IB Maths AI SL · Unit 5: Calculus
Exam-style IB Maths AI SL optimisation questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
1) Write down what to maximise/minimise. 2) Express in one variable. 3) Differentiate + set = 0. 4) Verify it's a max not a min. Every SL AI optimisation question follows this frame. Optimisation — the 4-step method overview →
$P(x) = -2x^2 + 120x - 500$. Find the exact number of components that maximises daily profit.
1. Derivative: $P'(x) = -4x + 120$.
2. Stationary point: $P'(x) = 0$.
3. Solve: $-4x + 120 = 0 \Rightarrow 4x = 120$.
4. Evaluate: $x = \mathbf{30}$ components.
$V(x) = 27x - 0.75x^3$ (cm$^3$). Find the exact positive $x$ that maximises the volume.
1. Derivative: $V'(x) = 27 - 2.25x^2$.
2. Set to zero: $27 - 2.25x^2 = 0$.
3. Isolate: $x^2 = \frac{27}{2.25} = 12$.
4. Square root: $x = \sqrt{12}$.
5. Positive length: $x \approx \mathbf{3.46\text{ cm}}$.
$A(r) = 2\pi r^2 + \frac{2000\pi}{r}$. Find the radius (3 s.f.) that minimises surface area.
1. Recognise: fastest via GDC graph.
2. Enter: $Y_1 = 2\pi x^2 + \frac{2000\pi}{x}$.
3. Window: $x > 0$.
4. Locate minimum: G-Solv $\to$ MIN.
5. Read $x$: $7.9370\ldots$
6. State: $\mathbf{r = 7.94\text{ cm}}$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
A company manufactures and sells \(x\) electronics units per week. Their weekly cost function, in euros, is given by \(C(x) = 0.5x^2 - 10x + 200\). The revenue from selling \(x\) units is modelled by \(R(x) = 40x\).
A cylindrical tin can must have a fixed volume of \(500 \text{ cm}^3\). The cost of the metal for the top and bottom circles is \(3 \text{ cents per cm}^2\), and the cost of the metal for the curved side is \(2 \text{ cents per cm}^2\).
Let \(r\) be the radius of the base. Show that the total cost of the can is given by \(C(r) = 6\pi r^2 + \frac{2000}{r}\).
Use differential calculus to find the exact value of \(r\) that minimizes the total cost.
A farmer wants to construct a rectangular enclosure against a straight river. He has exactly \(100 \text{ m}\) of fencing and will not use fencing along the riverbank. Let \(x\) be the width of the enclosure (perpendicular to the river).
There are 12 exam-style optimisation questions in the AI SL question bank (Paper 1: 4 · Paper 2: 8), graded 3 hard, 5 very hard, 4 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 4 · Paper 2: 8. Practise with your GDC — AI papers expect calculator methods throughout.
Open the AI SL Unit 5 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.