Course
MA1 B
Term
Spring 2026
Schedule
Tue / Thu Lecture + Fri Recitation
Lessons
0
Course overview
This is calculus from the proof side: the same objects from MA0, but rebuilt with the precision of real analysis. We begin with the real numbers and completeness, then move through sequences, limits, continuity, differentiation, Riemann integration, and series. The emphasis is on definitions, proofs, examples, and counterexamples: not just knowing the rules of calculus, but knowing why they work.
- Start with the real numbers and the completeness property that makes calculus possible.
- Build limits, continuity, and differentiability carefully enough that the main theorems can be proved rather than assumed.
- Finish by treating integration, convergence, power series, and Taylor expansion as connected parts of one theory.
Weekly rhythm
- Read the lesson page first and work through the examples in order.
- Open the linked alternate video only if you want a second explanation or a different pace.
- Do the exercises, then recitation or lab immediately after the notes so the ideas turn into practice.
- Attempt the homework last, without jumping straight to solutions.
Course themes
- Completeness of the real numbers and the basic language of sets, intervals, and compactness.
- Sequences, limits, continuity, and the role of examples and counterexamples in analysis.
- Differentiation from first principles, including the Mean Value Theorem and Taylor approximation.
- Riemann integration and the Fundamental Theorem of Calculus.
- Numerical series, uniform convergence, power series, and Taylor series.
Prerequisites
This course assumes the following background.
- Prerequisite: MCE A (Mathematical Thinking and Python)
- Prerequisite: MA0 A (Differentiation)
- Prerequisite: MA0 B (Integration)
Course direction
The course is meant to make later analysis feel like a continuation rather than a restart. MA0 teaches the working calculus; MA1B asks why it works, which hypotheses are doing the real work, and how to recognise the examples where familiar calculus intuition breaks.
References
Outside notes, textbooks, or course pages worth keeping around.
course inspiration
MIT 18.100A: Real Analysis
A close reference point for the proof-based real-analysis direction of this course.
Main Reference
Calculus by Michael Spivak
Main problem-set and exposition reference for proof-based single-variable calculus.
Alternate Reference
Understanding Analysis by Stephen Abbott
A gentler analysis reference with strong intuition for real numbers, sequences, continuity, differentiation, integration, and series.