An Optimization Problem
Here's an optimization problem the form of which (if not the setting) will be familiar to instructors:
The number of people, \( P \), visiting a certain beach on a particular day in January depends on the number of hours, \( x \) that the temperature is below 30 C according to the rule
\[ P = x^3 - 12x^2 + 21 x + 105 \]
where \( x \geq 0 \).
Find the value of \( x \) for the maximum and minimum number of people who visit the beach.
Source: MathsQuest Maths B Year 12 for Queensland (2003) p. 57
What is the point of such a problem?
- To exercise the differentiate-then-solve approach.
- To get students to distinguish between the maximum and minimum by checking \( d^2 P/dx^2 \) at each extremum.
The function is posed as cubic so that the differentiation step produces a quadratic. Students have learned how to solve the quadratic.
Problematic features of the problem
- Better just to graph the function. Then it's easy to spot the maximum and minimum and to consider the issues with the real maximum being at \( x=24 \).
- It's hard to imagine that any real problem would involve fitting a cubic function to a one-variable model. Cloudiness? Rain? Wind? Weekend or weekday? Temperature on previous days? The forecast?
- Why would more hours of cool temperatures cause the population to go down, but then go up? What's the mechanism?
- No connection to data for a system that certainly doesn't have a simple mechanism.
This problem teaches the wrong things!
- Should be: “Sketch out a graph that describes what you imagine the relationship is between temperature and number of visitors.”
- The problem gives no hint that precision of the model is an issue. As a rule, calculus books don’t touch on this even though it is an important application of derivatives: Conditioning of calculations.
- What's important about \( df /dx = 0 \) is not that it locates the extremum, but that near the extremum the value of the extremum is not sensitive to \( x \). (Unless there are active constraints, but this is not covered in calculus books.)
- What's important about \( d^2 f /dx^2 \) is not that it tells whether an extremum is a maximum or a minimum, but that it indicates how important precision in \( x \) is.