IA ideas library
Calculus IA ideas: optimisation, rates of change and area
Calculus ideas let you show mathematics that is clearly at the level of the course: finding a maximum or minimum, working out a rate of change, or calculating an area or volume that cannot be found with a formula. The best calculus IAs start from a real object or decision — a can, a road, a jump — and use calculus because it is the natural tool, not because it looks impressive.
Most of these ideas work for SL with differentiation and definite integrals; HL students can extend them with volumes of revolution, related rates, implicit differentiation or numerical methods.
Next step: shortlist ideas with the free IA topic explorer, which filters starting points by course, interest and how easy the data is to get. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.
How calculus explorations reach the top bands
Personal engagement (C)
Measure the real object yourself, choose the constraint that matters, and say why. Engagement shows when the optimum you find surprises you and you go and find out why reality differs.
Reflection (D)
Always compare the theoretical optimum with the real thing and explain the difference (manufacturing, strength, cost, comfort). That comparison is where Criterion D is earned.
Use of mathematics (E)
SL: A function set up from the situation step by step, differentiated or integrated correctly, with the maximum or minimum checked (second derivative or sign change) and the answer interpreted in context.
HL: HL calculus such as related rates, implicit differentiation, volumes of revolution, integration by parts or a numerical method with an error estimate, used because the problem needs it.
An HL version
Add a second constraint or variable, use an HL technique (volume of revolution, implicit differentiation, integration by parts) where it is genuinely needed, and estimate the error in any numerical method.
Calculus & optimisation ideas (10)
- Why aren't drinks cans the optimal shape?AA SLAI SLAA HLAI HLAccessible
- What is the best shape for a rain gutter?AA SLAI SLAA HLAI HLAccessible
- What is the volume of an egg?AA HLAI HLAA SLSolid
- Where should the 125 ml line go on a glass?AA HLAI HLSolid
- Designing a smooth roller-coaster drop with piecewise functionsAA SLAA HLAI SLAI HLAmbitious
- How big is my local lake? Estimating area with the trapezoidal ruleAI SLAI HLAA SLAA HLAccessible
- The lifeguard problem: what is the fastest route to a swimmer?AA SLAA HLAI SLAI HLSolid
- Will the sofa fit round the corner? The longest object through a corridorAA SLAA HLAI SLAI HLSolid
- Where is the best seat in the cinema?AA SLAA HLAI SLAI HLSolid
- How fast is a sprinter at each moment? Differentiating 100 m splitsAA SLAA HLAI SLAI HLSolid
Also relevant (30)
- Is a hanging chain a parabola? Catenary versus quadratic modelsAA SLAA HLAI SLAI HLSolid
- How long does a bottle take to drain? Testing Torricelli's lawAA SLAA HLAI SLAI HLSolid
- Why do stopping distances grow so fast? Modelling the Highway Code tableAA SLAI SLAccessible
- How long does a skydiver take to reach terminal velocity?AA HLAI HLAmbitious
- Estimating π by dropping needles: how fast does the estimate improve?AA HLAA SLAI SLAI HLSolid
- Is a satellite dish really a parabola, and where is its focus?AA SLAA HLSolid
- Measuring inequality: fitting Lorenz curves and computing the Gini coefficientAA SLAA HLAI SLAI HLSolid
- What price maximises profit at our school bake sale?AA SLAI SLAA HLAI HLSolid
- Where does e come from? Compounding more and more oftenAA SLAA HLSolid
- What is the best angle to shoot a free throw?AA SLAA HLAI SLAI HLSolid
- Why don't long jumpers take off at 45°?AA SLAA HLAI SLAI HLSolid
- When should a Formula 1 driver pit? Modelling tyre degradationAA SLAI SLAA HLAI HLAmbitious
- Where should a winger shoot from? Maximising the angle to goalAA SLAA HLAI SLAI HLSolid
- What is the best tilt for a solar panel at my home?AA SLAI SLAA HLAI HLSolid
- Planning the shortest delivery round: bounds for the travelling salesman problemAI HLSolid
- Why does 1 + 1/4 + 1/9 + … equal π²/6?AA HLAmbitious
- How does a calculator work out sin(x)?AA HLAA SLSolid
- How are letters drawn on a screen? Bézier curvesAA HLAA SLAI HLSolid
- What curve do straight lines make in string art?AA SLAA HLSolid
- What speed minimises the total cost of a long drive?AA SLAI SLAA HLAI HLSolid
- Estimating the area of a leaf with random pointsAA SLAI SLAA HLAI HLAccessible
- Why does the ancient square-root method double the correct digits?AA SLAA HLSolid
- Trapezoidal rule or Simpson's rule? Areas under the normal curveAA SLAA HLAI SLSolid
- When the computer gets it wrong: rounding error and cancellationAA HLAA SLAI HLAmbitious
- Measuring the area of an irregular field with trianglesAA SLAI SLAccessible
- How high did the lift go? Integrating a phone's accelerometerAA SLAA HLAI HLSolid
- The path of a bike valve: parametric equations of a cycloidAA HLAI HLAmbitious
- Modelling a bus journey between two stops from a phone's sensorsAA SLAI SLAI HLSolid
- Cutting a cake fairly when people value it differentlyAA SLAA HLSolid
- How much should you bid in a sealed-bid auction?AA HLAA SLAmbitious
Frequently asked questions
Is packaging optimisation too common for a Maths IA?
It is common, so a basic cylinder calculation will not stand out. Make it your own by measuring real products, adding a constraint that matters (material thickness, the lid, stacking in a box), or comparing several shapes.
Do SL students need integration by parts or substitution?
No. SL students can use the integration in their course, technology for harder integrals, or the trapezoidal rule. What matters is that you understand and explain every step you include.
Start your IA plan, free
Answer the planning questions: your question, the maths, the data and milestones back from your deadline. Then take it to your first teacher meeting.
Your plan is kept in this browser. A free account keeps it with your drafts on any device: Join free · Sign in
- Free IA self-check: guided questions give a rough band for each criterion. No upload.
- The A–E checklist as a PDF, free by email.
- Have a draft? AI feedback against A–E (Criterion A free), or a one-to-one review with Pete (€100).
Related types of IA: Optimisation & linear programming · Numerical methods & error analysis
Other topics: Modelling · Differential equations · Statistics · Probability · Geometry & Voronoi · Finance · Sport · Environment · Health & biology · Networks & graphs · Pure maths · Art & music · Games & puzzles · Optimisation · Simulation · Cryptography · Numerical methods · Fractals & chaos · Matrices & Markov · Trig & navigation · Kinematics · Voting & game theory · Surveys
By interest: Computer science · Physics and engineering · Psychology and social sciences · Geography and travel · Chemistry · Biology · Economics · Business management · Sports, exercise and health science · Environmental systems and societies · Music · Visual arts · Design technology · By course: AA SL · AA HL · AI SL · AI HL