Lesson 7: Optimisation Problems & Real-World Modelling
Applying differentiation methods, rules, optimisation, and related rates for NCEA Level 3 Calculus.
🎯 Ngā Whāinga Akoranga | Learning Intentions
Constructing mathematical models from real-world constraints to find maximum or minimum values (optimising volume, surface area, cost, distance, profit).
Formulate single-variable objective functions from word problems, differentiate to find optimal stationary points, and prove global max/min using boundary checks.
🎥 Media Anchor & Pedagogical Scaffold
Finding Absolute Extrema — Khan Academy
Video (6 min 24 sec, Khan Academy): A systematic guide to finding absolute maximum and minimum values of functions on closed intervals—the foundation for all optimisation problems. Combines first and second derivative tests with boundary checks.
🧠 1. Before Viewing (Activate & Predict)
How do industrial engineers use calculus to design packaging that minimises material cost while maximising container volume?
👁️ 2. During Viewing (Watch With a Job)
Watch the full 6¼-minute clip. In your calculus logbook, record:
- The Extreme Value Theorem process: To find absolute max/min on [a, b]: (1) evaluate f at endpoints a and b; (2) find all critical points (where f'(x) = 0) inside [a, b]; (3) evaluate f at each critical point; (4) compare all values and identify max and min.
- A worked example: Find the absolute maximum of f(x) = -x² + 4x on [0, 3]. (f'(x) = -2x + 4 = 0 → x = 2. Evaluate: f(0) = 0, f(2) = 4, f(3) = 3. Absolute max is 4 at x = 2.)
- Real-world link: Note how boundary points matter—the optimal solution might be at an endpoint (a practical constraint) rather than an interior critical point.
🗣️ 3. After Viewing & Kaiako Move (Process & Apply)
Kaiako Move: Work through classic NCEA Excellence problems: cylindrical tin can material minimization and fenced rectangular paddock maximization.
Immediate Task: Add to your Level 3 Calculus revision logbook (section 7): Optimisation Word Problem & Engineering Model Set.
⚡ Whakaoho | Do Now: From Words to One Variable (10 mins)
A rectangle has perimeter 20 m. Write its area in terms of one variable only. Do not maximise it yet. The hard part of optimisation is this step, not the differentiation.
📖 Activity 1: Optimising, Including at the Boundary (25 mins)
Finish your Do Now rectangle: maximise the area and justify the maximum with the second derivative. Then the case that separates Merit from Excellence: find the absolute maximum of f(x) = −x² + 4x on the interval [0, 3]. Solve f′(x) = −2x + 4 = 0 to get x = 2, then evaluate f(0) = 0, f(2) = 4 and f(3) = 3. Now change the interval to [0, 1] and repeat. The maximum moves to an endpoint, and no stationary point will tell you that. On a closed interval you must always check the endpoints.
📝 Activity 2: Exam-Style Practice — Model, Optimise, Interpret (20 mins)
An open-topped box is made from a 20 cm by 20 cm sheet by cutting squares of side x from each corner. Find the x that maximises volume. For Merit, state the domain of x and justify the maximum. For Excellence, explain why x = 10 is excluded and what the volume does as x approaches that value.
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 Alignment: Direct preparation for Level 3 Calculus (Apply differentiation methods in solving problems). Emphasise complete algebraic working and clear context conclusions for Merit/Excellence grades.