THE MEANING OF AN INCREMENT

You may have encountered this term before - an increment. Most likely this had happened in some course of algebra, which called for an "incremental approach". The term "increment" however has a direct relation to mathematics and, in fact, constitutes the bases of the department called calculus and math analysis. Most importantly, an increment constitutes the very foundation of an extremely important, fundamental mathematical application, which is a functional analysis. It is here, where the meaning of increment takes on itself a defining role in the formation of student's systemic knowledge of functions, without which, of course, there is no real mathematics and physics.

While the terms may sound somewhat intimidating, the reality behind them is very easily comprehended and it starts from the first very simple step - function cannot be viewed anymore as a static entity and ought to be seen as a very dynamic, ever-changing, dependency, be it a simple linear function or the most complex curvature.

Of course, we deal with increments on a daily bases: enough to look at such a situation as planning our daily activity. Indeed, if one of you wants to start some business meeting at, say 2 p.m. and wants to finish it at 4 p.m. then it becomes quite obvious that duration of this meeting will be  2 hours. Well, ask yourself a question: how did we know it? Do not rush here with a conventional answer, which all of you, of course, know. Let us arrive at this answer within proper mathematical framework.

We start with a simple thing - we'll give to the meeting's starting time of 2 p.m. a designation t₁, while the time of the meeting's end, that is 4 p.m., will be t₂ . Now we proceed with finding the duration of our business meeting. Obviously it will be  hours.
Simple, isn't it? But ask yourself a question, what is the physical (or mathematical) meaning of duration of this meeting. Well, we may state that it will be either time increment or the change in time. We may go even further in defining the physical essence of any change by stating that it is a difference between the end-value and a starting value of anything. How about change in distance from some point of reference on the ground: if you were, say 3 miles, from this point and then walked and positioned yourself 6 miles away from the same point. Obviously, the change in distance here was miles. Or in purely algebraic terms it was .
Another popular encounter with the increment (or as we already stated - the change) happens when you observe the change in temperature throughout the day. If at some point of time, say 2 p.m. ( t₁) the temperature outside was ( T₁), while at 5 p.m. ( t₂) it became  ( T₂), we will be able to state that:
  1. In duration of time  hours;
  2. The temperature has changed degrees, so it rose!
In mathematics (as well as in physics) the change of any parameter was named delta after Greek letter "Δ" (capital) or "δ" (lower case) and since then it became possible to represent any difference, which denotes change, as delta something, be it time, temperature or any other parameter. Thus we may state that the duration of the business meeting in the first example was , while the distance we walked away from point of reference was  , and in the last example we had simultaneously a change in time and in temperature .

Nowhere does change play a more important role than in functions. It turns seemingly static, boring entities such as straight lines or curvatures into dynamic, alive and surprising creatures, allowing us to analyze them, make forecasts about their behavior and use them to our advantage in virtually any technological field, since every physical process in the universe could be described as a function, that is a dependency, and functions, as we know, change.

The best way to start to investigate change is linear function, whose change is constant; rather than curvatures, where calculus apparatus is needed to investigate their complex changes. Since we talk about function, we have to remember that we will be talking about dependency, that is (in our case) about two variables, one of which will be independent and the other, the function itself, will be dependent. As we know, the independent variable in function is called the argument of given function, while dependent variable is, of course, function itself. As you know, in Cartesian System of Coordinates, the argument is represented as x , and consequently is plotted on the x-axis, while the function is y and plotted on the y-axis.

Let us review some basic linear function of form . For the first time we will be dealing not with a single point on the graph but with the whole segment of function. In order for us to understand how the linear function behaves itself we have to give our argument some increment or change. Look attentively at the picture to the left: we start at the point P with coordinates . But for now our focus will be on the x-coordinate (or argument) only. We begin to change our x-coordinate by means of moving to the right on the x-axis from and stop at the point with coordinate . It is quite obvious that what we have achieved here was a change in x-coordinates  or as we already know it, we obtained , which, of course was the difference between x-coordinates of the end-point and the starting point of change. In algebraic terms we obtained .

But look, while giving increment to our x-coordinates, we achieved another change - this time in function itself! The value of function at point was  , at point it became . So, not only argument changed, but the function itself changed also and the change in function was happening simultaneously with the change in argument. In this case we may state that for the change  of our function we obtained accorded to it change in function . But not only that! There was some very specific law, in accordance to which all those changes were happening - as is clear from the picture, was called run, while was called rise. And in order for us to understand this law we have to point our attention to an interactive model below.

Place your mouse' pointer on the gray dot on the function  , which was chosen freely, since it is of the linear form , click and drag this dot along the line of  function in any direction you want. What do you see?

You see an interaction of this function's rises and runs (or of the increments of argument with the increments of function itself). Observe attentively, with all changes happening, what will remain the same? Correct, that will be the relation of or, better known in mathematics as , which, of course, is the slope or  of our function. No matter what you do, this relation will always be 2 positive and will always be for this particular function, while in general we may state now confidently:

For each linear function, its slope ! 

Try, on your own, to analyze slopes of the linear functions shown on the picture to the left. Remember, in order to do this you have to give an argument some increment, see how function changes and then confirm its slope , by means of this now well known to us relation:

Graphs of linear functions of the form y = mx + b [click]
Properties of powers [click]
Did you know? [click]
The principle of energy conversion [click]
The meaning of an increment [click]
The hydraulic press [click]

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