Search

Email

A     B     C     D     E     F     G     H     I     J     K     L     M     N     O     P     Q     R     S     T     U     V     W     X     Y     Z




  Mobiles  

  F7     F9


 Bookstore

 Tests

 Formulary




  Mathematics - Functions 3



In the previous chapters we have explained the difference between a function with a function specification and a graphical representation of measured values. The latter are e.g. with such a device as shown above in order to be able to impose fines in the case of driving too fast.

It started with the recording of the time when a certain route was passed. If the time was below a limit, then fine was again due. In the last case, an average speed was determined over a certain distance, while in the system above the speed at a certain point.


And precisely the so-called 'derivative' is used. It would be the slope of a tangent at the indicated point. You have already seen straight line functions in chapter Functions 1:

f(x) = 2x   f(x) = 3x   f(x) = 4x  

There it was explained to you that the multiplier of x yields the slope of a straight line. At f (x) = 2x, the f(x) increases more slowly than f (x) = 4x. And as the slope in the graph above is Δs/Δt, it corresponds exactly to the velocity:

Δs
v=
Δt

Below is the diagram. From the point at which we want to determine the velocity, we continue by Δt and obtain a distance extended by Δs which has been returned. The average speed on this distance is represented by the straight line 1.


When a derivative is formed, t1 now slowly returns to t0, that is, let Δt become zero. As a result, the line 1 is slowly turned to the line 2, which would then be a tangent to the curve exactly at the point t0. We write it like this:

<


Thus, if we consider an infinitesimal range at the point S0 and t0 of the above function, the slope can be assumed to be linear. One proceeds from a difference, e.g. t1 - t 0, which can be made smaller and smaller. From this the 'differential calculus' has its name.

So now we have come from the average to the current speed. Geometrically, the secant, ie, a connection of two points of a circular arc, has changed into a tangent, with only one point of contact, exactly the one for which we need for calculating our velocity.


If you compare this image with the previous one, you might think it is the same with just another label. Right. And yet it is to be said here that it is a function. A curve, which is formed from measured values, actually consists only of a few support points, which are connected by a line.

A function, on the other hand, is exactly defined at every point in the curve. And only here, of course, can I slowly move from the point f(x + Δx) / x + Δx to f (x)/x, thus:


Imprint