matematicas visuales visual math

After viewing lineal, quadratic and cubic functions, we can play here with polynomials of higher degrees.

One interesting method of generating polynomial functions is using the so called Lagrange interpolating polynomials.

Polynomial Functions (4): Lagrange interpolating polynomial
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.

Our purpose here is to have a better understanding of the behavior of different polynomial functions and their derivatives.

THE CONCEPT OF DERIVATIVE OF A FUNCTION

The derivative of a function at a point can be defined as the instantaneous rate of change or as the slope of the tangent line to the graph of the function at this point. We can say that this slope of the tangent of a function at a point is the slope of the function.

The slope of a function will, in general, depend on x. Then, starting from a function we can get a new function, the derivative function of the original function.

The process of finding the derivative of a function is called differentiation.

The value of the derivative function for any value x is the slope of the original function at x.

We can review here several concepts that we have already studied. For example, the geometric interpretation of the derivative as the slope of the tangent:

Polynomials and derivative. Lagrange polynomials: geometric interpretation of derivative as the slope of the tangent line | matematicasVisuales

But how can we draw a tangent line?. We can use a magnifying glass!. If we look very near the point in the graph of the function we can see how the function resembles the tangent line. This tangent line is the best linear approximation of the function at that point:

Polynomials and derivative. Lagrange polynomials: the parabola resembles the tangent line when we look it very near | matematicasVisuales

There is a connection between the derivative and the increasing or decreasing of a (polynomial) function. If the derivative is positive in some interval then the function increases in this interval and if the derivative in negative in some interval then the function decreases in this interval:

Polynomials and derivative. Lagrange polynomials: increasing and decreasing functions and derivative | matematicasVisuales

A stationary point is a point where the derivative vanishes (the slope of the tangent is zero, the tangent is an horizontal line). In these points the function stops increasing or decreasing.

At a stationary point the function can change from increasing to decreasing. Then this stationary point is a local maximum (or relative minimum). The derivative changes from positive to negative.

Polynomials and derivative. Lagrange polynomials: stationary point, a local maximum | matematicasVisuales

In other cases, at a stationary point the function can change from decreasing to increasing. Then this stationary point is a local minimum (or relative minimum). The derivative changes from negative to positive.

Some times, stationary points are not maxima nor minima. They can be inflection points, points at which the sign of the curvature changes.

Polynomials and derivative. Lagrange polynomials: stationary point, an inflection point | matematicasVisuales

Inflection points are related with local maxima or minima of the derivative function. At these points the tangent crosses the curve.

Polynomials and derivative. Lagrange polynomials: inflection point and derivative extremum | matematicasVisuales

One simple and interesting idea is that when we translate up and down the graph of a function (we add or subtract a number from the original function) the derivative does not change. The reason is very intuitive and we can play with the following version of the mathlet to see this property. When you move the violet dot you are translating up and down the graph and the derivative is the same:

It is important to notice that the derivative of a polynomial of degree 1 is a constant function (a polynomial of degree 0). The derivative of a polinomial of degree 2 is a polynomial of degree 1. And the derivative of a polynomial of degree 3 is a polynomial of degree 2.

When we derive such a polynomial function the result is a polynomial that has a degree 1 less than the original function.

When we study the integral of a polynomial of degree 2 we can see that in this case the new function is a polynomial of degree 3. One degree more than the original function. And the integral of a polynomial function is a polynomial function of degree 1 more than the original function.

These results are related to the Fundamental Theorem of Calculus.

REFERENCES

Michael Spivak, Calculus, Third Edition, Publish-or-Perish, Inc.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.

MORE LINKS

Polynomial Functions (4): Lagrange interpolating polynomial
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 (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 (1): Linear functions
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)
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
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
Polynomials of degree 3 are cubic functions. A real cubic function always crosses the x-axis at least once.
Piecewise Linear Functions. Only one piece
As an introduction to Piecewise Linear Functions we study linear functions restricted to an open interval: their graphs are like segments.
Piecewise Constant Functions
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
A continuous piecewise linear function is defined by several segments or rays connected, without jumps between them.
Polynomial Functions (4): Lagrange interpolating polynomial
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.
Definite 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.
Monotonic functions are integrable
Monotonic functions in a closed interval are integrable. In these cases we can bound the error we make when approximating the integral using rectangles.
Indefinite integral
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.
Polynomial functions and integral (1): Linear functions
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
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)
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 (1)
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)
The Second Fundamental Theorem of Calculus is a powerful tool for evaluating definite integral (if we know an antiderivative of the function).
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.
Taylor polynomials (3): Square root
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
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
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
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.
Complex Polynomial Functions(2): Polynomial of degree 2
A polynomial of degree 2 has two zeros or roots. In this representation you can see Cassini ovals and a lemniscate.
Complex Polynomial Functions(3): Polynomial of degree 3
A complex polinomial of degree 3 has three roots or zeros.
Complex Polynomial Functions(4): Polynomial of degree n
Every complex polynomial of degree n has n zeros or roots.