Introduction
This section develops single-variable calculus rigorously (building on, and formalising, the intuitive treatment often used elsewhere) and connects it with analytic geometry — using algebra and coordinates to study geometric curves and surfaces.
1. Functions, limits, and continuity
The formal definition of a limit (the ε–δ definition: for every desired closeness ε to the limiting value, some sufficiently small neighbourhood δ around the point guarantees the function stays within that closeness) makes rigorous the intuitive idea of a function “approaching” a value. Continuity at a point requires the function’s value, its limit as that point is approached, and equality between the two, all to exist — a function can fail to be continuous by lacking any one of these three conditions (an undefined point, a limit that does not exist, or a limit that disagrees with the function’s actual value there).
2. Differentiation and its applications
The derivative, formally defined as a limit of a difference quotient, measures instantaneous rate of change. Its major applications include finding extrema (maximum and minimum values, located where the derivative is zero or undefined, then classified using the first or second derivative test), analysing concavity and points of inflection (via the second derivative), and — importantly for this syllabus — Taylor and Maclaurin series, which approximate a function near a point using a polynomial built from its successive derivatives evaluated at that point, with the approximation improving as more terms are included; this connects differentiation directly to the series and convergence topics of point 4.
3. Integration techniques and applications
Integration reverses differentiation (the Fundamental Theorem of Calculus) and is computed using a toolkit of techniques suited to different integrand forms: substitution (reversing the chain rule), integration by parts (reversing the product rule, ∫u dv = uv − ∫v du, particularly useful for products of different function types such as a polynomial times an exponential), partial fractions (decomposing a rational function into simpler fractions before integrating each separately), and trigonometric substitution (for integrands containing expressions like √(a²−x²), converted into trigonometric form to simplify the square root). Applications include computing areas between curves, volumes of revolution (via disk, washer, or shell methods), and arc length.
4. Sequences, series, and convergence
A sequence converges if its terms approach a fixed limiting value as the index grows without bound; a series (the sum of a sequence’s terms) converges if its sequence of partial sums converges. Because a series’ convergence cannot always be determined by inspection, several convergence tests are used depending on the series’ form: the ratio test (comparing consecutive term ratios, particularly effective for series involving factorials or exponentials), the comparison test (comparing against a known convergent or divergent series), and the integral test (comparing a series to the integral of a related continuous, decreasing function) — selecting the most appropriate test for a given series’ structure is the main practical skill this topic requires.
5. Plane analytic geometry: lines and conics
Analytic geometry represents geometric objects using algebraic equations in a coordinate system. A straight line is represented by a linear equation; its slope encodes direction, enabling calculations of angles between lines and conditions for parallel or perpendicular lines. Conic sections — the curves formed by intersecting a plane with a cone at different angles — comprise the circle, ellipse, parabola, and hyperbola, each with a characteristic algebraic equation form (recognisable by the pattern of squared and cross terms) and characteristic geometric properties (foci, directrices, eccentricity, the last of which numerically distinguishes the four types: e = 0 for a circle, 0 < e < 1 for an ellipse, e = 1 for a parabola, e > 1 for a hyperbola).
6. Three-dimensional coordinate geometry
Extending analytic geometry to three dimensions, points are represented by ordered triples, lines by parametric or symmetric equations (built from a point and a direction vector), and planes by an equation of the form Ax + By + Cz = D, where (A, B, C) is a vector normal (perpendicular) to the plane — this normal vector is the key tool for computing angles between planes, distances from a point to a plane, and lines of intersection between planes. Quadric surfaces (three-dimensional analogues of conic sections, such as ellipsoids, paraboloids, and hyperboloids) extend the conic classification of point 5 into three dimensions.
Key Takeaways
- The rigorous ε–δ definition of a limit underlies continuity, which requires a function’s value, its limit, and their equality to all hold at a point.
- Differentiation’s applications include locating and classifying extrema, analysing concavity, and building Taylor/Maclaurin polynomial approximations near a point.
- Integration techniques (substitution, by parts, partial fractions, trigonometric substitution) are chosen based on the integrand’s specific form; the same toolkit underlies area, volume, and arc-length applications.
- Series convergence is determined using tests (ratio, comparison, integral) suited to the series’ specific structure, since convergence is rarely obvious by direct inspection.
- Conic sections are classified by eccentricity in the plane; in three dimensions, planes are defined by a normal vector, and quadric surfaces generalise conics to three-dimensional geometry.