Then, for the arithmetic, some code can be short and, compared with cells in a spreadsheet, easier and with more control over the time steps, e.g., in Rexx with cf for customer fraction:
Say ' ==== Growth ===='
Say ' '
Say ' Customer'
Say ' Year Fraction'
max_years = 5
steps_per_year = 10 * 365
cf = 1 * ( 1 / 100 )
year = 1
k = 1 * ( 1 / 2000 )
Do Forever
Do i = 1 To steps_per_year
cf = cf + k * cf * ( 1 - cf )
End
Say Format(year,9) Format(100*cf,10,2) || '%'
If year = max_years Then Leave
year = year + 1
End
==== Growth ====
Customer
Year Fraction
1 5.89%
2 27.97%
3 70.66%
4 93.73%
5 98.93%
Adjust some of the constants in the program and can get more output, say, for each month, day, etc. The code above uses 10 steps per day.
For more, someone could use the calculus solution and compare.
In a sense, for the FedEx problem and the assumptions about what was driving the growth, the calculus solution is a smooth version of the somewhat more appropriate discrete time version.
But when I did the calculation at FedEx, my best source of arithmetic was an HP calculator in which case the calculus solution was a lot easier.
Of course, this FedEx calculation was just one example and there are many others.
My view from 10,000 feet up is that in business, at times some math can be an advantage if not the work of a steady job.
If some math is an advantage, then that advantage tends to go to the owners of the business. If a mathematician wants to get paid for some math they have in mind, maybe they should start a business and be the owner.
For the arithmetic for the calculus solution, in Rexx,
y.0 = ( 1 / 100 )
b = 1
k = ( 1 / 2000 )
Do t = 1 To 5
t1 = t * (10 * 365 )
e1 = RxCalcExp( b * k * t1, 16 )
y.t = ( y.0 * e1 ) / ( y.0 * ( e1 - 1 ) + b )
Say Format(t,9) Format(100*y.t,6,2) || '%'
End
==== Growth ====
Customer Fraction
Year Calculus Discrete
1 5.90% 5.89%
2 27.99% 27.97%
3 70.68% 70.66%
4 93.73% 93.73%
5 98.93% 98.93%