Chapter 1 — Calculus I
Lesson map
Lesson 01 published. Map, example, EX001.
flowchart TD
N0["Learning outcomes"]
N1["1.1 Why an engineer meets privacy early"]
N2["1.2 Rates of change"]
N3["1.4 Optimisation as trade-off"]
N4["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4
N4 --> EX["Worked example"]
EX --> Q["Exercise EX001"]
Logical drawing
flowchart LR
N0["Learning outcomes"]
N1["1.1 Why an engineer meets privacy early"]
N2["1.2 Rates of change"]
N3["1.4 Optimisation as trade-off"]
N4["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4 --> Q["EX001"]
[Learning outcomes]
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[1.1 Why an engineer meets privacy early]
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[1.2 Rates of change]
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v
[1.4 Optimisation as trade-off]
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v
[Synthesis]
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v
[Worked example] --> [Exercise EX001]
Learning outcomes
You will (1) read rates of change in data growth; (2) use limits to reason about anonymity as groups shrink; (3) connect calculus to privacy engineering.
1.1 Why an engineer meets privacy early
Zara Mbeki begins a five-year engineering course at the fictional Corwen Polytechnic, supervised by Prof. Anders Holm. Her placement partner is the fictional Pellmore Telehealth, a telemedicine provider with 300 staff and patients in the European Union and Brazil. Pellmore's app books appointments, runs video consultations and stores records. Prof. Holm explains that privacy and data protection will be her main engineering domain, and that every foundation course will be read through it, starting with calculus, because privacy problems often involve quantities that grow or shrink over time.
1.2 Rates of change
Pellmore's stored video grows by roughly forty gigabytes a day. The derivative of storage with respect to time is that daily rate, and integrating it over a year gives the total. Zara sees a privacy consequence immediately: if recordings are never deleted, the amount of sensitive health data at risk grows linearly forever. A retention rule turns the growth curve into a plateau, because data enter and leave at the same rate. She sketches both curves and labels the gap between them as avoidable exposure that a breach could reveal.
1.3 Limits and small groups
A limit describes what happens as a quantity approaches a value. Zara applies the idea to published statistics. If Pellmore reports the average waiting time for a clinic, the average describes many patients; as the group size approaches one, the average becomes one person's exact waiting time. Privacy protection that works for large groups therefore fails in the limit of very small groups. This intuition, she learns, is behind rules that suppress small cells in published tables and behind the formal guarantees she will meet later in differential privacy.
1.4 Optimisation as trade-off
Calculus also finds maxima and minima. Prof. Holm sets an exercise: the usefulness of a dataset rises with detail while privacy risk also rises. If both are modelled as smooth curves, the best level of detail is where the marginal gain in usefulness equals the marginal increase in risk. Zara realises the model is simplistic, because risk to people is not freely exchangeable for usefulness and law sets hard limits, but the habit of asking about marginal effects will help her argue for proportionate designs throughout the course.
Worked example
Zara models storage growth with and without retention, reasons about averages as group size shrinks, and frames detail versus risk as a marginal trade-off bounded by law.
Synthetic case. Corwen, Pellmore and the figures are synthetic.
Errors and limits
Smooth curves simplify discrete risks; legal limits are constraints, not terms in a trade-off.
Exercises
- EX001 — Retention plateau. Why does a retention rule turn storage growth into a plateau? Rubric: 3 = data enter and leave at equal rates, so the total stops rising;; 2 = mentions deletion;; 1 = says storage stops;; 0 = none.
- EX002 — Small-group limit. What happens to an average as its group size approaches one? Rubric: 3 = it becomes one person's exact value, revealing it;; 2 = says it gets less accurate;; 1 = wrong;; 0 = none.
- EX003 — Marginal trade-off limits. Why is the marginal trade-off model simplistic for privacy? Rubric: 3 = harm to people is not freely exchangeable and law sets hard limits;; 2 = mentions law;; 1 = accepts the model;; 0 = none.
Synthesis
Chapter 1 reads calculus through privacy. Chapter 2 uses discrete mathematics to count identities.