# U of T MAT135H1 Calculus I: Topics, Study Guide and Practice

> A study guide for MAT135H1 at the University of Toronto: limits, derivatives, linear approximation and differential equations, mapped to free practice.

Source: https://practicecalculus.com/courses/university-of-toronto-mat135h1
Updated: 2026-11-03

## At a glance

- **Course code:** MAT135H1 (Faculty of Arts and Science, St. George)
- **Hours:** 36 lecture hours and 12 tutorial hours
- **Prerequisites:** High school level calculus
- **Breadth:** The Physical and Mathematical Universes (5)
- **Covers:** Limits, asymptotes, continuity, derivatives, linear approximation, differential equations, slope fields and Euler's method
- **Next course:** MAT136H1 (Calculus II)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MAT135H1 is the first half of the University of Toronto's standard calculus sequence for science students, followed by MAT136H1.
- It is a differential calculus course: limits, continuity, derivatives, linear approximation and optimization.
- Unusually for a first calculus course, it also introduces differential equations, slope fields and Euler's method.
- It expects high school calculus, so it moves faster than a course that starts from zero.

**MAT135H1** is Calculus I at the University of Toronto's Faculty of Arts and Science, and with MAT136H1 it forms the default calculus sequence for most science students. It is a one-term course in differential calculus that leans heavily on applications and modelling. This guide maps each MAT135H1 topic to free practice, so you can learn every idea by working problems with instant feedback.

## What U of T MAT135H1 covers

The Arts and Science calendar describes MAT135H1 as an introduction to differential calculus, the study of how quantities change. Students use derivatives to approximate functions with straight lines, relate rates of change and find maximum and minimum values, with examples drawn from the social, biological and physical sciences. The course also stresses moving between algebraic, graphical, numerical and verbal descriptions of a problem.

The listed topics are limits, asymptotes, continuity, derivatives, linear approximation, differential equations and their solutions, slope fields and Euler's method. That lines up with a standard **Calculus 1** course minus integration, which waits for MAT136H1, plus an early look at differential equations.

## MAT135H1 topic map

| MAT135H1 topic | Practice | Read |
| --- | --- | --- |
| Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [How to evaluate limits](/blog/how-to-evaluate-limits), [limit laws](/rules/limit-laws) |
| Asymptotes and limits at infinity | [Limits at infinity](/learn/calculus-1/limits/limits-at-infinity) | [Limits at infinity](/blog/limits-at-infinity), [rational functions](/functions/rational-functions) |
| 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) |
| Derivative rules | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule), [product rule](/rules/product-rule) |
| Exponential growth models | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Exponential and log derivatives](/blog/derivatives-of-exponential-and-logarithmic-functions) |
| Linear approximation | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Linear approximation](/blog/linear-approximation) |
| Maxima and minima | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems), [curve sketching](/blog/curve-sketching-with-derivatives) |
| Related rates of change | [Related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Related rates](/blog/related-rates) |
| Differential equations | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Natural exponential function](/functions/natural-exponential-function) |

## MAT135H1 in the U of T calculus sequence

| Course | Approach | Main content |
| --- | --- | --- |
| MAT135H1 | Applied, modelling | Limits, derivatives, linear approximation, differential equations |
| [MAT136H1](/courses/university-of-toronto-mat136h1) | Applied, modelling | Integrals, series, differential equations |
| [MAT148H1](/courses/university-of-toronto-mat148h1) | Proof-based | Logic, induction, limits, differentiation |
| [MAT149H1](/courses/university-of-toronto-mat149h1) | Proof-based | Integration, series, Taylor's theorem |

The math department notes that programs requiring MAT135H1 and MAT136H1 also accept the proof-based pair. If you finish MAT135H1 with 70% or higher, the calendar lets you switch into MAT149H1 for your second term. Multivariable calculus with proofs comes later in [MAT237Y1](/courses/university-of-toronto-mat237y1).

## Example problems at the MAT135H1 level

**1. A limit with an asymptote.** Find $\lim_{x \to \infty} \dfrac{3x^2 + 1}{x^2 - 4}$.

Divide top and bottom by $x^2$:

$$\lim_{x \to \infty} \frac{3 + 1/x^2}{1 - 4/x^2} = \frac{3}{1} = 3$$

so $y = 3$ is a horizontal asymptote.

**2. A linear approximation.** Estimate $\sqrt{4.1}$ using the tangent line to $f(x) = \sqrt{x}$ at $x = 4$. Since $f'(x) = \frac{1}{2\sqrt{x}}$, $f'(4) = \frac{1}{4}$:

$$\sqrt{4.1} \approx 2 + \frac{1}{4}(0.1) = 2.025$$

**3. One step of Euler's method.** For $y' = y$ with $y(0) = 1$ and step size $h = 0.5$:

$$y_1 = y_0 + h \cdot y_0 = 1 + 0.5(1) = 1.5$$

The exact value is $y(0.5) = e^{0.5} \approx 1.649$, so Euler's method underestimates here because the solution curve bends upward.

## How to do well in MAT135H1

- **Make derivatives automatic.** The applied questions only feel manageable once the [power rule](/rules/power-rule) and [chain rule](/rules/chain-rule) take no thought. Short daily sets in [practice](/practice) build that speed.
- **Practice translating words into math.** Rates, approximations and models are central, so work through [related rates](/blog/related-rates) and [optimization](/blog/optimization-problems) until setting up the equation is routine.
- **Keep the formulas close.** The [calculus formula sheet](/formulas) puts every derivative rule on one page.
- **Learn from errors.** Our list of [common calculus mistakes](/blog/common-calculus-mistakes) covers the sign and algebra slips that cost the most marks.

> **Tip:** read every graph twice: once for the function and once for its slope. MAT135H1 often asks you to move between a graph, a formula and a sentence, and that habit makes those questions much easier.

## Frequently asked questions

### What is MAT135H1 at U of T?

MAT135H1 is Calculus I in the Faculty of Arts and Science. It covers differential calculus (limits, continuity, derivatives and their applications) with an emphasis on modelling in the life, social and physical sciences.

### What comes after MAT135H1?

The next course is MAT136H1, Calculus II, which covers integration, series and more differential equations. Students who earn 70% or higher in MAT135H1 can also move into MAT149H1, Calculus II with Proofs.

### Is MAT135H1 hard?

It assumes you have already seen calculus in high school, so the early material goes quickly and the focus is on understanding and applying ideas, not just computing. Regular practice with derivatives and word problems makes a big difference.

### What is the difference between MAT135H1 and MAT148H1?

MAT135H1 is an applied course built around modelling. MAT148H1, Calculus I with Proofs, takes a theoretical approach with logic, induction and proofs about limits and continuity. Both lead into a second calculus course.

### How should I prepare for MAT135H1?

Review functions, exponent and logarithm rules and trig values, then refresh limits and basic derivatives. The free Calculus 1 path on PracticeCalculus covers the same ground in short lessons.