The quantity of an object whether it is a bacteria, cat or even human population grows in different ways over a period of time. In general it is termed as growth of a variable. In many cases we can find a pattern in such growths. Suppose the growth is at a constant rate, it is called as arithmetical growth. If the growth is defined by some power of the variable, then the growth is a power function. On the other hand if the growth is defined by a constant with the variable in the exponent place, it is defined exponential growth or we can say that the growth as such. This growth is also referred as compound growth.
Suppose y is the quantity at any point of time x, the function of growth y = xa is a power function. If the same is in the form y = ax, then it is an exp^n function. (a is a constant in both cases). Thus the simplest exponential growth formula is y = akx, a and k being a constants
Let us see the pattern of a compound growth. It is easier to understand with an example. Consider one growth function as y = x2 and another growth function as y = 2x. The following table shows the results for the growth under these two conditions.
We find that the compound function grows faster between 0 and 2 but the power function grows faster between 2 and 4. But the difference of growths between the two is not appreciable. But for values of x more than 4, the compound function grows very steeply. For example when x reaches 10, the power growth goes only up to 100 whereas the compound growth shoots up to 1024! Thus, a compound growth is very steep after certain initial stage.
The above study can be clearer when you see the growth in graphical form. The graph of a compound growth is called as exponential graph.
Exponential growth is seen in many real life situations. The growth of money invested at a compound interest is a glaring example that anyone can understand. In scientific field, the growth of bacteria is a common example.
It may be noted that if the constant k in the exp^n formula is negative then there is a negative growth which is generally referred as exponential decay.
Suppose y is the quantity at any point of time x, the function of growth y = xa is a power function. If the same is in the form y = ax, then it is an exp^n function. (a is a constant in both cases). Thus the simplest exponential growth formula is y = akx, a and k being a constants
Let us see the pattern of a compound growth. It is easier to understand with an example. Consider one growth function as y = x2 and another growth function as y = 2x. The following table shows the results for the growth under these two conditions.
| x | y = x2 | y = 2x |
| 0 | 0 | 1 |
| 1 | 1 | 2 |
| 2 | 4 | 4 |
| 3 | 9 | 8 |
| 4 | 16 | 16 |
| 5 | 25 | 32 |
We find that the compound function grows faster between 0 and 2 but the power function grows faster between 2 and 4. But the difference of growths between the two is not appreciable. But for values of x more than 4, the compound function grows very steeply. For example when x reaches 10, the power growth goes only up to 100 whereas the compound growth shoots up to 1024! Thus, a compound growth is very steep after certain initial stage.
The above study can be clearer when you see the growth in graphical form. The graph of a compound growth is called as exponential graph.
Exponential growth is seen in many real life situations. The growth of money invested at a compound interest is a glaring example that anyone can understand. In scientific field, the growth of bacteria is a common example.
It may be noted that if the constant k in the exp^n formula is negative then there is a negative growth which is generally referred as exponential decay.
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