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

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.

MAT148H1 topic map

MAT148H1, MAT149H1 and the old MAT137Y1

After the pair, students can continue to 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

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

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 until they are reflexes, so proof questions get your full attention.
• Know the theorems by their hypotheses. For the Mean Value Theorem and 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 and 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.