U of T · MAT137
Calculus with Proofs
Rigorous, proof-based calculus — one of U of T's most demanding and rewarding first-year courses.
About this course
MAT137 is proof-based calculus: it takes the intuitive ideas of limits, derivatives, and integrals and places them on a rigorous logical foundation using formal definitions and proofs.
It's challenging not just because the material is difficult, but because it requires mathematical maturity — the ability to read, write, and understand proofs — that most students haven't developed in high school.
We prepare students to meet that challenge by building genuine understanding from the ground up: the real definition of a limit, why the derivative is what it is, what the Fundamental Theorem of Calculus actually says — and why.
This course moves fast. We recommend starting sessions early in the semester.
Topics covered
- Epsilon-delta definition of limits
- Continuity and its consequences (IVT, EVT)
- The derivative: definition and rules
- Mean Value Theorem and applications
- Riemann integrals from first principles
- Fundamental Theorem of Calculus (both parts)
- Sequences and series
- Power series and Taylor series with remainder estimates
What you get
Proof foundations
We build the mathematical maturity needed before course material becomes overwhelming.
Weekly sessions
Keep pace with the course and tackle challenging proofs together.
Exam strategy
How to read and write proofs under time pressure.
Ready to get started?
Book a free diagnostic session and find the right starting point.
