What does the calculus content in OCR Maths A cover?

It covers differentiation (first principles, the chain, product and quotient rules, standard derivatives, and implicit and parametric differentiation), the applications of differentiation (tangents and normals, stationary points, optimisation and connected rates), integration (indefinite and definite integrals, areas, and the trapezium rule), integration techniques (substitution, by parts and partial fractions), first-order differential equations solved by separation of variables, and numerical methods (change of sign, iteration and Newton-Raphson). Calculus is examined across all three papers and underpins much of mechanics.

Which calculus topics carry the most marks in OCR Maths A?

Differentiation and integration are the heaviest single topics, because they appear in long structured questions on every paper and feed into mechanics. Optimisation and area questions are reliable high-tariff items, differential equations bring exponential modelling, and numerical methods provide a self-contained block of marks. Fluency with the chain rule and the standard derivatives and integrals is the highest-value skill, since almost every calculus question begins there.

What techniques recur across the OCR calculus content?

The chain, product and quotient rules; the standard derivatives and integrals of powers, exponentials, logarithms and trigonometric functions; classifying stationary points with the second derivative; optimisation by expressing a quantity in one variable; integration by substitution and by parts; splitting a rational function into partial fractions before integrating; separating variables to solve a differential equation; and the change-of-sign, iteration and Newton-Raphson root-finding methods. These are automatic by exam day.

What is the most common calculus mistake in OCR Maths A?

Dropping the inside derivative in the chain rule, omitting the constant of integration, and treating a region below the axis as a negative area. Other frequent errors are reversing the quotient rule order, failing to justify the nature of an optimisation solution, forgetting to change the limits or the dx in a substitution, integrating before separating a differential equation, and omitting continuity in a sign-change argument. Working in degrees rather than radians breaks every trigonometric derivative and integral.

How should I revise the OCR calculus content?

Work topic by topic against the specification, drilling each technique until it is automatic, then practise mixed problems under timed conditions. Keep a sheet of standard results: the power, chain, product and quotient rules, the standard derivatives and integrals, the second-derivative test, the by-parts and trapezium formulae, and the Newton-Raphson formula. Always work in radians for trigonometric calculus, show every step so method marks survive, and finish optimisation questions by stating the actual optimal value.

EnglandMaths

OCR A-Level Maths A Pure calculus: differentiation, integration, differential equations and numerical methods

A deep-dive OCR A-Level Mathematics A guide to the calculus Pure content: differentiation and its applications, integration and integration techniques, differential equations, and numerical methods, with the standard results and techniques OCR repeats across all three papers.

Generated by Claude Opus 4.820 min readH240/1.07-1.09Updated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

What the calculus content demands
Differentiation and its applications
Integration and its techniques
Differential equations and numerical methods
How the calculus content is examined
Check your knowledge

What the calculus content demands

Calculus is the analytical heart of OCR A-Level Mathematics A (H240). Differentiation and integration appear in long structured questions on all three papers, and the techniques carry directly into mechanics (kinematics from velocity and acceleration, work from force). The examiners reward fluent technique with the standard rules and results, and the judgement to choose and combine them in unfamiliar multi-step problems.

This guide walks through the calculus topics in specification order, then sets out the exam patterns OCR repeats. Each topic has a matching dot-point page with worked exam questions; this overview ties them together.

Differentiation and its applications

Differentiation (1.07) covers first principles, the power, chain, product and quotient rules, the standard derivatives of $e^{kx}$ , $\ln x$ , $\sin x$ , $\cos x$ and $\tan x$ , and implicit and parametric differentiation. Applications of differentiation uses the derivative to find tangents and normals, decide where a function increases or decreases, locate and classify stationary points with the second derivative, find points of inflection, solve optimisation problems, and link connected rates of change with the chain rule.

Integration and its techniques

Integration (1.08) covers indefinite and definite integrals as the reverse of differentiation, the standard integrals (including $\int x^{-1}\,dx = \ln|x| + c$ ), the area under a curve and between two curves, and the trapezium rule for a numerical estimate. Integration techniques adds substitution, integration by parts, integration with partial fractions, the $\frac{f'(x)}{f(x)}$ logarithm pattern, and using a trigonometric identity to integrate $\sin^2 x$ or $\cos^2 x$ .

Differential equations and numerical methods

Differential equations forms a first-order equation from a described rate of change and solves it by separating the variables, then fits an initial condition for the particular solution, with growth, decay and cooling as the standard models. Numerical methods (1.09) locates a root by a change of sign, solves an equation by a fixed-point iteration $x_{n+1} = g(x_n)$ (with staircase and cobweb diagrams), and uses the Newton-Raphson method, while explaining when each method fails.

How the calculus content is examined

A typical OCR profile for the calculus topics:

Technique questions. Differentiating with the rules, integrating a standard function, evaluating a definite integral, or carrying out one iteration.
Multi-step problems. Optimisation with a constraint, the area between a line and a curve, a connected-rate problem, or solving a differential equation and interpreting it.
Method-and-justify items. Classifying a stationary point, justifying a maximum, or arguing a root by a sign change with continuity.
Synoptic links. Calculus woven into mechanics and into the exponential and trigonometric functions of the advanced pure content.

Check your knowledge

A mix of recall and technique questions covering the calculus content. Attempt them under timed conditions, then check against the solutions.

Differentiate $y = (2x + 1)^5$ . (2 marks)
Find $\displaystyle\int 6x^2\,dx$ . (1 mark)
State the second-derivative condition for a minimum. (1 mark)
Evaluate $\displaystyle\int_0^1 e^{2x}\,dx$ , leaving your answer in terms of $e$ . (2 marks)
Write the Newton-Raphson iteration formula. (1 mark)
Separate the variables in $\dfrac{dy}{dx} = 2xy$ . (2 marks)

Solutions

Step 1: Differentiate using the chain rule

The chain rule states that $\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}$ where $u$ is the inner function. Here $u = 2x + 1$ , so the derivative of the outer function $u^5$ gives the factor of 5, and the derivative of $u$ gives the factor of 2.

\dfrac{dy}{dx} = 5(2x + 1)^4 \cdot 2 = 10(2x + 1)^4

Step 2: Integrate a power of $x$

To integrate $x^n$ , increase the power by 1 and divide by the new power. For $6x^2$ , this gives $\dfrac{6x^3}{3} = 2x^3$ , plus an arbitrary constant of integration.

\int 6x^2\,dx = 2x^3 + c

Step 3: Identify the second-derivative condition for a minimum

At a stationary point (where $\dfrac{dy}{dx} = 0$ ), a positive second derivative means the curve is concave up at that point, which confirms a local minimum. A negative value would indicate a maximum.

\dfrac{d^2y}{dx^2} > 0

Step 4: Evaluate the definite integral of an exponential

Integrating $e^{2x}$ gives $\tfrac{1}{2}e^{2x}$ because the chain rule in reverse requires dividing by the derivative of the exponent, which is 2. Substituting the limits then gives the final value.

\left[\tfrac{1}{2}e^{2x}\right]_0^1 = \tfrac{1}{2}e^2 - \tfrac{1}{2} = \tfrac{1}{2}(e^2 - 1)

Step 5: State the Newton-Raphson iteration formula

Newton-Raphson refines an initial estimate of a root by following the tangent line to the $x$ -axis at each step. Each iteration uses the function value and its derivative at the current approximation.

x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}

Step 6: Separate the variables

To separate variables, rewrite the equation so all $y$ terms (including $dy$ ) are on one side and all $x$ terms (including $dx$ ) are on the other. Each side can then be integrated independently.

\dfrac{1}{y}\,dy = 2x\,dx

Sources & how we know this

OCR A Level Mathematics A (H240) specification — OCR (2017)