01 - Derivative
A function y = f (x) is represented by a curve on the Cartesian plane. Even a straight line is
a particular type of curve: a curve is a "slope" constant. A
any curve, however, has in general, point by point, a different slope.
01 Slope.
Let us now define quantitatively the concept of slope. Already we find the slope
indicated as a percentage of danger signs in the streets of the mountain.
The definition of slope in mathematics is similar to that used in road signs.
a slope of 10% has the following geometric meaning:
the slope is the vertical relationship between the catheter and catheter
horizontal triangle as shown. So, in this case, slope = 10% = 10/100 = 0.1.
If the catheter was 100 meters as vertical and horizontal, the slope would be 100%
or equal to 1, and corresponds to an angle of 45 °.
With the growth of the catheter will have vertical slopes increasing to infinity. For example,
a gradient of 700% or greater than 7, meaning that with the increase of the slope, the angle at the base (left) tends to draw ever closer to 90 degrees.
When the angle at the base will be 90 °, the slope will be infinite.
The slope is then the ratio of the catheter vertical and horizontal.
If this definition we report on a curve, we can define a point in the slope of the latter
the tangent to the curve at the specified point:
the curve represents the function y = f (x). The slope of the curve the point P is the slope of the tangent to the curve
drawn at the same point P.
Point by point, the slope is different in general.
02 - Derivative.
The slope of a curve at a point is called the derivative of the function at that point.
The concept of derivative is of fundamental importance and forms the basis of differential calculus. With
derived one can study the performance of a function or calculate the solutions of equations whose unknowns
are not just numbers but functions. These types of equations are called differential equations.
The derivative is itself a function as a point per punto, essa assume valori in corrispondenza della x.
La derivata della funzione y = f(x) è quindi una funzione della x e si indica con la scrittura y = f ' (x). Essa si chiama
anche derivata prima.
Essendo la derivata prima una funzione, se si fa la derivata di questa, si ottiene la derivata seconda y = f '' (x).
Dalla derivata seconda si ottiene la derivata terza e così via.
03 - Studio di funzione.
Vediamo ora alcuni esempi in cui di ciascuna funzione data definiremo punto per punto sia la derivata prima
che la derivata seconda.
Consideriamo la funzione y = x + 1. Essa è rappresentata nel piano cartesiano da una retta obliqua.
The slope of a line is obviously constant at every point where its derivative is constant. In this case
, forming a straight angle of 45 °, the derivative is equal to 1 at any point on the line. The first derivative function that is being so y = 1. The derivative of the first derivative is the derivative
second. It is 0 at any point because the first derivative, being a horizontal line,
has no slope at all points. The second derivative is then y = 0.
Now consider the function y = x ². It is represented by a parabola at any point at which the slope
is variable.
Nell'origine 0 la pendenza è nulla perché la tangente alla parabola è ivi orizzontale. A desta dell'origine, la pendenza
è positiva e cresce via via che ci si allontana dall'origine verso destra. A sinistra, invece, la pendenza è negativa e
cresce in valore assoluto più ci si allontana dall'origine verso sinistra.
Nei
punti in cui la derivata prima è nulla si hanno dei punti di massimo e minimo relativo.
Come si vede dall'esempio la derivata è uno strumento fondamentale per lo studio di una funzione, cioè
per individuarne il comportamento. Dove cresce, dove decresce e dove vi sono massimi e minimi relativi.
Se la derivata è positiva, the function is increasing there, if the derivative is negative, the function is decreasing.
If the derivative is zero, there can there be a maximum or a relative minimum.
The calculation of the derivative of a given function is feasible in principle forever. We show here some
formulas used for this purpose:
submode
y = k (where k is any number) y '= 0
y = kxy' = k
y = k x ² y '= 2 kx
y = ³ y k x 'k = 3 x ²
... ...
If the function y = x ³ - 3x ² +2 x calculating the first derivative is:
y ' = 3x ² - 6x + 2
while the second derivative:
y''= 6x - 6
Note that the derivative of a sum of terms is done by adding the derivatives of each term.
Also note that the first derivative of y = f (x) can be indicated simply by the symbol y 'while the second derivative y''
with the symbol.
04 - Differential Equations.
In physics we study the quantities that are measured by observing natural phenomena to derive
mathematical laws that express the interdependence. This, in brief, is the purpose and the method of physics.
Suppose a certain size y is represented by a function una variabile y = f(x), cioè
che la grandezza y vari in funzione della grandezza x. Supponiamo anche che questa grandezza sia tale che
la sua derivata sia uguale in ogni punto alla somma fra la x e la y. In sintesi si supponga che :
y ' = x + y.
Questa è una equazione differenziale. Una equazione in cui l'incognita non è un numero ma una funzione.
Le equazioni differenziali sono il cuore della fisica. Tutte le leggi fisiche note sono espresse in termini di
equazioni differenziali. Risolvendo queste equazioni si ricavano le grandezze fisiche nel loro variare in
funzione di altre grandezze fisiche.
La soluzione delle equazioni differenziali non è always possible in analytical terms, that is exactly.
In many cases you must then use methods of numerical approximation to the realizable computer.
The proposed above, assuming that the curve passes through 0, we obtain:
Note that the origin 0, the value of x + y is obviously 0 and hence y 'should be 0. The curve is sought
then tangent to the axis of x in 0.
Without going into detail, just mention that we have found the solution of differential
using a very simple method of numerical approximation based on the fact that the line
tangent to a curve at a point near that point is "confused" with it:
The derivative of the function P is Q'H / HP. If the point Q is very close to point P, the derivative can be approximated with
QH / HP because the points Q 'and Q tend to overlap. With this artifice can put
QH = (Q'H / HP) HP * and then from the point P is obtained, although approximate, the point
Q and similarly all subsequent paragraphs. You can get the curve right away.
This method is the basis of many techniques of approximation of differential equations feasible
to your computer.
Finish.
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