Lesson 1Complex numbers: algebra, polar form, Euler's formula, roots and basic complex equationsWi review complex numbers as extension a di real line. Students work wid algebraic operations, polar an exponential forms, Euler’s formula, roots a complex numbers, an simple complex equations weh 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 definitionsDis section review fundamental function families use inna modellin. Wi analyse polynomial, rational, exponential, logarithmic, an piecewise functions, focusin pon domains, ranges, graphs, transformations, an 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 basicsWi introduce probability an statistics tools fi engineering data. Students learn probability rules, combinatorial countin, discrete an continuous distributions, expected value, variance, an interpretation a 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 convergenceDis section cover sequences an infinite series, focusin pon convergence. Wi test series usin standard criteria, build power series, compute Taylor an Maclaurin expansions, an determine radius an interval a 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 infinityWi formalize limits an continuity to support rigorous calculus. Students apply limit laws, analyse one-sided limits, handle indeterminate forms, use L’Hôpital’s rule, an study limits at infinity an asymptotic behaviour a 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 valueWi study how definite integrals model accumulated quantities inna engineering. Topics include geometric area, volumes a revolution, work by variable forces, average values, an interpretin integral expressions inna 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 equationsDis section revisit Euclidean geometry an trigonometry fi exam use. Wi study triangle congruence, circle theorems, radian measure, trigonometric identities, inverse trig functions, an solvin trigonometric 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 approximationsWi apply derivatives to analyse an approximate functions. Topics include optimization inna one variable, curve sketchin usin first an second derivatives, related rates, linearization, an differential approximations fi 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 theoremWi develop differential calculus as a rate-a-change tool. Students learn derivative rules, chain an implicit differentiation, higher-order derivatives, an di Mean Value Theorem, wid emphasis pon symbolic skills an 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 modeling)Dis section introduce linear algebra tools use inna modellin. Wi solve linear systems, manipulate matrices, compute determinants, an interpret eigenvalues an eigenvectors inna simple mechanical, electrical, an 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 partsDis section focus pon computin antiderivatives an definite integrals. Wi apply di Fundamental Theorem a Calculus, substitution, an integration by parts, an interpret integrals as signed area an 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 transformationsDis section develop three-dimensional analytic geometry usin vectors. Wi practice vector operations, dot an cross products, equations a lines an planes, distances, projections, an basic coordinate transformations between frames.
Vector addition and scalar multiplicationDot product and projectionsCross product and geometryLines and planes in 3D spaceCoordinate changes and rotations
