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Identify critical, stationary and extreme points, classify local maxima and minima with the first and second derivative tests, and solve optimisation problems by finding absolute maxima and minima on a closed interval.
Read articleUse the sign of the first derivative to find where a function increases or decreases, prove that a function is increasing or decreasing on an interval, and solve applied problems on intervals of increase and decrease.
Read articleFind equations of tangents and normals, calculate the angle between two curves where they meet, solve related-rates problems, and apply derivatives to rates of change of geometric and physical quantities.
Read articleUse logarithmic differentiation for functions with variable powers and long products, find second-order derivatives of explicit and parametric functions, and choose the right differentiation technique for any function.
Read articleDifferentiate composite functions with the chain rule, find derivatives of implicit functions and of curves given in parametric form, and differentiate one function with respect to another.
Read articleExamine the continuity and differentiability of the modulus and greatest integer functions, derive the derivatives of exponential, logarithmic and inverse trigonometric functions, and relate continuity to differentiability.
Read articleTest a function for continuity at a point and over an interval, use the algebra of continuous functions for sums, products and quotients, and recognise removable, jump and other discontinuities.
Read articleDecide whether a system of linear equations is consistent and how many solutions it has, solve systems with a unique solution using the inverse of the coefficient matrix, and verify the solution.
Read articleFind the area of a triangle and test collinearity with determinants, form the adjoint of a square matrix and use its properties, and compute the inverse of a 2x2 or 3x3 matrix by the adjoint method.
Read articleEvaluate determinants of order 2 and 3, find minors and cofactors, use the properties of determinants to avoid full expansion, and solve problems such as factorising determinants with algebraic entries.
Read articleMultiply matrices after checking their orders and see why the product is not commutative, find non-zero matrices whose product is zero, and tell singular from non-singular matrices while proving that an inverse is unique.
Read articleClassify matrices by type and order, prove and use the properties of symmetric and skew-symmetric matrices, add, subtract and multiply matrices by scalars, and verify the properties of the transpose.
Read articleProve and use the co-function identities such as inverse sine plus inverse cosine equals pi by two, the inverse tangent addition and subtraction formulae, the double and triple angle forms, and solve equations with inverse functions.
Read articleDefine the six inverse trigonometric functions with their domains, ranges and principal branches, sketch their graphs, evaluate principal values, and convert one inverse trigonometric function into another.
Read articleClassify functions as one-one, many-one, into or onto, find when a function is invertible, form composite functions and inverses, and sketch a function with its inverse as reflections in the line y = x.
Read articleDefine relations on a set and recognise the empty, universal and identity relations, test a relation for the reflexive, symmetric and transitive properties, and decide when it is an equivalence relation.
Read articleDefine probability through the axioms on sets, find the probabilities of 'not', 'and' and 'or' events with set operations, and use the addition theorem for two and three events.
Read articleFind the range of ungrouped and grouped data, calculate the mean deviation about the mean for both kinds of data, and use measures of dispersion to compare how variable two data sets are.
Read articleSee the derivative as an instantaneous rate of change and the slope of a tangent, differentiate simple functions from first principles, use the sum, product and quotient rules, and apply the standard derivative formulae.
Read articleUnderstand limits through left-hand and right-hand limits, evaluate standard algebraic, trigonometric, exponential and logarithmic limits, resolve indeterminate forms, and combine limits with the fundamental theorems.
Read articleIdentify the coordinate axes, planes and octants in three dimensions and plot points, derive and use the distance formula in space, and apply the section and midpoint formulae to points dividing a segment.
Read articleUse the standard equation of a hyperbola and the relation between its semi-axes and eccentricity, read its transverse and conjugate axes, vertices, foci, latus rectum and directrices, and apply the focal property.
Read articleUse the standard equation of an ellipse and the relation between its semi-axes and eccentricity, read its vertices, foci, latus rectum and directrices, build equations from given data, and apply the focal property.
Read articleSee conics as sections of a double cone with a focus, directrix and eccentricity, derive and sketch the four standard parabolas, read their focus, vertex, directrix, axis and latus rectum, and build equations from a focus and directrix.
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