| Text and course set-up | The textbook and basic procedures | |
| Schedule/Grading | When and how grades are determined | |
| Assignments 1 | Beginnings to Exam 1 | |
| Assignments 2 | Exam 1 to Exam 2 | |
| Assignments 3 | Exam 2 to the Final | |
| Solutions | Solutions to assignments, quizzes, and exams |
Advanced Calculus is 'calculus revisited', but you will not be differentiating or integrating in this course. Our attention will focus instead on understanding the mathematical structures - real numbers, sequences, and functions - that are the 'raw material' of calculus. More fundamentally, we are concerned with the logical bases and proof methods of all mathematics. The best way to carefully learn definitions and logic is to write proofs, and that is what you will be doing a lot. Writing (and reading) are central to your success in this course.
The course begins with a rapid review of sets, functions, numbers, and absolute value, lingering a bit longer on induction proofs. Supremums, infimums, and nested intervals, introduced in Chapter 2, play a basic role in future developments. The rhythm of the course sets in when we study sequences in Chapter 3 - what they are, what it means for a sequence to have a limit, what monotone sequences, subsequences, and the Cauchy property have to do with anything. Once we are experts on sequences we repeat the process in Chapter 4 with limits of functions, relating them to limits of sequences. Chapter 5 discusses the special property of functions known as continuity. Here we encounter some familiar theorems from calculus, including the Intermediate Value Theorem. Finally in Chapters 6 and 7 we study the derivative and the integral, using all we have learned about supremums, sequences, limits, and continuity. The payoff is more favorite facts, from the Mean Value Theorem to the Fundamental Theorem of Calculus.
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