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

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 in the U of T calculus sequence

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.

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:

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}:

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

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 and chain rule take no thought. Short daily sets in practice build that speed.
• Practice translating words into math. Rates, approximations and models are central, so work through related rates and optimization until setting up the equation is routine.
• Keep the formulas close. The calculus formula sheet puts every derivative rule on one page.
• Learn from errors. Our list of 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.