 |
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.
|
 |
Power with natural exponents are simple and important functions. Their inverse functions are power with rational exponents (a radical or a nth root)
|
 |
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.
|
 |
Polynomials of degree 3 are cubic functions. A real cubic function always crosses the x-axis at least once.
|
 |
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.
|
Polynomial functions and derivative
|
 |
The derivative of a lineal function is a constant function.
|
 |
The derivative of a quadratic function is a linear function, it is to say, a straight line.
|
 |
The derivative of a cubic function is a quadratic function, a parabola.
|
 |
Lagrange polynomials are polynomials that pases through n given points. We use Lagrange polynomials to explore a general polynomial function and its derivative.
|
 |
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
|
 |
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.
|
 |
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 in a closed interval are integrable. In these cases we can bound the error we make when approximating the integral using rectangles.
|
 |
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 show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.
|
 |
Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.
|
Polynomial functions and integral
|
 |
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.
|
 |
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.
|
 |
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 tell us that every continuous function has an antiderivative and shows how to construct one using the integral.
|
 |
The Second Fundamental Theorem of Calculus is a powerful tool for evaluating definite integral (if we know an antiderivative of the function).
|
Taylor's Polynomials
|
 |
By increasing the degree, Taylor polynomial approximates the exponential function more and more.
|
 |
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
|
 |
By increasing the degree, Taylor polynomial approximates the sine function more and more.
|
 |
The function is not defined for values less than -1. Taylor polynomials about the origin approximates the function between -1 and 1.
|
 |
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
|
 |
The function has a singularity at -1. Taylor polynomials about the origin approximates the function between -1 and 1.
|
 |
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.
|
 |
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
|
 |
|
 |
|
 |
|
 |
|
 |
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
|
 |
|
 |
|
 |
|
 |
|
 |
|
 |
|