McGill MATH 222 Calculus 3: Topics, Study Guide and Practice

MATH 222 is Calculus 3 at McGill University, the third course in the main calculus sequence after MATH 141. It is a bridge course: it finishes the series story from Calculus 2 with Taylor's theorem, then extends derivatives and integrals to functions of several variables. This guide maps every MATH 222 topic to practice so you can learn each idea by working problems.

What McGill MATH 222 covers

McGill's Department of Mathematics and Statistics lists the topics as Taylor series and Taylor's theorem in one and several variables, a review of vector geometry, partial differentiation, directional derivatives, extrema of functions of two or three variables, parametric curves and arc length, polar and spherical coordinates, and multiple integrals. The course is worth 3 credits and runs in summer, fall and winter.

Compared with a standard sequence, MATH 222 contains the Taylor series and parametric and polar topics that many universities put in Calculus 2, plus the differential and integral core of Calculus 3. Vector calculus is left for MATH 248.

MATH 222 topic map

MATH 222 in McGill's calculus sequence

Students on the MATH 150 and 151 track skip MATH 222, since that pair covers its material.

Example problems at the MATH 222 level

1. A Taylor series. Starting from e^t = \sum_{n=0}^{\infty} \frac{t^n}{n!} and substituting t = -x^2:

2. Extrema of a function of two variables. For f(x, y) = x^2 - y^2, the only critical point is (0, 0). There D = f_{xx}f_{yy} - f_{xy}^2 = (2)(-2) - 0 = -4 < 0, so the origin is a saddle point.

3. A polar double integral. Over the disk x^2 + y^2 \le 4:

How to do well in McGill MATH 222

• Refresh series first. The course opens with Taylor series, so review the common Maclaurin series and the ratio test before the term starts.
• Get comfortable with vectors. If you have not taken MATH 133, work through the vectors and space lessons early.
• Remember the Jacobian factor. Changing to polar or spherical coordinates adds r or \rho^2\sin\phi; leaving it out is the most common lost mark.
• Practice and review. Short daily practice sets and the calculus formula sheet keep both the series and multivariable halves fresh.

Tip: Taylor's theorem in several variables is the same idea as in one variable: match the value, the first derivatives and the second derivatives at a point. Seeing it that way makes the second derivative test easy to remember.