Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Calculus — Topic 5.7
📋 Learning Intentions
- Identify the quantity to be optimised and express it as a function of a single variable using constraint equations.
- Find the derivative of the objective function and set it equal to zero to locate critical points.
- Determine whether a critical point is a local maximum or local minimum using the second derivative test or sign analysis of the first derivative.
- Justify that a critical point gives the global optimum within a given domain, including checking endpoints.
- Interpret the optimal value in context, stating appropriate units and confirming the result is physically meaningful.
- Use the GDC to graph the objective function and confirm the optimum numerically.
Real-World Introduction
A packaging company needs to manufacture 500 cm³ cylindrical cans using the least possible aluminium. Making the can taller saves on the circular ends but increases the curved surface; making it shorter does the opposite. There is exactly one radius that balances these competing effects and minimises total material — and calculus finds it.
Engineers at Airbus use the same principle to minimise fuel consumption by choosing the optimal cruising altitude. Farmers use it to enclose the maximum area with a fixed length of fencing. Economists use it to maximise profit given a cost constraint. In every case the mathematical structure is identical: one quantity to optimise, one or more constraints, and a derivative set to zero.
🔍 Why This Matters Optimisation is the mathematical engine behind engineering design, logistics, economics, and machine learning. Every time an algorithm finds the "best" solution — fastest route, cheapest design, highest yield — it is solving a version of the problem studied here.
Core Concepts & Definitions
Before differentiating anything, the problem must be translated into algebra. Two distinct functions appear in every optimisation problem: the quantity being optimised (the objective function ) and any equation linking the variables together (the constraint ). Recognising which is which is the first — and most important — step.
📖 Key Definition — Objective Function The objective function is the expression representing the quantity to be maximised or minimised. It is written in terms of a single variable after the constraint has been used to eliminate all other variables. For example, if area \( A = xy \) and the constraint gives \( y = 20 - 2x \), the objective function becomes \( A(x) = x(20 - 2x) = 20x - 2x^2 \).
The constraint is an equation that comes from the physical conditions of the problem — a fixed perimeter, a fixed volume, a fixed budget. It is used to reduce the objective function to one variable only. Without this reduction, differentiation is not possible in the standard