Not only is it necessary for us to be able to start in the study of algebra, geometry, trigonometry and analytical geometry, but also all the specific properties of real numbers, as well as some basic notations to represent them as a whole.
Graphical visualization plays an important role in the study of calculus. The question we can ask ourselves in the form of understanding and analysis is:
How can we express these graphics?
There are two ways: manually, using pencil and square millimeter on a sheet of notebook, and the other way is mediating automatic graphing devices such as calculators and computers that have the appropriate software. My recommendation is that we first master the manual graphing, since through this process we will understand the use of the equations, the corresponding clearances and the use and management of the two-dimensional and three-dimensional system.
To explain the use of some software in particular, I have chosen the 6.0 geogebra, by which I have graphically represented the intersection between the line and the parabola . The image shows a clear example of one of the applications of the integral, which is area calculation between the intersection between two curves, the image was made in 6.0 geogebra and edited in Microsoft PowerPoint 2010:
With the basic knowledge preliminary to the calculation we can be prepared to start any topic of calculation, taking into account that the calculation is the foundation for many of the mathematical branches and for most of the knowledge of the modern world and today.
Aspects considered for the study of mathematics prior to calculation
The ability to solve equations and inequalities is crucial for understanding the calculation; it is because of this that it is necessary to study the aspects related to the set of real numbers.
The real number system consists of a set R of elements called real numbers and two operations called addition and multiplication. The real numerical system can be described completely by means of a set of axioms, being understood by axiom to the word used to indicate a formal proposition that is considered true without demonstration. Following the axioms are theorems that are propositions that often cite the axioms and that require a formal demonstration.
The importance of starting the study of the precalculus with the set of the real numbers lies in the fact that here the whole base is born, the fundamental operations of the set of the whole numbers (Z) together with the operations of rational numbers (Q) are fundamental pillar before falling into the study of equations and inequalities, in many cases we as teachers make the mistake of starting directly with some subjects of geometry, algebra or trigonometry without first starting with the set of real numbers and the most fundamental operations.
The creation of analytical geometry is attributed to René Descartes (1596-1650), a French mathematician and philosopher. In his book Geometry, published in 1637, Descartes established the union of algebra and geometry through a system of rectangular Cartesian coordinates (named in his honor). This coordinate system uses ordered pairs of real numbers.
Any two real numbers form a pair, and when the order in which the numbers appear is significant, it is called an ordered pair. If X is the first real number and Y is the second, this ordered pair is denoted by (X, Y).
The set of all the ordered pairs of real numbers is called the number plane, which is denoted by , and each pair ordered (X, Y) is a point on the number plane. The Cartesian system seeks to express a two-dimensional system, for which two lines are chosen in the geometric plane, a horizontal called the x-axis, and the other vertical called the y-axis. The point of intersection of the axes x and y receives the name of origin, and is denoted by O. It is established that the positive sense of the x-axis is to the right of the origin, and the positive sense of the y-axis is to the top of the origin.
The importance of the Cartesian plane is that we can express algebraic quantities, from which we can superimpose and obtain geometric figures, such as straight lines, parabolas, triangles, rectangles, hyperbolas, circles. All this importance that we have today is what Descartes was looking for in antiquity, that is to say, to be able to unite algebra and geometry in a whole, and for that I create this ideal system, which in turn has been so useful and beneficial. to the field of engineering and science, are sciences such as physics where we can see what has been able to nourish this important system, in turn the Cartesian system succeeded in sowing the algebraic, geometric and trigonometric bases for what today we know as infinitesimal calculus.
Definition of the equation of a graph
An equation of a graph is an equation that is satisfied by the coordinates of those, and only those points of the graph.
From this definition, it follows that an equation of a graph has the following properties:
If a point P (X, Y) is on the graph, then its coordinates satisfy the equation.
If a point P (X, Y) is not in the graph, then its coordinates do not satisfy the equation.
Under this concept of defining the equation of a graph that appears on page 1161, from Larson's 7th edition calculation book, we can conclude that if we have an equation, we can derive a series of coordinate points from them using p. (X, Y) that result in the consequence of a graph, this means that any point that is on the graph, that is to say that it belongs to the graph satisfies the equation, and that any ordered point or pair ( X, Y) that does not belong to the graph does not satisfy the equation.
In order to understand this statement I will carry out a practical exercise, for this we have the following:
Statement: Graph in the Cartesian plane the following equation: y = 2x-4
- If we equal X = 0, we would already have the first point that we would graph in the Cartesian system of the ordered pair (X, Y), it would be (0, Y ), we would need to calculate y, for this we say that:
If X = 0 → y =2(0) -4 → y = 0-4 → y =-4.
Therefore the first ordered pair (X, Y) would be equal to (0, -4).
- If we match Y = 0 we would already have the second point that we would graph in the Cartesian system of the ordered pair (X, Y), it would be (X, 0) , we would need to calculate X, for this we say that:
Yes Y = 0 → 2x-4 = 0 →2x = 4 → x = → x = 2.
Therefore the second ordered pair (X, Y) would be equal to (2,0).
If we analyze the axiom of Euclidean geometry that says that a single and single line passes through two points, which means that having the points P1 (0, -4) and P2 (2.0), the coordinates expressed for P1 are : x = 0 and y = -4, for P2 they are: x = 2 and y = 0.
To graph the straight line, use the software geogebra 6.0 compatible with Windows 7 and in turn perform the edition of the graphic with the image editor in Microsoft PowerPoint 2010:
Analysis of the importance of the study of elementary mathematics prior to calculation
As a teacher of calculation in the Food Engineering training program of the Universidad Experimental Sur del Lago (UNESUR), I have noticed that when students start in the first semester of the program, and come across the curricular unit of calculus I a clash of ways to perceive and understand these new learnings, since their habit is to learn superficially without going deeply into the new topics that are taught, these new topics are only learned if between the students and the teacher there is a willingness to teach and learn on the basic one that the calculation is based, and this is not more than the subjects on algebra, geometry, trigonometry among other points more. That is why in this publication he took the example of using the equation of a line, finding his points of cuts with the Cartesian axes, and graphing using free software such as Geogebra 6.0, all with the intention of encouraging teachers and students to start with a basic study of mathematics prior to the calculation, to then be able to address as more security issues related to the calculation as limit and continuity, derived from real variable functions and definite and indefinite integrals.
Calculation book with analytical geometry. Author: Louis Leithold. Editorial Oxford. 7th edition. Mexico 1998.
Bibliography recommended for reading and understanding of the calculation
Book of calculation and analytical geometry. Author: Larson and Hostetler. Volume I.