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Pure mathematics: calculus

6 dot points across 6 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you use the derivative to find tangents, classify stationary points, and solve optimisation and connected-rate problems?

Tangents and normals, increasing and decreasing functions, stationary points and their nature using the second derivative, points of inflection, optimisation, and connected rates of change.
A focused answer to the OCR A-Level Mathematics A applications of differentiation content, covering tangents and normals, increasing and decreasing functions, stationary points and their classification by the second derivative, points of inflection, optimisation problems, and connected rates of change.
11 min answer →

How do you form a differential equation from a rate of change and solve it by separating the variables?

Forming first-order differential equations from a context, solving them by separation of variables, finding particular solutions from initial conditions, and interpreting the solution in modelling.
A focused answer to the OCR A-Level Mathematics A differential equations content, covering forming a first-order differential equation from a described rate of change, solving by separation of variables, applying an initial condition to find a particular solution, and interpreting the result in growth, decay and cooling models.
11 min answer →

How do you find the rate at which a quantity changes, from first principles and using the standard rules?

Differentiation from first principles, the power rule, the chain, product and quotient rules, derivatives of standard functions including exponentials, logarithms and trigonometric functions, and implicit and parametric differentiation.
A focused answer to the OCR A-Level Mathematics A differentiation content, covering differentiation from first principles, the power rule, the chain, product and quotient rules, derivatives of exponential, logarithmic and trigonometric functions, and implicit and parametric differentiation.
12 min answer →

How do you integrate harder functions using substitution, integration by parts, and partial fractions?

Integration by substitution, integration by parts, integration using partial fractions, and integrating expressions of the form f prime over f and products reducible by a trigonometric identity.
A focused answer to the OCR A-Level Mathematics A advanced integration content, covering integration by substitution, integration by parts, integrating with partial fractions, the f prime over f logarithm pattern, and using trigonometric identities to integrate products such as sine squared.
12 min answer →

How do you reverse differentiation to find areas under and between curves, and integrate the standard functions?

Indefinite and definite integrals as the reverse of differentiation, the integrals of standard functions, the area under a curve and between two curves, and the trapezium rule for numerical integration.
A focused answer to the OCR A-Level Mathematics A integration content, covering indefinite and definite integrals as the reverse of differentiation, the integrals of standard functions, the area under a curve and between two curves, and the trapezium rule for estimating an integral numerically.
12 min answer →

How do you locate and approximate the roots of an equation when an exact solution is not available?

Locating roots by change of sign, iterative methods of the form x equals g of x, the Newton-Raphson method, and the conditions under which these numerical methods fail.
A focused answer to the OCR A-Level Mathematics A numerical methods content, covering locating roots by a change of sign, fixed-point iteration of the form x equals g of x with staircase and cobweb diagrams, the Newton-Raphson method, and the situations in which each method fails.
11 min answer →