Calculus Applications in Environmental Modelling
Using differentiation to understand and optimise environmental systems — he tātai taiao
Learning intentions — Level 3 Calculus (4 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
DOC modelling estimates the brown kiwi population P in a protected sanctuary can be modelled by:
(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 represents a smoothed model of annual NZ greenhouse gas emissions over 15 years (t = 0 to t = 15), where positive values represent net emissions increasing.
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 in the context of NZ 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:
Curriculum alignment
- Level 3 Calculus (4 credits) — Apply differentiation methods in solving problems: Apply rules of differentiation to polynomial, exponential, trigonometric, and other functions. Find and interpret rates of change, stationary points, and graphs of gradient functions.
- Mathematical reasoning: Construct and communicate mathematical arguments using correct notation and terminology, including justification using second derivative.
- Physical World / Environmental science (cross-curricular): Model physical and environmental phenomena using mathematical functions. Interpret rate of change in the context of real-world systems.
NZ Curriculum Level 8 (Year 13). NCEA Level 3 Mathematics with Calculus — (external, 4 credits). Mātauranga Māori integration through kaitiakitanga and mahinga kai contexts.
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: Aligned to NCEA Level 3 and Te Mātaiaho. 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: Draw a curve with at least one clear local maximum and one local minimum, plus a section of rapid increase, so students practise all three cases. A sine-like curve shifted up works well.
- 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.