Calculus Applications in Environmental Modelling
Using differentiation to understand and optimise environmental systems — he tātai taiao
Learning intentions — Level 3 Calculus (AS 91578, external, 6 credits)
Mātauranga context — He Tātai Taiao
He tātai taiao — mathematical modelling of the natural world
Māori navigators, tohunga, and kaitiaki have always observed patterns in the natural world — the movement of stars, the rise and fall of tides, the growth of kaimoana populations. Calculus is a modern tool for quantifying these same patterns. When we differentiate a population model, we are asking the same question a kaitiaki asks: how fast is this changing, and is it sustainable?
Instantaneous rate of change
f′(x) gives the rate of change at a specific point — not an average, but the exact speed of change at that instant.
Optimisation
Setting f′(x) = 0 finds stationary points — where a quantity is at its maximum, minimum, or plateau.
Gradient function sketch
Where f(x) increases, f′(x) > 0. Where f(x) has a peak, f′(x) = 0. Where f(x) decreases, f′(x) < 0.
Second derivative test
f″(x) > 0 at a stationary point → local minimum. f″(x) < 0 → local maximum.
Problem 1 — Kiwi population recovery
Kiwi population model Kaitiakitanga
In this scenario, the brown kiwi population P in a fenced sanctuary is modelled by the function below. The model is illustrative — it is not real sanctuary data.
(a) Find P′(t), the rate of change of the population.
(b) Calculate the rate of change at t = 3 years. Interpret your answer in context.
In context, P′(3) = _____ means the population is increasing/decreasing at a rate of _____ kiwi per year at t = 3.
(c) Find the time(s) when the population is growing fastest. Show full working and justify your answer using f″(t).
(d) After how many years does the sanctuary kiwi population start to decline? Justify using calculus.
Problem 2 — Estuary pollution decay
Pollution concentration model Environmental Science
After an industrial spill, the concentration of a pollutant in a Northland estuary is modelled by:
Recall: if C(t) = Aekt, then C′(t) = Akekt
(a) Find C′(t). What does the sign of C′(t) tell you about the pollutant over time?
(b) Find the rate at which the concentration is decreasing at t = 5 days. Give your answer to 3 significant figures.
(c) The safe level for swimming is 20 mg/L. Using algebra (not calculus), find when the estuary reaches this level. Show full working.
(d) Excellence extension: Is the rate of decrease speeding up or slowing down over time? Use C″(t) to justify your answer, and explain what this means for the estuary ecosystem.
Problem 3 — Optimising a mahinga kai wetland
Wetland design optimisation Mahinga Kai
A hapū is restoring a rectangular mahinga kai (food gathering) wetland on land bordered by a straight waterway. The waterway forms one side, so only 3 sides need fencing. They have 240 metres of fencing available.
Let the dimension parallel to the waterway = x metres, and the perpendicular dimension = y metres.
(a) Write an expression for y in terms of x using the fencing constraint.
(b) Write an expression for the area A(x) of the wetland in terms of x only.
(c) Differentiate A(x) and find the value of x that maximises the area. Show full working.
(d) Calculate the maximum area and the corresponding value of y. State your answer in context.
(e) Verify this is a maximum using the second derivative test.
(f) Merit/Excellence: The hapū also wants the wetland to be at least 40 m wide (perpendicular to the waterway). What is the maximum area now? How does this constraint change the solution?
Problem 4 — Gradient function sketch
The graph below shows an illustrative function f(t) over 15 years (t = 0 to t = 15). It is not real data and it is not a published emissions projection — it is a shape to practise on. Read it purely as a quantity that rises and falls over time.
Sketch the gradient function f′(t) on the second set of axes. Mark clearly where f′(t) = 0, where f′(t) > 0, and where f′(t) < 0.
Teacher: sketch the function on this copy before distributing
Explain what f′(t) = 0 means for the quantity f(t) — and then, in one sentence, what the same statement would mean if f(t) were a country's annual greenhouse gas emissions:
Self-assessment against Level 3 Calculus
| Skill | Not yet | Achievement | Merit | Excellence |
|---|---|---|---|---|
| Differentiate polynomial functions | ☐ | ☐ | ☐ | ☐ |
| Differentiate exponential functions | ☐ | ☐ | ☐ | ☐ |
| Interpret derivatives in context | ☐ | ☐ | ☐ | ☐ |
| Optimisation (max/min) | ☐ | ☐ | ☐ | ☐ |
| Sketch gradient functions | ☐ | ☐ | ☐ | ☐ |
The problem I found most challenging and why:
Hononga Marautanga · Curriculum Alignment
Curriculum alignment for this handout has not yet been verified against the live curriculum statements. A generated placeholder that stood here was removed on 2026-08-29 because it matched no real statement.
Paearu Angitu · Success Criteria
By the end of this activity you will be able to demonstrate:
- Achievement: Apply correct differentiation rules and find numerical answers.
- Merit: Interpret results in context with clear written reasoning.
- Excellence: Show insightful extension, generalise your findings, or modify constraints with justification.
Kaiako Planning Snapshot
Resources already provided. What to print: this handout (3–4 pages). Self-contained — no additional photocopying required.
Classroom use for kaiako: 3–4 lessons (see pacing below). Use as an NCEA practice assessment or teacher-led investigation sequence. Linked next step: Te Wānanga to generate a full lesson plan, or save resources to My Kete.
NZ pedagogy basis: Anchored to The New Zealand Curriculum (2007), Mathematics and Statistics, Level 8, and to NCEA Level 3 Achievement Standard AS 91578. (The live Te Mātaiaho corpus ends at Phase 4, Years 9–10, so a senior course is not anchored there.) Suitable for NZ kura and schools.
For teachers and ākonga:
Inclusion: ESOL / ELL ākonga — mathematical language is defined in context. UDL / neurodiverse learners — problems are chunked with explicit scaffolding. ADHD-friendly: clear step structure, short sub-tasks within each problem.
Differentiation: Entry — Problems 1 and 3 (polynomial). On-level — Problem 2 (chain rule with exponential). Extension — Problem 3(f), Problem 2(d), constraint modification. Scaffold down with worked examples; stretch by removing structured supports.
Pacing: Lesson 1: Problem 1. Lesson 2: Problem 2. Lessons 3–4: Problems 3–4. Self-assessment can be peer-reviewed.
Teacher notes
- Level 3 Calculus alignment: Problems 1–3 cover Achievement and Merit criteria. Problem 3(f) and Problem 2(d) target Excellence (insightful extension, constraint modification).
- Calculator use: Graphic calculators permitted. Students should verify by-hand differentiation with GC but show algebraic working clearly.
- Context integrity: The kiwi population figures are illustrative. If students want to use real DOC data, direct them to the DOC website — this makes an excellent extension research task.
- Problem 4: You supply the curve. Draw one with at least one clear local maximum and one local minimum, plus a section of rapid increase, so ākonga practise all three cases. A sine-like curve shifted up works well. Because the curve is yours, the problem deliberately does not claim to show real emissions data; if you want the real thing, use the published national greenhouse gas inventory and say where it came from.
- Differentiation: Problems 1 and 3 are polynomial — straightforward application. Problem 2 requires chain rule with ekx. Ensure students have covered this before assigning Problem 2.
- Pacing: 3–4 lessons. Problem 1 (lesson 1), Problem 2 (lesson 2), Problem 3 + Problem 4 (lesson 3–4). Self-assessment can be peer-reviewed.