Lesson 1Complex numbers: algebra, polar form, Euler's formula, roots and basic complex equationsWe look at complex numbers as an extension of real numbers. You'll practise algebraic operations, polar and exponential forms, Euler’s formula, roots of complex numbers, and simple equations related 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 part reviews main function types used in modelling. We check out polynomial, rational, exponential, logarithmic, and piecewise functions, looking at domains, ranges, graphs, transformations, and inverses.
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 stats tools for engineering data. You'll learn rules, counting combos, discrete and continuous distributions, expected value, variance, and how to read basic stats.
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 handles sequences and series, especially convergence. We test with standard methods, build power series, do 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 make limits and continuity solid for proper calculus. You'll use limit laws, one-sided limits, indeterminate forms, L’Hôpital’s rule, and limits at infinity plus asymptotic function behaviour.
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, revolution volumes, variable force work, averages, and real problem integrals.
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 equationsWe revisit Euclidean geometry and trig for exams. Study triangle congruence, circle theorems, radians, trig identities, inverse trig, 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. Covers one-variable optimisation, curve sketching with first and second derivatives, related rates, linearisation, and approximations.
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 for rates of change. Learn rules, chain and implicit differentiation, higher derivatives, Mean Value Theorem, stressing symbols 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)Intro to linear algebra for modelling. Solve systems, handle matrices, determinants, and interpret eigenvalues/eigenvectors in basic mechanical, electrical, 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 partsFocus on antiderivatives and definite integrals. Use Fundamental Theorem, substitution, parts, see 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 transformationsBuild 3D analytic geometry with vectors. Practise operations, dot/cross products, line/plane equations, distances, projections, basic coordinate changes.
Vector addition and scalar multiplicationDot product and projectionsCross product and geometryLines and planes in 3D spaceCoordinate changes and rotations
