One of the jokes among engineering students comes in the form of this formula:
Productivity = Result / Effort
Therefore, as effort approaches zero, productivity approaches infinity.
However, this joke assumes that result and effort are independent of each other. It ignores the fact that results is in fact a function of effort.
To understand this, let us have a look at a graph of effort against time of a typical student:
The hardworking student would have a graph that looks more like the following:
On the other hand, another type of students have a more exotic-looking graph, marked by periods of sporadic inspiration and scolding by supervisor, like so:
In all cases, what matters most is the total effort spent, which can be found by looking at the area under the graph. To simplify matters, let us call this cumulative effort just ‘effort’ from now on.
We can now see how effort correlates to result. Typically, the result against effort graph looks like the following:
There are three regions in the graph:
- Region of minimum efficiency: Usually found at the start of the work, where requirements are still not clear and the person does not have a clear direction to take.
- Region of maximum efficiency: When the person knows what to do.
- Region of saturation: When nothing more can be done no matter how much more effort is spent.
In almost every case, a student should strive to achieve the point in between region of maximum efficiency and the region of saturation.
It is clear from the graph that as effort approaches zero, result also approaches zero. Intuitively, we may think that the formula for productivity then becomes the following:
Unfortunately, this doesn’t make much sense mathematically.
This problem can be solved by observing that the graph indicates that as effort approaches zero, the gradient of the result against effort graph also approaches zero, i.e.:
Hence, we can utilize L’Hôpital’s rule, and the formula for productivity becomes:
It is clear that if no effort is done, productivity will be 0 as well. Therefore, the statement that “as effort approaches zero, productivity approaches infinity” is false. The key to the problem is in fact the result. No matter how much (or how little) effort a student put, the result is the one that determines how productive a student is.
In other words, if a student puts in a lot of effort, but comes up with a little result, he is considered unproductive. If he puts in a lot of effort and comes up with a reasonable result, he may be reasonably productive. If he puts in a little effort and comes up with a very good result, then he can be considered as productive.
This case study is also a good example of how a student can spend so much time proving something so trivial using an overly-complicated method, which reduces his productivity tremendously.
Unproductive,
hendri
