Lesson 1Complex numbers: algebra, polar form, Euler's formula, roots and basic complex equationsWe look at complex numbers as an extension of the real numbers. Learners handle algebraic operations, polar and exponential forms, Euler’s formula, roots of complex numbers, and simple complex equations for 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 part reviews basic function types used in modelling. We examine polynomial, rational, exponential, logarithmic, and piecewise functions, looking at 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 cover probability and statistics tools for engineering data. Learners pick up probability rules, counting combinations, discrete and continuous distributions, expected value, variance, and 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 part covers sequences and infinite series, focusing on convergence. We test series with standard methods, build power series, work out Taylor and Maclaurin expansions, and find 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 set out limits and continuity for proper calculus. Learners use limit laws, check one-sided limits, sort indeterminate forms, apply L’Hôpital’s rule, and study limits at infinity and function asymptotes.
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 see how definite integrals model built-up quantities in engineering. Topics cover area, volumes of revolution, work by varying forces, average values, and reading integral expressions in real scenarios.
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 goes back over Euclidean geometry and trigonometry for exams. We cover triangle congruence, circle theorems, radian measure, trig identities, inverse trig functions, and solving trig equations.
Triangle congruence and similarityCircle theorems and chordsRadian measure and arc lengthCore trigonometric identitiesSolving trigonometric equationsLesson 8Applications of derivatives: optimization, curve sketching, related rates, linearization and approximationsWe use derivatives to analyse and approximate functions. Topics include one-variable optimisation, curve sketching with first and second derivatives, related rates, linearisation, and 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 build differential calculus as a tool for rates of change. Learners master derivative rules, chain and implicit differentiation, higher-order derivatives, and the Mean Value Theorem, stressing symbolic work and meanings.
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 for modelling. We solve linear systems, work with matrices, calculate determinants, and explain eigenvalues and eigenvectors in basic 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 is about finding antiderivatives and definite integrals. We use the Fundamental Theorem of Calculus, substitution, and integration by parts, seeing 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 builds 3D analytic geometry with vectors. We do vector operations, dot and cross products, equations of lines and planes, distances, projections, and basic coordinate changes between frames.
Vector addition and scalar multiplicationDot product and projectionsCross product and geometryLines and planes in 3D spaceCoordinate changes and rotations
