A-Level Maths: Applications of Differentiation

Knowing how to differentiate is half the story; A-Level papers are really testing what you do with the derivative. This quiz covers the application layer: curve analysis, optimisation and rates of change, the contexts where calculus earns its exam weight.

The curve-analysis questions work through the standard toolkit on concrete functions: locating stationary points of x⁴ − 4x³ + 10 and classifying them with the second derivative, finding the points of inflection of x⁴ − 6x² + 5 where concavity genuinely changes, determining the interval on which x³ − 6x² + 9x + 5 is convex (f'' > 0 gives x > 2), and identifying which functions are concave everywhere — a question that requires checking second derivatives across whole domains rather than at points. The inflection-condition question states the A-Level test precisely: f''(a) = 0 together with a sign change in concavity.

The optimisation questions use the two classic setups: the open-topped box cut from a 12 cm square sheet, where the volume expression x(12 − 2x)² must be constructed before any calculus happens — the modelling step that carries its own marks — and the maximum area of a rectangle with fixed perimeter, resolved at the square.

Rates of change appear in the exam's favourite form: the inflating spherical balloon, connecting dV/dt to dr/dt through the chain rule at a specific radius. Kinematics completes the set with a particle whose position is t³ − 6t² + 9t, asking when the velocity vanishes — the bridge into the mechanics paper.

Every explanation shows the differentiation, the algebra and the interpretation as separate steps, mirroring how method marks are awarded. Every computed value has been checked.

• Locate and classify stationary points with the second derivative test
• Find points of inflection via concavity sign changes
• Determine intervals of convexity and concavity
• Construct and optimise models: boxes, areas, fixed perimeters
• Connect rates of change through the chain rule and apply calculus to kinematics

Topic scope follows section G (Differentiation — applications) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: tangents and normals, stationary points, convexity and inflection, increasing/decreasing functions, optimisation and rates of change.