First-year calculus tutoring

If you are stuck in calculus right now

First-year calculus is the course that catches out more capable students than any other. Lectures move at a pace that assumes you already read the section, tutorials skip the middle steps, and the problem sets are noticeably harder than the examples you were shown. If you are sitting in front of a question with no idea how to start, that is a normal experience in this course — and it is fixable.

What usually goes wrong is not the calculus. It is everything underneath it: algebra that is slow instead of automatic, trigonometry half-remembered, and a habit from high school of matching a question to a memorised method. Calculus punishes that habit because the questions are deliberately not in the shape of the examples.

What first-year calculus covers

Across institutions the mathematics is remarkably consistent, even where the course numbering and applications differ. A differential calculus course builds from limits and continuity, to the derivative as a rate of change and as the slope of a tangent, to the rules that let you differentiate anything you can write down — product, quotient and chain rules, implicit differentiation, and derivatives of exponential, logarithmic and trigonometric functions. From there it moves into applications: related rates, optimisation, linear approximation, and curve sketching using the first and second derivatives.

An integral calculus course reverses the question. You start with antiderivatives and the definite integral as a limit of sums, meet the Fundamental Theorem of Calculus that ties the two halves of the subject together, and then spend most of the term on technique: substitution, integration by parts, partial fractions, trigonometric integrals, and improper integrals. Applications include areas, volumes of revolution, average values, arc length, and simple differential equations. Many courses close with sequences and series, convergence tests, and Taylor series.

Where students commonly get stuck

Limits treated as plugging in. Students learn to substitute the value and stall the moment they hit an indeterminate form. The fix is recognising that a limit asks about behaviour near a point, not at it, and knowing the small set of algebraic moves — factoring, conjugates, dividing by the highest power — that reveal it.

The chain rule inside everything else. Chain rule on its own is fine. Chain rule buried inside a quotient, inside an implicit differentiation, inside a related rates problem is where marks disappear. We drill layered differentiation until identifying the inner function is automatic.

Word problems. Related rates and optimisation are the two topics students most often abandon, because the calculus is easy and the setup is not. Both are really translation exercises: name the quantities, write the relationship, then differentiate. We teach the setup as its own skill rather than hoping it appears.

Choosing an integration technique. In integral calculus the single biggest source of lost marks is not executing a method badly — it is picking the wrong one, then spending fifteen exam minutes on it. Recognition can be trained, and it is mostly pattern exposure.

Series convergence. Students memorise a list of tests without a decision procedure for choosing among them, and often confuse whether a sequence, a series, or its terms are converging.

How sessions work

Sessions are one-to-one and online over Zoom, using an interactive whiteboard so both of us can write. That matters in calculus, where the useful conversation happens mid-problem rather than at the end. You bring your problem set, your lecture notes, or a past midterm, and we work through real questions rather than generic examples — with you doing the writing as soon as you are able to.

A typical session starts with a question you could not finish, digs into exactly which step broke, repairs that, and then tests the repair on a fresh problem. Before exams we shift to mixed timed practice, because in an exam the hard part is deciding what kind of question you are looking at.

Which institutions we support

We support students taking first-year calculus at UBC and SFU, and we work with students at any college or university, in Canada or abroad. If your institution is not listed, that changes nothing about how we work — bring your course materials and we will work from those.