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

> A study guide for MAT149H1 (formerly part of MAT137Y1) at U of T: integration, improper integrals, series, power series and Taylor's theorem, with practice.

Source: https://practicecalculus.com/courses/university-of-toronto-mat149h1
Updated: 2026-11-03

## At a glance

- **Course code:** MAT149H1 (the calendar lists MAT137Y1 as the previous course number)
- **Hours:** 36 lecture hours and 12 tutorial hours
- **Prerequisites:** MAT148H1, MAT158H1, or MAT135H1 or MAT130H1 with 70% or higher
- **Exclusions:** MAT137Y1, MAT137Y5, MAT137H5 with MAT139H5, MATA37H3
- **Covers:** Integration and its applications, the Fundamental Theorem of Calculus, improper integrals, sequences and series, power series, Taylor's theorem
- **Next course:** MAT237Y1 (Multivariable Calculus with Proofs), with linear algebra
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- MAT149H1 is the second half of U of T's proof-based calculus sequence, and the calendar lists MAT137Y1 as its previous course number.
- It covers integration, the Fundamental Theorem of Calculus, improper integrals, sequences and series, power series and Taylor's theorem.
- Students can enter from MAT148H1, or from MAT135H1 or MAT130H1 with at least 70%.
- The calendar suggests students coming from MAT135H1 or MAT130H1 prepare by studying proof techniques.

**MAT149H1** is Calculus II with Proofs at the University of Toronto, the second half of the proof-based sequence that begins with [MAT148H1](/courses/university-of-toronto-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 topic | Practice | Read |
| --- | --- | --- |
| Riemann sums and the integral | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums), [definite integral properties](/rules/definite-integral-properties) |
| Fundamental Theorem of Calculus | [Derivatives of integrals](/learn/calculus-1/integrals/fundamental-theorem-part-1) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1), [FTC part 2](/theorems/fundamental-theorem-of-calculus-part-2) |
| Integration techniques | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [u-substitution](/blog/u-substitution), [integration by parts](/blog/integration-by-parts) |
| Applications of integration | [Volumes with disks](/learn/calculus-1/applications-of-integrals/volumes-disks) | [Volumes of revolution](/blog/volumes-of-revolution), [Mean Value Theorem for integrals](/theorems/mean-value-theorem-for-integrals) |
| Improper integrals | [Integrands that blow up](/learn/calculus-2/improper-integrals-and-applications/infinite-discontinuities) | [Improper integrals](/blog/improper-integrals) |
| Sequences | [Sequences and their limits](/learn/calculus-2/sequences-and-series/sequences) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Series and convergence tests | [Comparison tests](/learn/calculus-2/sequences-and-series/comparison-tests) | [Convergence tests](/blog/convergence-tests), [geometric series](/blog/geometric-series) |
| Power series | [Power series and radius of convergence](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series](/blog/power-series), [ratio test](/rules/ratio-test) |
| Taylor's theorem | [Estimating with Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/estimating-with-taylor) | [Taylor's theorem](/theorems/taylors-theorem), [Taylor series](/blog/taylor-series) |

## MAT149H1 vs MAT136H1

| Course | Entry | Style |
| --- | --- | --- |
| MAT149H1 | MAT148H1, MAT158H1, or MAT135H1/MAT130H1 at 70%+ | Proof-based integration and series |
| [MAT136H1](/courses/university-of-toronto-mat136h1) | MAT135H1 or an equivalent | Applied integration, series and differential equations |
| [MAT237Y1](/courses/university-of-toronto-mat237y1) | A first-year calculus sequence plus linear algebra | Proof-based multivariable calculus |

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

$$F'(x) = \sin\big((x^2)^2\big)\cdot 2x = 2x\sin(x^4)$$

**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:

$$\sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right) = 1$$

**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:

$$|\text{error}| \le \frac{0.5^5}{120} \approx 0.00026$$

## How to do well in MAT149H1

- **Master the tests before proving them.** Work through the [choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) lesson until you can name the right test at a glance.
- **Learn the key series by heart.** The [geometric series formula](/rules/geometric-series-formula) and common Maclaurin series show up in nearly every proof.
- **Keep the formulas visible.** The [calculus formula sheet](/formulas) lists integrals and series on one page.
- **Mix practice and proof.** Daily sets in [practice](/practice) plus a weekly review of proofs, guided by [how to prepare for a calculus exam](/blog/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.

## Frequently asked questions

### What is MAT149H1?

MAT149H1 is Calculus II with Proofs at the University of Toronto. It develops integration and infinite series with a theoretical, proof-based approach, while still practicing computation and problem solving.

### Can I take MAT149H1 after MAT135H1?

Yes, if you earned 70% or higher in MAT135H1 (or MAT130H1). The calendar recommends that those students build proof skills first, for example through the Preparing for University Math program or the material in MAT147H1.

### Is MAT149H1 the same as the second half of MAT137Y1?

The calendar lists MAT137Y1 as the previous course number for MAT149H1 and the two exclude each other, so MAT149H1 now covers the integration and series part of what was MAT137Y1.

### Is MAT149H1 hard?

Series and Taylor's theorem are where many students find proof-based calculus hardest, because the arguments involve inequalities and limits at the same time. Regular practice with convergence tests makes the proofs far easier to follow.

### What comes after MAT149H1?

The usual next step is MAT237Y1, Multivariable Calculus with Proofs, which also requires a linear algebra course such as MAT223H1 or MAT240H1.