# U of T MAT148H1 Calculus I with Proofs: Study Guide and Practice

> A study guide for MAT148H1 (formerly MAT137Y1) at the University of Toronto: logic, induction, limits, continuity and derivatives, with free practice.

Source: https://practicecalculus.com/courses/university-of-toronto-mat148h1
Updated: 2026-11-03

## At a glance

- **Course code:** MAT148H1 (the calendar lists MAT137Y1 as the previous course number)
- **Hours:** 36 lecture hours and 12 tutorial hours
- **Prerequisites:** High school level calculus
- **Exclusions:** MAT137Y1, MAT137Y5, MAT137H5 with MAT139H5, MATA30H3 with MATA31H3
- **Covers:** Logic and quantifiers, sets, induction, limits and continuity, the Mean Value Theorem, transcendental functions, the inverse function theorem, differentiation and applications
- **Next course:** MAT149H1 (Calculus II with Proofs)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MAT148H1 is the first half of U of T's proof-based calculus sequence, and the calendar lists MAT137Y1 as its previous course number.
- It starts with logic, sets and induction before proving results about limits, continuity and derivatives.
- It still covers computation and problem solving, not only theory.
- It is followed by MAT149H1, Calculus II with Proofs.

**MAT148H1** is Calculus I with Proofs at the University of Toronto, the first half of the proof-based calculus sequence in the Faculty of Arts and Science. The calendar lists **MAT137Y1** as its previous course number, so if you are searching for MAT137, this is where its first half now lives. This guide maps each MAT148H1 topic to practice, so you can make the computations automatic and spend your energy on proofs.

## What U of T MAT148H1 covers

The calendar describes MAT148H1 as a conceptual course for students interested in the theoretical foundations of mathematics, while still giving attention to computation and problem-solving techniques. It recommends the course for future study in computer science, economics, mathematics, physics and statistics.

The listed topics are logic and quantifiers, sets, mathematical induction, limits and continuity, the Mean Value Theorem and its applications, elementary transcendental functions including trigonometric functions, the inverse function theorem, and differentiation with applications. That is the differential half of a standard **Calculus 1** course, taught with proofs, plus an introduction to mathematical reasoning. Integration is saved for [MAT149H1](/courses/university-of-toronto-mat149h1).

## MAT148H1 topic map

| MAT148H1 topic | Practice | Read |
| --- | --- | --- |
| Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [What is a limit?](/blog/what-is-a-limit), [limit laws](/rules/limit-laws) |
| Limits by squeezing | [One-sided and infinite limits](/learn/calculus-1/limits/one-sided-and-infinite-limits) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity](/blog/continuity), [Intermediate Value Theorem](/theorems/intermediate-value-theorem) |
| The derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Differentiability | [When derivatives fail](/learn/calculus-1/the-derivative/differentiability) | [Absolute value function](/functions/absolute-value-function) |
| Differentiation rules | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule), [quotient rule](/rules/quotient-rule) |
| Transcendental functions | [Sine, cosine and tangent](/learn/calculus-1/derivative-rules/trig-derivatives) | [Derivatives of trig functions](/blog/derivatives-of-trig-functions), [natural logarithm](/functions/natural-logarithm) |
| Inverse functions | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Inverse function rule](/rules/inverse-function-rule), [arctangent](/functions/arctangent) |
| Mean Value Theorem | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem), [Rolle's theorem](/theorems/rolles-theorem) |
| Applications of derivatives | [Absolute extrema](/learn/calculus-1/applications-of-derivatives/absolute-extrema) | [Extreme Value Theorem](/theorems/extreme-value-theorem), [curve sketching](/blog/curve-sketching-with-derivatives) |

## MAT148H1, MAT149H1 and the old MAT137Y1

| Course | Approach | Content |
| --- | --- | --- |
| MAT137Y1 (previous number) | Proof-based | Listed as the previous number of both MAT148H1 and MAT149H1 |
| MAT148H1 | Proof-based | Logic, induction, limits, continuity, derivatives |
| [MAT149H1](/courses/university-of-toronto-mat149h1) | Proof-based | Integration, sequences and series, Taylor's theorem |
| [MAT135H1](/courses/university-of-toronto-mat135h1) | Applied | Differential calculus with modelling |

After the pair, students can continue to [MAT237Y1](/courses/university-of-toronto-mat237y1), Multivariable Calculus with Proofs, together with a linear algebra course.

## Example problems at the MAT148H1 level

**1. An induction proof.** Show $1 + 2 + \cdots + n = \frac{n(n+1)}{2}$ for all $n \ge 1$. The case $n = 1$ gives $1 = 1$. If it holds for $n$, then

$$1 + \cdots + n + (n+1) = \frac{n(n+1)}{2} + (n+1) = \frac{(n+1)(n+2)}{2}$$

which is the statement for $n + 1$.

**2. An epsilon-delta limit.** Prove $\lim_{x \to 2} (3x - 1) = 5$. Given $\varepsilon > 0$, choose $\delta = \varepsilon/3$. If $0 < |x - 2| < \delta$, then

$$|(3x - 1) - 5| = 3|x - 2| < 3\delta = \varepsilon$$

**3. The Mean Value Theorem.** For $f(x) = x^3$ on $[0, 2]$, the average slope is $\frac{8 - 0}{2} = 4$. Setting $f'(c) = 3c^2 = 4$ gives $c = \frac{2}{\sqrt{3}} \approx 1.155$, which lies in $(0, 2)$ as the theorem promises.

## How to do well in MAT148H1

- **Separate computing from proving.** Drill derivative rules in [practice](/practice) until they are reflexes, so proof questions get your full attention.
- **Know the theorems by their hypotheses.** For the [Mean Value Theorem](/theorems/mean-value-theorem) and [Intermediate Value Theorem](/theorems/intermediate-value-theorem), learn exactly what must be true before you can use them.
- **Write out quantifiers.** Many proofs fail because "for every" and "there exists" get swapped. Say each statement in words before writing it.
- **Keep a reference handy.** The [calculus formula sheet](/formulas) and [how to study calculus](/blog/how-to-study-calculus) help you plan weekly review.

> **Tip:** before proving a limit, compute it. Knowing the answer tells you what inequality you are aiming for.

## Frequently asked questions

### What is MAT148H1?

MAT148H1 is Calculus I with Proofs in U of T's Faculty of Arts and Science. It takes a conceptual approach to limits, continuity and differentiation, building the proofs behind the rules while still practicing computation.

### Did MAT148H1 replace MAT137Y1?

The Arts and Science calendar lists MAT137Y1 as the previous course number for both MAT148H1 and MAT149H1, and the courses exclude each other. So the former full-year MAT137Y1 now corresponds to the two half-courses MAT148H1 and MAT149H1.

### Is MAT148H1 hard?

It is demanding because writing proofs is a new skill for most first-year students. The calculus itself is the same as in other first courses, so getting the computations automatic frees up time to focus on proofs.

### MAT148H1 vs MAT135H1: which should I take?

MAT135H1 is applied and modelling based. MAT148H1 is theoretical and suits students heading toward mathematics, computer science, economics, physics or statistics. Programs that accept MAT135H1 and MAT136H1 also accept MAT148H1 and MAT149H1.

### How do I prepare for MAT148H1?

Review high school calculus so derivative computations are easy, and get comfortable reading statements with quantifiers such as "for every" and "there exists". Practicing limits carefully is the best bridge into the epsilon-delta definition.