Lesson 1Complex numbers: algebra, polar form, Euler's formula, roots and basic complex equationsWe revise complex numbers as extensions of the real line. Students practise algebraic operations, polar and exponential forms, Euler’s formula, roots of complex numbers, and simple complex equations relevant to oscillatory systems.
Algebra of complex numbersModulus, argument, and conjugatePolar and exponential formsEuler’s formula and rotationsRoots and basic complex equationsLesson 2Functions and their properties: polynomial, rational, exponential, logarithmic, and piecewise definitionsThis section revises fundamental function families used in modelling. We analyse polynomial, rational, exponential, logarithmic, and piecewise functions, focusing on domains, ranges, graphs, transformations, and inverse relationships.
Domain and range analysisPolynomial and rational graphsExponential growth and decayLogarithmic functions and inversesPiecewise and step functionsLesson 3Probability and statistics basics: probability rules, discrete and continuous distributions, expected value, variance, combinatorics basicsWe introduce probability and statistics tools for engineering data. Students learn probability rules, combinatorial counting, discrete and continuous distributions, expected value, variance, and interpretation of basic statistical summaries.
Sample spaces and eventsAddition and multiplication rulesCombinatorics and counting methodsDiscrete and continuous variablesExpectation, variance, and spreadLesson 4Sequences and series: convergence tests, Taylor and Maclaurin series, power series representation and radius of convergenceThis section covers sequences and infinite series, focusing on convergence. We test series using standard criteria, build power series, compute Taylor and Maclaurin expansions, and determine radius and interval of convergence.
Limits of sequences and behaviorSeries convergence conceptsComparison and ratio testsPower series and convergence radiusTaylor and Maclaurin seriesLesson 5Limits and continuity: limit laws, indeterminate forms, L'Hôpital's rule, limits at infinityWe formalise limits and continuity to support rigorous calculus. Students apply limit laws, analyse one-sided limits, handle indeterminate forms, use L’Hôpital’s rule, and study limits at infinity and asymptotic behaviour of functions.
Limit laws and computationsOne-sided limits and continuityRemovable and jump discontinuitiesIndeterminate forms and algebraic tricksL’Hôpital’s rule and limits at infinityLesson 6Applications of integrals: area, volume by revolution, work, accumulation problems, average valueWe study how definite integrals model accumulated quantities in engineering. Topics include geometric area, volumes of revolution, work by variable forces, average values, and interpreting integral expressions in real problems.
Area between curves and axesVolumes by disks and washersShell method for volumesWork by variable forcesAverage value of a functionLesson 7Euclidean geometry and trigonometry: triangle properties, circle theorems, trigonometric identities, solving trig equationsThis section revisits Euclidean geometry and trigonometry for exam use. We study triangle congruence, circle theorems, radian measure, trigonometric identities, inverse trig functions, and solving trigonometric equations.
Triangle congruence and similarityCircle theorems and chordsRadian measure and arc lengthCore trigonometric identitiesSolving trigonometric equationsLesson 8Applications of derivatives: optimisation, curve sketching, related rates, linearisation and approximationsWe apply derivatives to analyse and approximate functions. Topics include optimisation in one variable, curve sketching using first and second derivatives, related rates, linearisation, and differential approximations for estimates.
Critical points and extremaFirst and second derivative testsCurve sketching strategiesRelated rates word problemsLinearization and differentialsLesson 9Differential calculus: derivative rules, implicit differentiation, higher-order derivatives, mean value theoremWe develop differential calculus as a rate-of-change tool. Students learn derivative rules, chain and implicit differentiation, higher-order derivatives, and the Mean Value Theorem, with emphasis on symbolic skills and interpretations.
Limit definition of derivativeBasic derivative rulesChain rule applicationsImplicit differentiation methodsHigher derivatives and MVTLesson 10Linear algebra essentials: systems of linear equations, matrices, determinants, eigenvalues (basic concepts relevant to modelling)This section introduces linear algebra tools used in modelling. We solve linear systems, manipulate matrices, compute determinants, and interpret eigenvalues and eigenvectors in simple mechanical, electrical, and population models.
Gaussian elimination methodsMatrix operations and inversesDeterminants and Cramer’s ruleEigenvalues and eigenvectors basicsLinear models and applicationsLesson 11Integral calculus: antiderivatives, definite integrals, Fundamental Theorem of Calculus, substitution and integration by partsThis section focuses on computing antiderivatives and definite integrals. We apply the Fundamental Theorem of Calculus, substitution, and integration by parts, and interpret integrals as signed area and accumulated change.
Antiderivatives and familiesDefinite integrals as areaFundamental Theorem of CalculusSubstitution and change of variableIntegration by parts strategiesLesson 12Vectors and analytic geometry: vector operations, dot and cross product, lines and planes in 3D, coordinate transformationsThis section develops three-dimensional analytic geometry using vectors. We practise vector operations, dot and cross products, equations of lines and planes, distances, projections, and basic coordinate transformations between frames.
Vector addition and scalar multiplicationDot product and projectionsCross product and geometryLines and planes in 3D spaceCoordinate changes and rotations
