Calculus 1 Course

Lesson 1 • Types of Elementary Functions

Surveys polynomial, rational, exponential, logarithmic, and trigonometric functions. Provides the function library needed for limit and derivative work.

Lesson 2 • Transformations and Symmetry

Covers shifts, reflections, stretches, and symmetry of graphs. Enables quick sketching of transformed functions used in limit analysis.

Lesson 3 • Combining and Composing Functions

Introduces arithmetic combinations and composition of functions. Prepares students for the chain rule and composite function differentiation.

Lesson 1 • Precise Definition of a Limit

Introduces the epsilon-delta definition for rigorous limit proofs. Strengthens conceptual depth and prepares students for theoretical coursework.

Lesson 2 • Intuitive Concept of a Limit

Introduces limits through tables and graphs before formal definitions. Grounds abstract notation in observable function behaviour.

Lesson 3 • Limit Laws and Algebraic Techniques

Presents limit laws and algebraic strategies for exact evaluation. Connects graphical intuition to precise computational methods.

Lesson 4 • Limits Involving Infinity

Examines infinite limits and limits at infinity to describe asymptotic behaviour. Directly supports curve sketching and improper integral concepts.

Lesson 1 • The Difference Quotient and Tangent Lines

Connects average rate of change to the slope of secant lines. Motivates the derivative as the limiting slope of tangent lines.

Lesson 2 • Derivative as a Function

Extends the derivative concept from a point to a function. Students graph derivatives and identify where functions are differentiable.

Lesson 3 • Computing Derivatives from the Definition

Applies the limit definition to compute derivatives of basic functions. Builds procedural fluency before shortcut rules are introduced.

Lesson 1 • Derivatives of Trigonometric Functions

Establishes derivatives of all six trigonometric functions. Enables differentiation of models involving periodic behaviour.

Lesson 2 • The Chain Rule

Presents the chain rule for differentiating composite functions. Unlocks differentiation of the broadest class of elementary expressions.

Lesson 3 • Derivatives of Exponential and Logarithmic Functions

Derives rules for exponential and logarithmic differentiation. Completes the elementary function toolkit for applied modelling.

Lesson 4 • Power, Constant, and Sum Rules

Introduces the core shortcut rules for polynomials and sums. Replaces limit-based computation with efficient algebraic differentiation.

Lesson 1 • Solving Classic Related Rates Problems

Applies the setup process to standard geometric scenarios. Builds pattern recognition for common related rates problem types.

Lesson 2 • Implicit Differentiation Fundamentals

Differentiates equations not solved for y by treating y as a function of x. Expands the scope of curves students can analyse.

Lesson 3 • Derivatives of Inverse Trigonometric Functions

Uses implicit differentiation to derive inverse trig derivatives. Adds essential functions for integration and physics applications.

Lesson 1 • Linear Approximation and Differentials

Uses tangent lines to approximate function values and introduces differentials. Connects derivative meaning to practical estimation methods.

Lesson 2 • Mean Value Theorem and Its Consequences

States and applies the Mean Value Theorem and Rolle's Theorem. Connects derivative values to global function behaviour.

Lesson 3 • Curve Sketching

Synthesises domain, intercepts, asymptotes, and derivative information into accurate graphs. Develops analytical visualisation skills for complex functions.

Lesson 4 • First and Second Derivative Tests

Uses sign changes of f prime and f double prime to classify extrema and concavity. Enables complete qualitative analysis of function graphs.

Lesson 5 • Optimisation Problems

Formulates and solves real-world optimisation problems using derivatives. Translates constraints into objective functions and applies extrema techniques.

Lesson 1 • Area Under a Curve and Riemann Sums

Approximates area using left, right, and midpoint Riemann sums. Motivates the definite integral as the limit of these approximations.

Lesson 2 • The Definite Integral

Defines the definite integral formally and states its properties. Establishes the notation and rules used in all integration work.

Lesson 3 • The Fundamental Theorem of Calculus

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

Lesson 4 • Net Change and Accumulation

Interprets definite integrals as net change and total accumulation in context. Bridges abstract integration to applied rate-of-change problems.

Lesson 1 • Area Between Curves

Computes area enclosed between two functions using definite integrals. Applies integration to a fundamental geometric measurement problem.

Lesson 2 • Average Value and Applied Integrals

Defines average value of a function and applies integrals to work and motion. Connects integration to practical physical and engineering contexts.

Lesson 3 • Substitution Rule for Indefinite Integrals

Introduces u-substitution as the reverse of the chain rule. Extends the range of integrable functions beyond basic antiderivative rules.

Lesson 4 • Volumes by Disk and Washer Methods

Calculates volumes of solids of revolution using disk and washer cross-sections. Extends area concepts into three-dimensional geometric applications.