Lesson 1Complex numbers: algebra, polar form, Euler's formula, roots and basic complex equationsWe go over complex numbers as going beyond the real numbers. Learners handle algebra 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 goes over basic function types used in modelling. We look at polynomial, rational, exponential, logarithmic, and piecewise functions, focusing on domains, ranges, graphs, changes, and inverse links.
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 bring in probability and statistics tools for engineering data. Learners pick up probability rules, counting combinations, discrete and continuous distributions, expected value, variance, and reading basic stats 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 endless series, stressing convergence. We test series with usual methods, make 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, deal with unclear forms, apply L’Hôpital’s rule, and study limits at infinity and function behaviour at ends.
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 learn how definite integrals show built-up amounts in engineering. Topics cover area shapes, revolution volumes, work by changing forces, average values, and reading integral terms in real issues.
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 part looks back at Euclidean geometry and trigonometry for exams. We study triangle matches, circle rules, 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 check and guess functions. Topics include one-variable optimization, curve drawing with first and second derivatives, linked rates, linearization, and small change guesses 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 change-rate tool. Learners get derivative rules, chain and hidden differentiation, higher derivatives, and Mean Value Theorem, stressing symbol 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 part brings in linear algebra tools for modelling. We solve line systems, work matrices, find determinants, and read 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 part focuses on finding antiderivatives and definite integrals. We use Fundamental Theorem of Calculus, substitution, and parts integration, and read integrals as signed area and built-up 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 part builds three-D analytic geometry with vectors. We do vector operations, dot and cross products, line and plane equations, distances, projections, and basic frame changes.
Vector addition and scalar multiplicationDot product and projectionsCross product and geometryLines and planes in 3D spaceCoordinate changes and rotations
