Calculus Course

Lesson 1 • Review of Functions and Their Properties

Covers domain, range, composition, and inverse functions as prerequisites for calculus. Establishes the function language used throughout the entire course.

Lesson 2 • Algebraic Techniques for Evaluating Limits

Teaches substitution, factoring, rationalisation, and conjugate methods for computing limits. Provides tools to resolve indeterminate forms without calculus.

Lesson 3 • Continuity and Discontinuity

Defines continuity at a point and on an interval, classifying removable, jump, and infinite discontinuities. Prepares students for differentiability requirements.

Lesson 4 • Introduction to the Limit Concept

Introduces limits intuitively through tables and graphs before formal definition. Connects informal understanding to the precise epsilon-delta framework.

Lesson 1 • Basic Differentiation Rules

Introduces the power, constant, sum, and difference rules for rapid differentiation. Reduces repetitive limit computation to systematic algebraic procedures.

Lesson 2 • Product and Quotient Rules

Extends differentiation to products and quotients of functions with formal proofs. Enables differentiation of rational and mixed-type functions.

Lesson 3 • The Derivative as a Limit

Derives the derivative using the difference quotient and interprets it as instantaneous rate of change. Anchors all differentiation rules in the limit definition.

Lesson 4 • Derivatives of Exponential and Logarithmic Functions

Derives and applies differentiation rules for natural and general exponential and logarithmic functions. Completes the elementary function differentiation toolkit.

Lesson 1 • Concavity and the Second Derivative

Introduces concavity, inflection points, and the Second Derivative Test for classifying extrema. Completes the analytical toolkit for full curve analysis.

Lesson 2 • Mean Value Theorem and L'Hopital's Rule

Proves and applies the Mean Value Theorem and uses L'Hopital's Rule for indeterminate limits. Bridges theoretical results with practical limit evaluation.

Lesson 3 • Linear Approximation and Differentials

Uses the tangent line to approximate function values and introduces the differential notation. Connects derivative meaning to practical estimation techniques.

Lesson 4 • Optimisation Problems

Formulates and solves optimisation problems using derivatives and the Extreme Value Theorem. Develops modelling skills for maximising or minimising real quantities.

Lesson 5 • Implicit Differentiation and Related Rates

Differentiates implicitly defined relations and links rates of change of related variables. Provides tools for problems where explicit function form is unavailable.

Lesson 1 • The Fundamental Theorem of Calculus

States and proves both parts of the Fundamental Theorem, linking differentiation and integration. Enables exact evaluation of definite integrals using antiderivatives.

Lesson 2 • Substitution Rule for Integration

Introduces u-substitution as the integration counterpart of the chain rule. Extends the range of integrable functions to composite expressions.

Lesson 3 • Riemann Sums and the Definite Integral

Constructs the definite integral as the limit of Riemann sums using left, right, and midpoint approximations. Provides the geometric and analytical foundation for integration.

Lesson 4 • Antiderivatives and Indefinite Integrals

Defines antiderivatives and the indefinite integral, introducing the constant of integration. Establishes integration as the reverse process of differentiation.

Lesson 1 • Trigonometric Integrals

Evaluates integrals of powers and products of sine, cosine, tangent, and secant. Builds specialised techniques for trigonometric function families.

Lesson 2 • Partial Fraction Decomposition

Decomposes rational functions into partial fractions for integration. Covers distinct linear, repeated linear, and irreducible quadratic factor cases.

Lesson 3 • Trigonometric Substitution

Eliminates square roots in integrands using sine, tangent, and secant substitutions. Solves integrals involving expressions of the form a^2 minus x^2 and related forms.

Lesson 4 • Integration by Parts

Derives and applies the integration by parts formula, selecting u and dv strategically. Handles products of polynomial, exponential, and logarithmic functions.

Lesson 1 • Average Value and Probability Applications

Computes the average value of a function and applies integration to continuous probability distributions. Connects integration to statistical and analytical applications.

Lesson 2 • Volumes by Slicing and Disks

Computes volumes of solids of revolution using the disk and washer methods. Connects cross-sectional area integration to three-dimensional geometry.

Lesson 3 • Arc Length and Surface Area

Derives and applies formulas for arc length of curves and surface area of revolution. Extends integration to one-dimensional and two-dimensional geometric measures.

Lesson 4 • Work, Force, and Physical Applications

Models work done by variable forces and fluid pressure using definite integrals. Demonstrates integration as a tool for physical quantity accumulation.

Lesson 1 • Power Series and Radius of Convergence

Defines power series, determines radius and interval of convergence, and performs term-by-term operations. Prepares students for Taylor and Maclaurin series construction.

Lesson 2 • Sequences and Their Limits

Defines sequences formally, examines convergence, and applies limit laws to sequences. Provides the foundation for understanding infinite series behaviour.

Lesson 3 • Alternating Series and Absolute Convergence

Analyses alternating series using the Alternating Series Test and distinguishes absolute from conditional convergence. Completes the convergence classification framework.

Lesson 4 • Taylor and Maclaurin Series

Constructs Taylor and Maclaurin series for standard functions and estimates error using the remainder theorem. Enables function approximation and series-based computation.

Lesson 5 • Infinite Series and Partial Sums

Introduces infinite series as limits of partial sums and examines geometric and telescoping series. Establishes the framework for all subsequent convergence analysis.

Lesson 1 • Partial Derivatives and the Gradient

Computes partial derivatives, higher-order partials, and the gradient vector. Connects partial differentiation to directional rates of change.

Lesson 2 • Multivariable Chain Rule and Implicit Differentiation

Extends the chain rule to compositions of multivariable functions and implicit relations. Enables differentiation in parametric and implicitly defined multivariable settings.

Lesson 3 • Functions of Several Variables

Introduces functions of two and three variables, their domains, and level curves and surfaces. Builds spatial intuition needed for multivariable differentiation and integration.

Lesson 4 • Optimisation of Multivariable Functions

Finds critical points of functions of two variables and classifies them using the second derivative test. Introduces constrained optimisation via Lagrange multipliers.