Home / Learn / Mathematics

Calculus from Start

Full curriculum

1. Module 1 The Infinite and the Infinitesimal: Archimedes' Method of Exhaustion This module explores the ancient Greek method of approximating areas and volumes using inscribed and circumscribed polygons, laying the groundwork for limits.
   • Archimedes' Calculation of the Area of a ParabolaDiscover how Archimedes used a method of summing infinitely many infinitesimally small pieces to find the area under a parabolic segment.
   • Approximating Pi: Archimedes' PolygonsExamine Archimedes' use of inscribed and circumscribed polygons to bound the value of pi, illustrating the concept of convergence.
   • The Method of Indivisibles: Cavalieri's PrincipleUnderstand how Cavalieri's principle, a precursor to integration, relates the volumes of solids by comparing cross-sectional areas.
   • Zeno's Paradoxes and the Problem of MotionAnalyze Zeno's paradoxes of motion, which highlighted the conceptual difficulties in understanding infinite processes and continuity.
2. Module 2 The Dawn of the Calculus: Newton and Leibniz This module introduces the independent development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century.
   • Newton's Fluxions and FluentsExplore Newton's approach to calculus, focusing on the concepts of 'fluxions' (rates of change) and 'fluents' (changing quantities).
   • Leibniz's Differentials and IntegralsExamine Leibniz's notation and conceptualization of calculus using infinitesimals (differentials) and their summation (integrals).
   • The Tangent Problem: Finding Instantaneous SlopeUnderstand the geometric problem of finding the slope of a tangent line to a curve at a single point, a key motivation for calculus.
   • The Area Under a Curve Problem: Early IntegrationInvestigate the problem of finding the area bounded by a curve and the x-axis, a central problem solved by integration.
3. Module 3 The Derivative: Quantifying Change This module delves into the concept of the derivative, the fundamental tool for measuring instantaneous rates of change.
   • The Limit of a Difference QuotientLearn how the concept of a limit is used to define the derivative from the average rate of change.
   • The Power Rule for PolynomialsMaster the basic rule for differentiating polynomial functions, a foundational technique.
   • Derivatives of Trigonometric FunctionsCalculate the derivatives of sine, cosine, and tangent functions, essential for modeling periodic phenomena.
   • Interpreting the Derivative: Velocity and AccelerationApply the derivative to understand physical concepts like instantaneous velocity and acceleration from position functions.
4. Module 4 The Chain Rule and Implicit Differentiation This module covers advanced differentiation techniques for composite functions and implicitly defined curves.
   • The Chain Rule: Differentiating CompositionsUnderstand and apply the chain rule for differentiating functions that are compositions of other functions.
   • Product and Quotient RulesLearn the rules for differentiating products and quotients of functions.
   • Implicit Differentiation of EquationsDifferentiate equations where y is not explicitly defined as a function of x.
   • Related Rates ProblemsSolve problems where multiple quantities are changing with respect to time and their rates are related.
5. Module 5 The Integral: Accumulating Quantities This module introduces the concept of the integral as a way to accumulate quantities and find areas.
   • The Antiderivative: Reversing DifferentiationExplore the concept of the antiderivative, the reverse operation of differentiation.
   • The Definite Integral: Area Under a CurveUnderstand the definite integral as a tool for calculating the precise area under a curve between two points.
   • The Fundamental Theorem of Calculus, Part 1Discover how the derivative of an accumulating function relates to the original function.
   • The Fundamental Theorem of Calculus, Part 2Learn how to evaluate definite integrals using antiderivatives, a cornerstone of calculus.
6. Module 6 Techniques of Integration: Substitution and Beyond This module covers essential techniques for evaluating more complex integrals.
   • Integration by Substitution (u-substitution)Master the technique of substitution to simplify and solve integrals.
   • Integrals of Exponential and Logarithmic FunctionsCalculate integrals involving e^x and ln(x).
   • Integrals of Trigonometric FunctionsEvaluate integrals of trigonometric functions, including those requiring trigonometric identities.
   • Numerical Integration: The Trapezoidal RuleExplore methods for approximating definite integrals when analytical solutions are difficult or impossible.
7. Module 7 Applications of Integration: Areas, Volumes, and Work This module demonstrates how integrals are used to solve a variety of real-world problems.
   • Area Between CurvesCalculate the area enclosed between two or more curves.
   • Volumes of Revolution: Disk and Washer MethodsFind the volumes of solids generated by revolving a region around an axis.
   • Volumes of Revolution: Shell MethodUse the shell method to calculate volumes of revolution, particularly useful for certain geometries.
   • Work Done by a Variable ForceApply integration to calculate the work done when a force is not constant.
8. Module 8 Sequences and Series: Infinite Sums This module introduces the study of infinite sequences and the sums of their terms, known as series.
   • Understanding Infinite SequencesDefine and analyze the behavior of infinite sequences.
   • Convergence and Divergence of SeriesDetermine whether an infinite series converges to a finite sum or diverges.
   • The Geometric SeriesAnalyze the properties and convergence of geometric series, a fundamental type of series.
   • Power Series and Taylor SeriesExplore series that represent functions, leading to approximations and expansions.
9. Module 9 Multivariable Calculus: Functions of Several Variables This module extends calculus concepts to functions involving more than one independent variable.
   • Partial DerivativesLearn to compute partial derivatives, which measure the rate of change with respect to one variable while holding others constant.
   • The Gradient VectorUnderstand the gradient as a vector indicating the direction and magnitude of the greatest rate of increase of a multivariable function.
   • Double Integrals over Rectangular RegionsExtend the concept of integration to find volumes under surfaces.
   • Change of Variables: Polar CoordinatesUse coordinate transformations, such as polar coordinates, to simplify integration in two dimensions.
10. Module 10 Applications of Multivariable Calculus: Optimization and Physics This module showcases advanced applications of multivariable calculus in optimization and physical sciences.
    • Optimization with Lagrange MultipliersSolve constrained optimization problems using Lagrange multipliers.
    • Vector Fields and Line IntegralsIntroduce vector fields and the concept of line integrals to calculate work or flux along a curve.
    • Green's Theorem: Connecting Line and Area IntegralsUnderstand Green's theorem, which relates a line integral around a simple closed curve to a double integral over the plane region it encloses.
    • Stokes' Theorem and the Divergence TheoremExplore higher-dimensional theorems that generalize Green's theorem to relate integrals over surfaces and their boundaries.

This course was generated by STRATA's AI at the request of a learner, then reviewed for our community guidelines. Content is for general learning.