3. Exponential growth and decay
Discrete exponential growth
Geometric sequences (seen in Core 2) are sequences which exhibit exponential growth or decay (depending on the common ratio). The amount by which the sequence will increase at a given stage is proportional to the value of the sequence. (Consider compound interest, an example of a geometric sequence).
The reason this is "discrete" exponential growth is simply that there are precise time units. For the example briefly mentioned before, a bank will pay interest annually, or possibly more frequently, however there is not a continuous flow of interest into the account. Unlike the geometric sequences, it is often useful to consider the start point to be "0", as this implies that nothing has happened, and therefore is a more natural way of expressing the growth.
Exponential growth occurs if the common ratio is greater than 1, this is relatively easy to understand, as it would be expected that for any real number (other than 0), there would be a magnitudinal increase resulting from a multiplication by a number greater than 1.
Continuous exponential growth
In a lot of situations, growth does not occur at set intervals, and therefore the idea presented previously does not fit the situations as well.
These situations fit the following:
Where "t" is the time, "a", and "b" are constants. The value of "b" determines whether there is exponential growth or decay, if it is greater than 1 then there is exponential growth, and if , there is exponential decay.
Graphs of exponential growth
Logarithms can be used in conjunction with the exponential growth and decay. Consider that the graph of an exponential function will have an exponentially increasing gradient, and that it can be expressed as:
Logarithmic manipulation yields:
This is in the form of:
Hence one can deduce that for a truly exponential function, the graph of against "t" would be a straight line. (Of course, in reality there will most likely be some small deviations [exponential growth and decay are not exact things in real life], however there should be a notable line).
Transformations of the growth graph
A series of transformations can be applied to a graph of exponential growth to map it onto another section (hence the whole graph can be "built up" from a single piece).
Translating by "n" in the positive x-direction:
Read these other OCR Core 3 notes:
- Successive transformations
- Exponential growth and decay
- Extending differentiation and integration