CEGEP Calculus and Linear Algebra Tutoring in Montreal

Three courses, three different kinds of difficulty

The mathematics in the Science program is three courses, each built on a competency the ministry sets for every college. Whatever number your college prints on the outline, the competency code tells you which course it is.

• Calculus I (201-NYA or 201-SN2), competency 0M02. Differential calculus in 75 hours of class: limits, including infinite limits and limits at infinity; continuity; the derivative from its definition; the product, quotient and chain rules and implicit differentiation; the derivatives of exponential, logarithmic, trigonometric and inverse trigonometric functions; L'Hospital's rule for $\frac{0}{0}$ and $\frac{\infty }{\infty }$; the full analysis of a function; related rates; and optimization.
• Calculus II (201-NYB or 201-SN3), competency 0M03. Integral calculus, and more ground in fewer hours: 60 hours of class against Calculus I's 75. Antiderivatives and substitution, the definite integral and the Fundamental Theorem, integration by parts and trigonometric substitution among the usual techniques, the indeterminate forms that must be rewritten before L'Hospital's rule applies, improper integrals, areas, volumes of revolution by disks and shells, separable differential equations, and infinite series: convergence, power series and Maclaurin series.
• Linear Algebra (201-NYC or 201-SN4), competency 0M04. Linear algebra and vector geometry: matrices, determinants, and systems solved by Gauss, Gauss-Jordan and the inverse matrix; then vectors in the plane and in space, the dot, cross and triple scalar products, and the equations of lines and planes. It is also the one of the three whose standard asks, in so many words, for “rigorous proofs of propositions.”

All three share one criterion: “demonstration of rigorous mathematical reasoning through the use of concepts, properties and theorems.” A correct number is not the whole answer; the reasoning that reaches it is part of what is assessed.

Where the marks go in the first term

In Calculus I, the marks tend to go in five places.

• The algebra the course assumes. The program lists factoring, algebraic fractions, rationalizing, polynomial division, the laws of exponents and logarithms, and the unit circle as previously acquired, so the course does not stop to teach them again. A gap there shows up as a limit that will not simplify or a derivative left half-finished.
• Two things that are new. Domains involving roots and logarithms, and the inverse trigonometric functions arcsine and arctangent, are listed as concepts to be acquired in the course itself, at the same pace as everything else.
• The analysis of a function. It is six jobs, not one: domain and intercepts, asymptotes, increase and decrease, extrema, concavity and inflection points, and a sketch that has to agree with all five. One sign error early on carries through every part after it.
• Related rates and optimization. Students who can do the differentiating often lose the marks before it, in turning a sentence into a relation, or into a function of one variable.
• Writing the reasoning down. Notation is assessed (“correct use of mathematical terminology and syntax”), and so is naming the theorem behind a step and checking that its conditions hold.

Calculus II and Linear Algebra have their own trap, and it is the same one in different clothes: choosing. Which technique settles this integral, which test settles this series, which product of vectors answers this question. Practice that names the method in its heading does not train the choice.

How sessions work

I'm Marius, and my own day-to-day work in Montreal is statistical modelling and computational analysis, so this is mathematics I use rather than only teach. Every student starts with a short written diagnostic on the prerequisites their course builds on. For Calculus II that includes the differentiation rules, because the program builds antiderivatives directly on them. The diagnostic separates what is genuinely solid from what only looks solid, so sessions aim at the real gap instead of re-teaching what you already know.

After that, sessions follow the material your course is actually testing. If you want work between sessions, you get a practice set written for exactly the topic we covered, and I mark it before we next meet, reading each error for what caused it. A short written recap goes to whoever books after every session, the student or a parent: what we covered, what was handled unaided, what has been assigned and what comes next. Behind it sits a standing record: the route through the course with dates, what each session established, and a short list of recurring slips to catch.

Practice problems, summaries and formula sheets for your course and its exams are included in the session rate. Before a final, we work through past-exam-style problems until the method is second nature.

I meet students in person downtown and at public libraries such as the Grande Bibliothèque, serving families in Westmount, Outremont, Town of Mount Royal, Hampstead, Côte-Saint-Luc, NDG, Nuns' Island and the West Island. Online sessions run on Zoom with a shared digital whiteboard. Sessions are booked and paid one at a time.

When students usually get in touch

The most useful moment is early: the first weeks of Calculus I, when a gap in the algebra first turns into lost marks rather than a slow homework night. The other common moments are when the first test comes back lower than expected; the weeks before a final; the start of Calculus II, for a student who finished Calculus I unsure of the rules the new course is built on; and the point where Linear Algebra turns from matrices to lines and planes in space, where a student used to calculus suddenly has to picture things in three dimensions.

Still in Secondary 5? The Science program asks for Secondary V Mathematics in the Technical and Scientific (TS) or Science (SN) option, and the Secondary 4 and 5 math tutoring page covers that level. Took AP Calculus AB in high school? The AP Calculus tutoring page explains how it lines up with Calculus I and II.

Every course topic named on this page has a free practice worksheet in the series below, written to the same three competencies, and trying one is a quick way to find which topic to bring to a first session.

Free practice worksheets

Every practice sheet is free to print, with no sign-up.

• CEGEP Calculus I worksheets: 9 sets
• CEGEP Calculus II worksheets: 8 sets
• CEGEP Linear Algebra worksheets: 7 sets

For parents

• Signs your teen is falling behind in CEGEP Calculus I
• The A student who gets a C in Calculus I: what changed

Topics explained, with a self-check

• Continuous but not differentiable: corners, cusps and vertical tangents
• Logs, powers or exponentials: which grows fastest as x → ∞?
• The IVT: proving a root exists without finding it
• The derivative of an integral with x in the limits
• The squeeze theorem, and why sin x / x tends to 1
• Where the derivative of arctan x comes from
• Why the derivative of ln x is 1/x, and why ln|x| has the same one

Common questions

Tutoring at other levels

• High school math tutoring in Montreal
• Secondary 4 and 5 math tutoring in Montreal
• AP Calculus AB tutoring and exam preparation in Montreal
• University math and science courses