What this subject covers¶
This subject is single-variable calculus, together with sequences and series and an introduction to ordinary differential equations. It matches a typical first-year university sequence. Multivariable calculus is a separate subject, which will build on this one and on linear algebra.
The core text is written for first-year students of engineering and science. Readers who want full rigour, for example mathematics majors, can open the rigorous track wherever it appears; how to read this site explains how the two layers work.
Exponential and logarithmic functions are introduced early, in the preliminaries, as functions you already know. Their derivatives come from one stated fact about , which an optional page in the chapter on integrals proves by constructing as an integral.
Chapters¶
The chapters are planned in this order. Each one will be linked here once it is written.
Preliminaries: real numbers, inequalities, functions, and the elementary functions (polynomial, exponential, logarithmic and trigonometric).
Limits: what a limit is, the limit laws, and limits involving infinity.
Continuity: continuous functions, and the intermediate and extreme value theorems.
Derivatives: the derivative, the rules for computing it, and related rates.
Applications of derivatives: extreme values, the mean value theorem, curve sketching, optimisation, L’Hôpital’s rule, Newton’s method and Taylor polynomials.
Integrals: antiderivatives, Riemann sums, the definite integral, the fundamental theorem of calculus and substitution.
Integration techniques: integration by parts, trigonometric integrals and substitutions, partial fractions, and numerical integration.
Applications of integrals: areas between curves, volumes, arc length, average values, work and centre of mass.
Improper integrals: integrals over unbounded intervals and of unbounded functions, and how to compare them.
Sequences and series: sequences, convergence tests, power series and Taylor series.
Introduction to differential equations: slope fields, separable and linear first-order equations, modelling, Euler’s method, and second-order linear equations with constant coefficients.
Prerequisite map¶
Every planned topic of this subject and the topics it builds on. It is generated from the curriculum and the pages each time the site is built.
Arrows point from a topic to the topics that build on it. Dashed boxes are planned topics that have not been written yet (70 of 82); a written topic is coloured by its status and links to its page.
The chapters. An arrow means that the later chapter builds on the earlier one (an arrow that a longer path already implies is left out).
1. Preliminaries.
2. Limits. Rounded boxes are prerequisites from other chapters.
3. Continuity. Rounded boxes are prerequisites from other chapters.
4. Derivatives. Rounded boxes are prerequisites from other chapters.
5. Applications of Derivatives. Rounded boxes are prerequisites from other chapters.
6. Integrals. Rounded boxes are prerequisites from other chapters.
7. Integration Techniques. Rounded boxes are prerequisites from other chapters.
8. Applications of Integrals. Rounded boxes are prerequisites from other chapters.
9. Improper Integrals. Rounded boxes are prerequisites from other chapters.
10. Sequences and Series. Rounded boxes are prerequisites from other chapters.
11. Introduction to Differential Equations. Rounded boxes are prerequisites from other chapters.
Suggested paths¶
| Path | Chapters | Notes |
|---|---|---|
| Engineering Calculus I | 1–6 | skip the rigorous track; the tier C rigorous exercises are optional |
| Engineering Calculus II | 7–11 | |
| Mathematics major, single-variable | all, including the rigorous track and extension pages | |
| Quick review before multivariable calculus | 2, 4, 5 (mean value theorem, Taylor), 6, 10 (Taylor series) |