Chapter 1 — Calculus I
Lesson map
Lesson 01 published. Map, example, EX001.
flowchart TD
N0["Learning outcomes"]
N1["1.1 The case and why calculus"]
N2["1.2 Rates of change"]
N3["1.3 Reading rates from data"]
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 The case and why calculus"]
N2["1.2 Rates of change"]
N3["1.3 Reading rates from data"]
N4["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4 --> Q["EX001"]
[Learning outcomes]
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[1.1 The case and why calculus]
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[1.2 Rates of change]
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v
[1.3 Reading rates from data]
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v
[Synthesis]
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v
[Worked example] --> [Exercise EX001]
Learning outcomes
You will (1) describe change with rates and limits; (2) model token supply growth; (3) read rates from data.
1.1 The case and why calculus
Jonas Weber is an engineering student at the fictional Avalon Polytechnic University, supervised by Dr Amara Nwosu. Over five years he will help the fictional Brightwater Community Energy Cooperative, about 400 member households, design and secure a system that records solar energy credits as tokens on a permissioned Ethereum-compatible ledger. Every person, contract and number in this book is synthetic. Calculus may seem distant from blockchain security, but engineers constantly reason about rates: how fast credits are issued, how quickly balances change, and how fast an abnormal flow must be noticed before it causes harm.
1.2 Rates of change
A derivative describes how fast one quantity changes with respect to another. If the cooperative issues credits in proportion to energy generated, total supply S(t) grows at rate dS/dt equal to generation in kilowatt-hours per hour times credits per kilowatt-hour. On a sunny afternoon the rate is high; at night it is zero. Jonas models supply as the integral of issuance minus redemptions. This gives a baseline for normal behaviour: if supply ever rises faster than generation could explain, something is wrong, whether a meter fault, a software error or misuse of the minting function.
1.3 Reading rates from data
Real data are discrete: the ledger records events, not smooth curves. Jonas estimates rates using differences over fixed intervals, such as credits minted per hour, and smooths noise with moving averages. He plots a synthetic week and marks the maximum plausible issuance rate from the cooperative's installed solar capacity of 1.2 megawatts. Any hour exceeding that bound deserves investigation. Calculus here is a practical tool: it turns a physical limit into a numerical threshold that a monitoring system can check automatically, which later chapters will build into the design.
Worked example
Jonas models supply as the integral of issuance minus redemption and sets an hourly issuance ceiling from the 1.2 MW installed capacity.
Synthetic case. Jonas Weber, Dr Amara Nwosu, Avalon Polytechnic University, Brightwater Community Energy Cooperative and all data are synthetic.
Errors and limits
Treating any rise in token supply as normal ignores physical limits that should bound issuance.
Exercises
- EX001 — Cooperative size. How many households does Brightwater serve? Rubric: 3 = about 400;; 2 = 400;; 1 = 4,000;; 0 = none.
- EX002 — Supply model. How does Jonas model supply? Rubric: 3 = integral of issuance minus redemptions;; 2 = integral;; 1 = constant;; 0 = none.
- EX003 — Issuance bound. What sets the issuance ceiling? Rubric: 3 = installed solar capacity of 1.2 MW;; 2 = capacity;; 1 = price;; 0 = none.
Synthesis
Rates and integrals turn physical limits into checkable bounds for token systems.