Calculus 2 Course

Lesson 1 • Limits and Continuity Refresher

Reviews limit laws, one-sided limits, and continuity conditions. Establishes the analytical foundation required for integration and series convergence.

Lesson 2 • Fundamental Theorem of Calculus

Connects differentiation and integration through both parts of the Fundamental Theorem. Builds the conceptual bridge to definite and indefinite integral computation.

Lesson 3 • Differentiation Rules and Applications

Revisits product, quotient, and chain rules alongside implicit differentiation. Ensures derivative fluency needed for integration techniques and differential equations.

Lesson 1 • Improper Integrals

Extends definite integration to unbounded intervals and discontinuous integrands using limits. Introduces convergence and divergence concepts critical for series analysis.

Lesson 2 • Trigonometric Substitution

Eliminates radicals in integrands by substituting trigonometric expressions. Covers all three standard cases and back-substitution to original variables.

Lesson 3 • Integration by Parts

Derives and applies the integration by parts formula using the LIATE heuristic. Connects to the product rule and handles repeated and cyclic applications.

Lesson 4 • Partial Fraction Decomposition

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

Lesson 1 • Area Between Curves

Computes areas enclosed by two or more curves using vertical and horizontal slices. Reinforces integral setup skills and intersection-point analysis.

Lesson 2 • Volume by Cylindrical Shells

Computes volumes of revolution using the shell method as an alternative to disks. Highlights when shells are more efficient than washers.

Lesson 3 • Volumes by Disk and Washer Methods

Generates solids of revolution and computes their volumes using disk and washer cross-sections. Builds spatial reasoning for three-dimensional integral applications.

Lesson 4 • Arc Length and Surface Area

Derives and applies formulas for arc length of curves and surface area of revolution. Connects differential geometry concepts to definite integral computation.

Lesson 1 • Fundamentals of Differential Equations

Defines ODEs, their order, and degree, and classifies them by linearity. Establishes vocabulary and conceptual framing for all subsequent ODE work.

Lesson 2 • Slope Fields and Qualitative Analysis

Constructs slope fields to visualise solution behaviour without explicit formulas. Develops geometric intuition that complements analytic solution methods.

Lesson 3 • Separable Differential Equations

Solves ODEs by separating variables and integrating both sides independently. Applies the technique to exponential growth, decay, and mixing problems.

Lesson 1 • Special Sequences and Growth Rates

Compares growth rates of polynomial, exponential, and factorial sequences. Prepares students for ratio and root tests in the series chapters.

Lesson 2 • Defining and Visualising Sequences

Introduces sequences as functions on positive integers and explores explicit and recursive definitions. Provides the conceptual entry point for infinite series.

Lesson 3 • Monotone and Bounded Sequences

Classifies sequences as monotone increasing or decreasing and identifies bounds. Applies the Monotone Convergence Theorem to guarantee convergence.

Lesson 1 • Series and Partial Sums

Defines infinite series as limits of partial sum sequences and introduces geometric series. Establishes the divergence test as the first convergence screening tool.

Lesson 2 • Integral and p-Series Tests

Uses the Integral Test to link series convergence to improper integrals. Applies results to classify p-series and estimate remainder bounds.

Lesson 3 • Ratio and Root Tests

Applies the Ratio and Root Tests to series with factorial and exponential terms. Identifies inconclusive cases and selects alternative tests accordingly.

Lesson 4 • Alternating Series and Absolute Convergence

Tests alternating series using the Alternating Series Test and estimates error bounds. Distinguishes absolute from conditional convergence.

Lesson 1 • Taylor and Maclaurin Series

Constructs Taylor series from successive derivatives and identifies standard Maclaurin series. Connects polynomial approximation to exact function representation.

Lesson 2 • Taylor Polynomials and Error Bounds

Uses Taylor polynomials to approximate function values and quantifies error with Taylor's Remainder Theorem. Applies bounds to achieve desired accuracy.

Lesson 3 • Differentiation and Integration of Power Series

Differentiates and integrates power series term by term within the interval of convergence. Generates new series representations from known ones.

Lesson 4 • Power Series Fundamentals

Defines power series centred at a point and determines radius and interval of convergence. Applies the Ratio Test to find convergence boundaries.

Lesson 1 • Arc Length and Surface Area Parametrically

Derives arc length and surface area formulas for parametric curves. Extends Chapter 3 results to the parametric setting.

Lesson 2 • Introduction to Polar Coordinates

Converts between polar and Cartesian coordinates and graphs standard polar curves. Establishes the coordinate system for polar calculus.

Lesson 3 • Parametric Curves and Calculus

Defines parametric equations and computes slopes and concavity of parametric curves. Connects parametric derivatives to standard Cartesian interpretations.