Chapter 6 — Calculus II

Chapter 6 — Calculus II

Lesson map

Lesson 06 published. Map, example, EX016.

flowchart TD
  N0["Learning outcomes"]
  N1["6.1 Accumulation"]
  N2["6.2 Series and repeated effects"]
  N3["6.3 Long-run behaviour"]
  N4["Synthesis"]
  N0 --> N1 --> N2 --> N3 --> N4
  N4 --> EX["Worked example"]
  EX --> Q["Exercise EX016"]

Logical drawing

flowchart LR
  N0["Learning outcomes"]
  N1["6.1 Accumulation"]
  N2["6.2 Series and repeated effects"]
  N3["6.3 Long-run behaviour"]
  N4["Synthesis"]
  N0 --> N1 --> N2 --> N3 --> N4 --> Q["EX016"]

[Learning outcomes]
    |
    v
[6.1 Accumulation]
    |
    v
[6.2 Series and repeated effects]
    |
    v
[6.3 Long-run behaviour]
    |
    v
[Synthesis]
    |
    v
[Worked example] --> [Exercise EX016]

Learning outcomes

You will (1) use integrals and series for accumulation; (2) model fee accumulation and decay; (3) estimate long-run behaviour.

6.1 Accumulation

Integrals measure accumulation over time. Jonas uses them to model how credits accumulate in a reserve contract that receives a small fee on each transfer. If transfers occur at rate r(t) and the fee is a fraction f, the reserve after time T is the integral of f times r(t) from zero to T, minus any withdrawals. He compares this model with a synthetic month of data and finds the reserve slightly smaller than predicted. Investigating the gap reveals rounding down on each fee, a small difference that adds up over thousands of transfers.

6.2 Series and repeated effects

Many small repeated effects form series. Jonas computes the cumulative rounding loss from per-transfer truncation over many transfers and shows that it is bounded by the number of transfers times one unit, a simple bound that lets the cooperative decide whether the loss matters. The EthTrust specification warns that rounding can be abused when an operation can be repeated, so Jonas notes that the design must ensure rounding never favours the person initiating a transaction. Series analysis makes repeated small effects visible before they become disputes.

6.3 Long-run behaviour

Jonas also models credits that expire after a year to encourage use, which produces exponential-like decay of old credits. Using integrals, he estimates the steady-state supply when issuance and expiry balance. He checks the model against simulated data and finds agreement within two percent. Long-run models help the board understand what the system will look like in five years, and give security monitoring a reference: a supply far from the steady state indicates either changed behaviour or a fault. He records the model's assumptions.

Worked example

The reserve model reveals truncation loss bounded by one unit per transfer, and a steady-state supply model matches simulation within 2%.

Synthetic case. Data and models are synthetic; EEA EthTrust is a real source.

Errors and limits

Ignoring rounding direction in repeated operations can let users gain value at others' expense.

Exercises

• EX016 — Reserve formula. How is the reserve modelled? Rubric: 3 = integral of fee fraction times transfer rate minus withdrawals;; 2 = integral;; 1 = constant;; 0 = none.
• EX017 — Rounding bound. What bounds cumulative truncation loss? Rubric: 3 = number of transfers times one unit;; 2 = one unit each;; 1 = vague;; 0 = none.
• EX018 — Rounding direction. Which way must rounding never favour? Rubric: 3 = the person initiating the transaction;; 2 = initiator;; 1 = cooperative;; 0 = none.

Synthesis

Integrals and series reveal cumulative effects such as rounding and expiry that shape system behaviour.

Continue studying — chapter 06 of 50

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