This fourth mathematics lesson is devoted to the examination of some functions.
We will first see what is meant by the term "function". This is the subject of the first paragraph.
This notion acquired, we will make a reminder on the coordinates of a point. Indeed, the functions being graphically represented, it is necessary to draw them, to know how to situate in a plane some of their particular points. After reviewing what arithmetic is called directly (or inversely) proportional quantities, we will discuss in the other paragraphs the examination of the functions themselves.
As we have already said, our mathematics lessons do not constitute a mathematics course as such, but contain the knowledge necessary for a good understanding of the other theoretical lessons. This is why the functions studied are those most often related to certain electrical, electrotechnical or electronic laws.
We have endeavored to give an example of application, either in the theory itself or in complementary examples. As always, we invite you to redo each example.
1. - CONCEPT OF FUNCTION
Often the law of dependence between two sizes can be specified. Thus, when a pedestrian walks at a constant speed of 6 km / h, the distance traveled depends on the time during which he walked. If we represent the speed by the symbol "v", the distance traveled by "d", the time of the journey by "t", we can write :
d = v . t
Knowing that v = 6 and knowing the time "t", we can calculate the distance "d". It is said that distance is a function of time.
We know that at the terminals of a given resistance R, the voltage U existing at its terminals depends on the intensity I of the current flowing through this resistance. We know the relationship between these three magnitudes :
U = RI
We will say that the voltage U is, for a given resistance R, as a function of the intensity I of the current. By definition, we will :
A quantity is a known function of another quantity when, knowing a value of the second, we can calculate the corresponding value of the first.
Numeric example applied to the relation:
U = RI
or
R = 10 ohms
If
I = 1 ampère
U = 10 . 1 = 10 Volts
If
I = 2 ampères
U = 10 . 2 = 20 Volts
If
I = 3 ampères
U = 10 . 3 = 30 Volts
Generalize our formula of Ohm's law in a mathematical form.
As we had agreed that the value of the resistance was constant, we replace the letter "R" with "a". As for the other two quantities U and I, replace them by y and x respectively. The relation U = RI can then be written :
y = ax
Thus, each value of x corresponds to a value of y that we know how to calculate. It is said that the number y is a function of the number x.
When a number is a function of another one can, knowing one, calculate the other.
The number x, to which we give the values we want, is called the variable. The number y, which is calculated, is called the function.
Mathematically, to express that a quantity (y) is a function of a variable x, we write :
y = f (x)
which reads : y equals f of x where y is a function of x.
Let's draw two perpendicular oriented axes (also called rectangular) x'Ox and y'Oy (Figure 1).
2. 1. - REMINDER OF DEFINITIONS
The axis x'x is called the axis of the abscisses or axis of x ; the axis y'y is called the order axis or axis of y.
The x-axis and the y-axis form the coordinate axes. The point 0 is the origin of the coordinates.
2. 2. - POSITIONING A POINT IN RELATION TO ITS COORDINATES
To a pair of any two algebraic numbers, example x = - 4 and y = 5, corresponds to a point A of the plane that we obtain in the following way :
x being equal to -4, count on x'x, starting from 0 and in the negative direction, 4 graduations ;
y being 5, count on y'y, starting from 0 and in the positive direction, 5 graduations ;
by the two marked markings, lead the parallels to the axes ; they intersect at point A.
We say that the numbers - 4 and + 5 are the coordinates of point A ; - 4 is his abscissa, + 5 is his ordinate.
2. 3. - DEFINITION OF COORDINATES OF A POINT LOCATED IN THE PLAN
Let a point B (Figure 1) lie in the plane.
By this point, even the parallels to the axes of the coordinates. The intersection of these parallels with the different axes determines the coordinates of the point.
Thus, Figure 1, the point B has for coordinates x = 6 and y = - 3, which is written : B (6, - 3).
2. 4.- SCALE OF GRADUATIONS
To simplify, we have carried divisions having the same length on both axes. However, it is possible to focus on one of the axes of the divisions of different length from that carried on the other, that is to say to adopt different scales for the two graduations.
For example, let us take again the representation of the law of OHM that we saw in the previous lesson.
Let the circuit of Figure 2-a. A generator delivers in a fixed resistance of 100 ohms. If the voltage supplied by the generator successively takes the values of 100, 200, 300 ... volts, the corresponding current will have the value 1, 2, 3 ... amperes (application of the relation I = U / R).
Ampere and volt are two corresponding units. If one adopts for scale 1 cm = 1 ampere, it would be unreasonable to take 1 cm = 1 volt because that would impose a graph of several meters (100 V = 100 cm = 1 m, 200 V = 200 cm = 2 m, etc.). Consequently, to graphically represent this relation, we will adopt as scale 1 cm = 100 volts and 1 cm = 1 ampere. Moreover, on the same axis, one can find two different scales.
Let's take the example where we have to measure the intensity of the current passing through a receiver R as a function of the value and the polarities of the voltage applied to its terminals (the current is the function, the voltage is the variable). Either the assembly of Figure 3-a.
Suppose that the measurements of the voltage U give values of a few volts and that the corresponding values of the current I1 of the milliampere order are recorded.
We give the graph Figure 3-b by plotting the voltages on the x-axis positive (1 cm = 1 V) and the intensities on the positive y-axis (1 cm = 1 mA).
Modify the assembly as shown in Figure 3-c by reversing the polarities of the generator and the significant increase in the value of the resistance that will be called R'.
As before, voltages of a few volts are always measured but the polarities are reversed. If, for example, it is appropriate to take the point B as a reference potential, the point A has become negative with respect to this point B. We therefore have negative voltages and their values are to be plotted on the negative x axis with the same scale, 1 cm = 1 volt.
The corresponding intensities of current I2 are measured, which are this time a few microamperes and a direction opposite to that of I1. We therefore report these values on the negative y-axis and adopt 1 cm scale = 1 µA, that is to say a scale 1 000 times larger than previously. We have :
depending on the values to be measured, choose scales ;
according to the sign of the quantities considered, the corresponding quadrants are used, in this case the first and the third (Figure 3-e).
3. - GRAPHIC
REPRESENTATION OF DIRECTLY PROPORTIONAL SIZES
Definition:
Two quantities are directly proportional when the various values of one are proportional to the corresponding values of the other.
We have seen for example that the distance traveled by a pedestrian walking at a constant speed depended on the time during which he walked : the distance traveled is proportional to the walking time. Likewise, the intensity of the current flowing through a determined resistance receiver depends on the voltage applied across this receiver. Let's do a digital application of this last example.
Either a 10 ohms resistor receiver across which a voltage is applied which successively takes the values of 10 V, 20 V, 30 V, etc ... the intensity therethrough will respectively take the values of 1 A, 2 A, 3 A, etc ... since I = U / R.
The voltage and the intensity are therefore two directly proportional quantities because, when the values of the voltage are multiplied by 2, 3, 4, etc., the intensity values are also multiplied by 2, 3, 4, etc. ...
Note that the quotient of two corresponding values of the quantities considered is constant. We have indeed :
(10 / 1) = (20 / 2) = (30 / 3) etc... = 10
This quotient (10 in our example) is called the coefficient of proportionality. More generally, "y" and "x" being the corresponding measures of two proportional quantities, "a" the coefficient of proportionality, these measures are related by the relation :
y / x = a
where we draw :
y = ax
In other words, two quantities are directly proportional when the measure of one "y" is obtained by multiplying the corresponding measure of the other "x" by a constant number, called coefficient of proportionality.
Let's take our 10 ohms resistance receiver and let it run through a current to which we will give arbitrary values. Let's calculate the different corresponding values of the voltage that we will graphically represent.
The voltage across the receiver is given by the relation :
U = RI which is of the form y = ax
We can therefore write the following equivalences :
U = y = the function
R = 10 =the coefficient of proportionality
I = x =the variable
Let x (thus I) be different values and calculate the corresponding values of y (hence U).
Arbitrary values of x (I)
1
2
3
4
5
Calculated values of y (U = RI)
10
20
30
40
50
Let's draw two coordinate axes, 0x and 0y (Figure 4).
Graduate as it should be 0x in amperes and 0y in volts adopting as scale 1 A = 1 cm and 10 V = 1 cm. Let's place the points A (1,10) ; B (2.20) ; C (3.30) ; D (4.40) ; E (5.50).
We note that these points are on the same line (delta)
that passes through the origin 0 and that we call representative line of the function y = 10 x hence the rule :
The representative curve of the function y = ax is a line passing through the origin.
We recall that in mathematics, the word "curve" is synonymous with "line". So do not be shocked if you are told that a "curve" is a "right".
4. 2. - PRACTICAL PRACTICE OF THE RIGHT REPRESENTING THE FUNCTION y = ax
As we have just seen, the line y = ax passes through the origin 0.
As to position a line, just know two of its points, we have to determine a second point, the first being known (the origin 0). We then take an arbitrary value of x multiplied by the coefficient of proportionality. The product obtained gives us the ordinate of the second point, the abscissa being obviously the arbitrary value of x chosen.
Keeping the previous data, for example, we take x = 4 from which y = 10 x 4 = 40 and we obtain the point P1 (4.40) (Figure 5).
We then join the point 0 to the point P1 and we obtain the line 1 which is the line of the function y = 10 x.
Remark : For the accuracy of the drawing, it is advisable to take a value of x as large as possible, compatible with the size of the graph.
If we had chosen for x the arbitrary value : 0.5 we would have obtained the point P2 (0.5 ; 7). All that is required is a slight error in positioning this point so that the information given by the reading of the curve is false. Thus, if we locate P2 (Figure 5) not with its exact coordinates (0.5 ; 7) but with a slight error (about 0.5 ; 6.5), for the higher values of x, the absolute error becomes important. By drawing the line 2
we see that at the value of x = 4 no longer corresponds y = 40 but 35. The error which corresponds to a difference of 12.5% is not negligible.
4. 3. - GENERALIZATION
Represent on the same graph (Figure 6) the following functions y = ax :
y = - 4x
y = - x
y = - 0,5x
y = 0,5x
y = x
y = 3x
To draw the corresponding lines, we must determine two points : one is already formed by the intersection of the axes x'x and y'y, it is the point 0.
To get the second, give x a numeric value :
y = - 4x
for x = 2
y = - 4 (2) = - 8 = P1
y = - x
for x = 5
y = 1 (- 5) = - 5 = P2
y = - 0,5x
for x = 6
y = - 0,5 (6) = - 3 = P3
y = 0,5x
for x = 6
y = 0,5 (6) = 3 = P4
y = x
for x = 6
y = 1 (6) = 6 = P5
y = 3x
for x = 2
y = 3 (2) = 6 = P6
These points being now perfectly defined by their coordinates, P1 (2 ; - 8) ; P2 (5 ; 5) ; P3 (6 ; - 3) etc ... we can place them in the plane and draw the lines representing each function by passing them through the origin 0 and the corresponding points P (Figure 6).
Let's look at Figure 6. We notice that in the function y = ax :
1 - When the coefficient "a" is positive (a > 0) ;
- the right is in the first and third quadrants,
- when x increases, y also increases : x and y vary in the same direction.
It is said then that the function is increasing.
2 - When the coefficient "a" is negative (a < 0) :
- the right is in the second and fourth quadrants,
- when x increases, y increases (x and y vary in opposite direction).
It is said that the function is decreasing.
In summary, the function y = ax is increasing for a > 0 and decreasing for a < 0.
4. 4.- SLOPE OF THE RIGHT y = ax
Continuing to observe Figure 6, we make the following observation : when the coefficient "a" increases in absolute value, the angle formed by the abscissa axis (x'x) and the corresponding line also increases. Thus, the angle x 0 P6 (function y = 3 x) is greater than the angle x 0 P5 (function y = x).
The coefficient "a" of "x" in the relation y = ax is called slope or angular coefficient, of the line y = ax.
In summary, the slope of the line y = ax increases as the absolute value of the coefficient a of x.
Remark : It is shown in trigonometry that "a" is the tangent of the angle formed by the abscissa with the line of the considered function.
Recall on the coordinates of a point
(delta)
that passes through the origin 0 and that we call representative line of the function y = 10 x hence the rule :
designates this angle, we have the relation :
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