OpenCyber AcademyLearning outcomes
You will (1) use rates of change to describe security behaviour over time; (2) interpret derivatives as speeds of change in reporting; (3) relate accumulation to totals of exposure.
1.1 Rates in human factors
Aiko Tanabe is an engineering student at the fictional Rivenhall Technical University. Her supervisor, Prof. Joanna Kerr, links every foundation course to a running case: the fictional Greyford Water utility, whose six hundred staff include control room operators, field engineers and office teams. In calculus, Aiko learns that a derivative describes how fast something changes. At Greyford, the number of staff reporting suspicious emails changes over weeks after a campaign. The rate of change tells her whether reporting is still rising, levelling off or falling back, which matters more to planning than any single weekly count.
1.2 Reading a curve
Aiko models weekly reports after a campaign as rising quickly at first and then levelling. The derivative is large in the first weeks and shrinks towards zero. She learns to read where the curve bends, because that is when the campaign's effect is fading and a reminder might help. She also sees that a straight line would mislead: it would predict reports growing for ever. Choosing a shape that fits how people actually respond is the first modelling decision in any human factors analysis, and calculus gives her the vocabulary to describe it.
1.3 Accumulation and exposure
Integration adds up a rate over time. Aiko uses it to estimate total exposure: if a phishing email sits unreported in inboxes, the number of people who could click grows each hour until someone reports it and it is removed. The area under the curve of unreported inboxes over time measures exposure. Faster reporting shrinks that area. This simple idea links her mathematics course to Greyford's goal of reducing how long malicious emails remain available, a measure that does not depend on blaming whoever clicked.
Worked example
Aiko describes reporting after a campaign as a curve, reads its rate of change to find when the effect fades and uses accumulation to define exposure time.
Synthetic case. Rivenhall, Greyford and all people and figures are synthetic.
Errors and limits
Smooth curves simplify noisy weekly data. Exposure models ignore how attractive each email is.
Exercises
- EX001 — Rate meaning. What does the derivative of weekly reports tell Greyford?
Show marking criteria
Rubric: 3 = whether reporting is rising, levelling off or falling;; 2 = trend;; 1 = vague;; 0 = none. - EX002 — Straight-line flaw. Why would a straight line mislead for reports after a campaign?
Show marking criteria
Rubric: 3 = it predicts reports growing for ever;; 2 = unbounded;; 1 = vague;; 0 = none. - EX003 — Exposure area. How does integration define exposure?
Show marking criteria
Rubric: 3 = area under unreported inboxes over time;; 2 = area;; 1 = count;; 0 = none.
Synthesis
Rates and accumulation give precise language for how behaviour changes and how long people remain exposed.