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 Exponential growth of logs"]
N3["Lab — Log growth and storage budget"]
N4["1.3 When linear estimates fail"]
N5["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4 --> N5
N5 --> 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 Exponential growth of logs"]
N3["Lab — Log growth and storage budget"]
N4["1.3 When linear estimates fail"]
N5["Synthesis"]
N0 --> N1 --> N2 --> N3 --> N4 --> N5 --> Q["EX001"]
[Learning outcomes]
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[1.1 The case and why calculus]
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[1.2 Exponential growth of logs]
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v
[Lab — Log growth and storage budget]
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v
[1.3 When linear estimates fail]
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v
[Synthesis]
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v
[Worked example] --> [Exercise EX001]
Learning outcomes
You will (1) use rates of change to reason about growing security workloads; (2) model exponential growth of log volume; (3) recognise when a linear estimate fails.
1.1 The case and why calculus
Sofia Mbeki is a first-year student in the five-year engineering programme at the fictional Kestrel Polytechnic University. Her industry partner throughout the course is the fictional Corvia Regional Transit, which runs buses and trams, a contactless fare system, a passenger app and depot networks. Her mentor there is security architect Daniel Ferreira. Every person, organisation and number here is synthetic. Sofia wonders why an architecture course starts with calculus. Daniel's answer is practical: architects size systems that grow, and derivatives describe how fast things change. A log pipeline, a certificate fleet or a rule base that grows faster than expected becomes a security problem long before anyone notices.
1.2 Exponential growth of logs
Corvia's fare-system logs grow about four per cent per month as new devices are installed. Sofia models volume as V(t) = V0 e^(kt), where k is the monthly growth rate in continuous form. The derivative dV/dt = kV shows that the increase each month is proportional to the current volume, so absolute growth accelerates. The lab below computes the volume after two years from a synthetic starting point and the time until the current storage budget is exceeded. The result shows the budget runs out more than a year earlier than a straight-line estimate predicts, which matters because dropped logs blind the security monitoring team.
Lab — Log growth and storage budget
# Synthetic log volume: V(t) = V0 * e^(k t), k from 4%/month; compare with tangent-line forecast.
import math
V0, budget = 8.0, 20.0 # TB at month 0, storage budget in TB
k = math.log(1.04)
exp_v = lambda t: V0 * math.exp(k * t)
lin_v = lambda t: V0 + k * V0 * t # tangent line at t=0
m = next(t for t in range(200) if exp_v(t) > budget)
ml = next(t for t in range(200) if lin_v(t) > budget)
print(f"month 24: exponential {exp_v(24):.1f} TB, linear {lin_v(24):.1f} TB")
print(f"budget {budget} TB exceeded: exponential month {m}, linear month {ml}")
Run with python3 lab_growth.py. Verified output:
month 24: exponential 20.5 TB, linear 15.5 TB
budget 20.0 TB exceeded: exponential month 24, linear month 39
1.3 When linear estimates fail
A linear estimate assumes a constant increase per month and is accurate only over short periods. Sofia compares the tangent line at month zero, the derivative-based linear forecast, with the exponential model. For the first few months they agree closely; by month twenty-four they diverge by a large margin. Architects often plan capacity from last year's increase, which is a linear view of an exponential process. The lesson is general: whenever a quantity grows in proportion to itself, such as connected devices or stored events, use a growth model and revisit it as real data arrive.
Worked example
Sofia fits a four per cent monthly growth model to synthetic log volumes, computes when storage is exceeded and shows Daniel the gap between linear and exponential forecasts.
Synthetic case. Kestrel Polytechnic, Corvia Regional Transit, Sofia, Daniel and all figures are invented.
Errors and limits
Real growth is rarely smooth; new device rollouts cause jumps. Models must be refitted to measured data.
Exercises
- EX001 — Proportional growth. What does dV/dt = kV tell Sofia about log growth? Rubric: 3 = increase each month is proportional to current volume, so absolute growth accelerates;; 2 = proportional;; 1 = constant growth;; 0 = none.
- EX002 — Budget month. In the lab, in which month is the storage budget first exceeded under the exponential model? Rubric: 3 = month 24, versus month 39 for the linear forecast;; 2 = month 24;; 1 = vague;; 0 = none.
- EX003 — Linear planning. Why is planning from last year's increase risky for log storage? Rubric: 3 = it applies a linear view to an exponential process and underestimates growth;; 2 = underestimates;; 1 = accurate;; 0 = none.
Synthesis
Calculus lets architects see how fast security workloads grow and avoid linear plans for exponential systems.