Course Description
Calculus 1 develops the ideas that make quantitative science possible: how to describe change precisely (the derivative) and how to add up infinitely many infinitesimal pieces (the integral), together with the machinery of functions, limits and continuity that makes both ideas rigorous.
The course covers functions, limits, continuity and their applications; mathematical induction; introduction to the derivative; differentiation; applications of the derivative; definite integrals; antiderivatives and integration; applications of the definite integral; indeterminate forms; improper integrals; numerical integration; sequences and series of numbers; and Taylor series expansions of elementary functions.
The official course description also lists vector analysis as a topic. The week-by-week schedule supplied for this term (see the Calendar) runs the sequences-and-series material through Week 16 and leaves Week 17 without an assigned reading, so this site treats Week 17 as a review week rather than inventing a vector-analysis lesson that was not scheduled. Check with your instructor if vector analysis is examined this term.
Learning Outcomes
On successful completion of this course a student will be able to:
- Determine the domain, range and graph of a function, combine and compose functions, and evaluate limits using limit laws, one-sided limits and the formal behaviour of limits at infinity.
- Decide where a function is continuous, and use continuity and the Intermediate Value Theorem to reason about a function's behaviour.
- Compute derivatives using the definition and the differentiation rules (product, quotient, chain, implicit), and interpret a derivative as a rate of change and as a tangent slope.
- Apply derivatives to related rates, linear approximation, extreme values, the Mean Value Theorem, curve sketching and optimisation problems.
- Evaluate definite and indefinite integrals using the Fundamental Theorem of Calculus, substitution, integration by parts, trigonometric methods and partial fractions.
- Apply definite integrals to area, volume, arc length and surface area, work, and other accumulation problems, and evaluate improper integrals and indeterminate forms.
- Determine the convergence or divergence of sequences and series using the standard convergence tests, and construct and use Taylor and Maclaurin series to represent and approximate functions.
Course Structure
Part I — Limits and Derivatives
Weeks 1–7. Functions and graphs, limits and continuity, the derivative, differentiation rules, and applications of the derivative, closing with the definite integral.
Part II — Integration and Its Applications
Weeks 9–13. The Fundamental Theorem of Calculus, substitution, volumes and arc length, transcendental functions, integration techniques and improper integrals.
Part III — Sequences, Series and Taylor Series
Weeks 14–16. Sequences, infinite series, convergence tests, power series and Taylor series, with applications to non-elementary integrals.
Lecture Notes
| Week | Topic | Status |
|---|---|---|
| 1 | Functions, Graphs & Trigonometry · self-check | Published |
| 2 | Rates of Change & Limits · self-check | Published |
| 3 | One-Sided Limits, Continuity & Infinity · self-check | Published |
| 4 | The Derivative & Differentiation Rules · self-check | Published |
| 5 | Implicit Differentiation & Related Rates · self-check | Published |
| 6 | Applications of the Derivative · self-check | Published |
| 7 | The Definite Integral · self-check | Published |
| 8 | Midterm Examination | Scheduled |
| 9 | Fundamental Theorem & Substitution · self-check | Published |
| 10 | Volumes, Arc Length & Surface Area · self-check | Published |
| 11 | Work, Inverse & Exponential Functions · self-check | Published |
| 12 | L’Hôpital’s Rule & Integration Techniques I · self-check | Published |
| 13 | Integration Techniques II & Improper Integrals · self-check | Published |
| 14 | Sequences & Infinite Series · self-check | Published |
| 15 | Series Convergence Tests · self-check | Published |
| 16 | Power Series & Taylor Series · self-check | Published |
| 17 | Course Review | Published |
| 18 | Final Examination | Scheduled |
Prerequisites
None. This is the first calculus course; a working comfort with algebra and the trigonometric functions from secondary school is assumed and is reviewed in Week 1.
Read the syllabus for grading and MyLab setup, then begin with Week 1: Functions, Graphs & Trigonometry.