Example 5: Mozart's Symphony Production

The following table gives the number  N  of symphonies that Mozart wrote at age  x.

The value of the function, read directly from the table, tells us, e.g., that Mozart wrote  8  symphonies when he was  16  and no symphonies when he was  20. The change in value depends on the period of Mozart's life that one considers. Between ages  7  and  8,  DN = N(8) – N(7) = 3 – 0 = 3, which tells us that Mozart's symphony production went up by  3  symphonies during that year. Between ages  24  and  26,  DN = N(26) – N(24) = 1 – 1 = 0, which tells us that the average change in symphony production over that two-year interval was zero. The rates of change during these intervals will also be averages. Between  7  and  8  Mozart's symphony production increased at an average rate of  DN/Dx = 3/1 = 3  symphonies per year. Between  24  and  26  the average rate was  DN/Dx = 0/2 = 0  symphonies per year.

It would be incorrect to conclude, however, that the zero rate of change between  x = 24  and  x = 26  means that Mozart continued to produce the one symphony per year that he was producing at the beginning of that interval. The table shows that his production actually fell and then rose again.

It would also be incorrect to conclude, from the average rate of  3  symphonies per year between  x = 7  and  x = 8,  that Mozart worked at this rate throughout the entire year. He may have had an inspired burst of creativity over a shorter period of only months, then leveled off.