Calculus Course for Beginners

Lesson 1 • Essential Algebraic Techniques

Reviews factoring, rational expressions, and exponent rules critical for simplifying calculus expressions. Prevents algebraic errors from obscuring calculus concepts.

Lesson 2 • Exponential and Logarithmic Functions

Examines growth/decay models, the natural base e, and logarithm laws. These functions appear repeatedly in differentiation and integration chapters.

Lesson 3 • Functions and Their Representations

Defines functions formally and explores four representations: verbal, algebraic, numerical, and graphical. Connects abstract definitions to visual intuition.

Lesson 4 • Numbers, Sets, and the Real Line

Covers real number properties, interval notation, and absolute value. Establishes the numeric language used throughout every calculus concept.

Lesson 1 • Continuity of Functions

Defines continuity at a point and on an interval, classifies discontinuities, and states the Intermediate Value Theorem. Prepares students for differentiability conditions.

Lesson 2 • Formal Definition of a Limit

Presents the epsilon-delta definition and uses it to verify limit values rigorously. Develops the logical precision underlying all calculus proofs.

Lesson 3 • Limits Involving Infinity

Analyzes infinite limits and limits at infinity to describe vertical and horizontal asymptotes. Connects limit behaviour to the long-run shape of function graphs.

Lesson 4 • Computing Limits Algebraically

Applies limit laws, factoring, rationalisation, and substitution to evaluate limits exactly. Directly enables derivative computation in the next chapter.

Lesson 1 • Implicit Differentiation and Higher Derivatives

Differentiates implicitly defined relations and computes higher-order derivatives. Prepares students for related rates and concavity analysis.

Lesson 2 • The Derivative as a Rate of Change

Connects average rate of change to the instantaneous rate via the difference quotient limit. Grounds the derivative in physical and geometric meaning.

Lesson 3 • The Chain Rule

Presents the chain rule for differentiating composite functions and applies it to nested expressions. Unlocks differentiation of exponential, logarithmic, and trig composites.

Lesson 4 • Basic Differentiation Rules

Introduces power, constant, sum, and difference rules for efficient differentiation. Reduces reliance on the limit definition for standard polynomial functions.

Lesson 1 • Linear Approximation and Differentials

Uses the tangent line to approximate function values near a known point. Introduces differentials as a tool for estimating error and sensitivity.

Lesson 2 • Related Rates

Uses implicit differentiation with respect to time to relate changing quantities. Develops a systematic problem-solving strategy for dynamic real-world scenarios.

Lesson 3 • Extreme Values and Critical Points

Identifies absolute and local extrema using the Extreme Value Theorem and critical points. Provides the analytical basis for optimisation problems.

Lesson 4 • Curve Sketching with Derivatives

Applies first and second derivative tests to determine monotonicity, concavity, and inflection points. Integrates limit and derivative knowledge into complete graph analysis.

Lesson 5 • Optimisation Problems

Translates word problems into objective functions and uses calculus to find optimal values. Applies the full derivative toolkit to practical decision-making scenarios.

Lesson 1 • Substitution Rule for Integration

Applies u-substitution as the integration counterpart of the chain rule. Expands the class of integrals students can evaluate analytically.

Lesson 2 • Area and Riemann Sums

Approximates area under a curve using left, right, and midpoint Riemann sums. Motivates the definite integral as the limit of increasingly refined partitions.

Lesson 3 • Fundamental Theorem of Calculus

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

Lesson 4 • Antiderivatives and Indefinite Integrals

Introduces antidifferentiation as the reverse of differentiation and establishes integral notation. Builds the conceptual bridge between derivatives and integrals.

Lesson 1 • Trigonometric Substitution

Replaces algebraic radicals with trig substitutions to simplify integrands. Handles integrals involving square roots of quadratic expressions.

Lesson 2 • Improper Integrals

Extends integration to unbounded intervals and functions with infinite discontinuities using limits. Introduces convergence and divergence of improper integrals.

Lesson 3 • Integration by Parts

Derives the integration-by-parts formula from the product rule and applies it systematically. Handles products of polynomials, exponentials, and logarithms.

Lesson 4 • Partial Fraction Decomposition

Decomposes rational functions into simpler fractions for integration. Covers distinct linear, repeated linear, and irreducible quadratic factors.

Lesson 1 • Work, Force, and Physical Applications

Models physical quantities such as work, hydrostatic force, and centre of mass using integrals. Demonstrates calculus as a tool for engineering and physics problems.

Lesson 2 • Average Value and Mean Value Theorem for Integrals

Defines the average value of a function over an interval and proves the integral Mean Value Theorem. Connects integration to statistical and physical averaging concepts.

Lesson 3 • Area Between Curves

Computes area enclosed between two functions by integrating their difference. Requires careful identification of intersection points and integration bounds.

Lesson 4 • Arc Length and Surface Area

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

Lesson 1 • Power Series and Radius of Convergence

Defines power series, finds radii and intervals of convergence, and differentiates or integrates term by term. Bridges series theory to function representation.

Lesson 2 • Convergence Tests for Series

Applies integral, comparison, ratio, and root tests to determine series convergence. Equips students with a systematic toolkit for classifying series behaviour.

Lesson 3 • Infinite Series and Convergence

Introduces partial sums, geometric series, and the divergence test. Establishes the framework for determining whether an infinite sum has a finite value.

Lesson 4 • Taylor and Maclaurin Series

Constructs Taylor and Maclaurin series for standard functions and estimates error with the remainder term. Provides polynomial approximations central to applied mathematics.