U of T MAT149H1 Calculus II with Proofs: Study Guide and Practice

MAT149H1 is Calculus II with Proofs at the University of Toronto, the second half of the proof-based sequence that begins with MAT148H1. The calendar lists MAT137Y1 as its previous course number, so this course now holds the integration and series material that MAT137 students used to meet in the winter. This guide maps each topic to practice so you can master the computations and focus on the proofs.

What U of T MAT149H1 covers

The Arts and Science calendar describes MAT149H1 as a conceptual course for students interested in the theoretical foundations of mathematics, with attention to computation and problem-solving techniques as well. Its topics are integration and its applications, the Fundamental Theorem of Calculus, improper integrals, sequences and series, power series, and Taylor's theorem with applications.

That corresponds to the integral half of Calculus 1 and the series half of Calculus 2, taught with full justification. Expect to prove why a series converges, not just to apply a test.

MAT149H1 topic map

MAT149H1 vs MAT136H1

MAT136H1 includes differential equations and computer algebra; MAT149H1 instead goes deeper into sequences, series and Taylor's theorem.

Example problems at the MAT149H1 level

1. Differentiating an integral. If F(x) = \int_0^{x^2} \sin(t^2)\,dt, the Fundamental Theorem and the chain rule give

2. A convergence proof by comparison. For n \ge 1, 0 < \dfrac{1}{n^2 + n} < \dfrac{1}{n^2}. Since \sum \frac{1}{n^2} is a convergent p-series, \sum \dfrac{1}{n^2 + n} converges. In fact it telescopes:

3. Bounding a Taylor error. Using \sin x \approx x - \frac{x^3}{6} at x = 0.5, Taylor's theorem bounds the error by \frac{|x|^5}{5!}, since every derivative of \sin is at most 1 in size:

How to do well in MAT149H1

• Master the tests before proving them. Work through the choosing a test lesson until you can name the right test at a glance.
• Learn the key series by heart. The geometric series formula and common Maclaurin series show up in nearly every proof.
• Keep the formulas visible. The calculus formula sheet lists integrals and series on one page.
• Mix practice and proof. Daily sets in practice plus a weekly review of proofs, guided by how to prepare for a calculus exam, keep both skills sharp.

Tip: when a series proof stalls, write out the first few partial sums. Seeing the numbers often suggests the comparison or bound you need.