UBC MATH 200 Calculus III: Topics, Study Guide and Practice

MATH 200 is Calculus III at UBC Vancouver, the multivariable calculus course that follows MATH 101 or one of its equivalents. It moves calculus from curves to surfaces: geometry in three dimensions, derivatives of functions of several variables, optimization, and integrals over regions. This guide maps every MATH 200 topic to practice so you can master each idea by solving problems.

What UBC MATH 200 covers

The UBC Vancouver calendar lists the topics as analytic geometry in two and three dimensions, partial and directional derivatives, the chain rule, maxima and minima, the second derivative test, Lagrange multipliers, and multiple integrals with applications. It is a 3-credit course with three lecture hours per week.

That is the differential and integral core of a standard Calculus 3 course. The calendar description does not include vector fields, line integrals or the theorems of Green and Stokes; at UBC those appear in courses such as MATH 217.

MATH 200 topic map

MATH 200 vs MATH 253 vs MATH 217

UBC's credit exclusion lists determine which of these can be combined, so check them for your program. The usual route is MATH 100, then MATH 101, then MATH 200.

Example problems at the MATH 200 level

1. Partial derivatives. For f(x, y) = x^3 y^2 + e^{xy}:

2. The second derivative test. For f(x, y) = x^2 + xy + y^2 - 3x, setting f_x = 2x + y - 3 = 0 and f_y = x + 2y = 0 gives (2, -1). With D = f_{xx}f_{yy} - f_{xy}^2 = 4 - 1 = 3 > 0 and f_{xx} = 2 > 0, it is a local minimum, f(2, -1) = -3.

3. A double integral. Over the triangle 0 \le y \le x \le 1:

How to do well in UBC MATH 200

• Sketch everything. Draw surfaces, level curves and regions of integration. Most errors on multiple integrals come from wrong limits.
• Reuse your single-variable skills. Partial derivatives are ordinary derivatives with other variables held constant, so keep the chain rule sharp with practice.
• Learn the Lagrange setup cold. Write \nabla f = \lambda \nabla g with the constraint every time; the Lagrange multipliers guide walks through it.
• Keep the formulas visible. The calculus formula sheet and Calculus 1 vs 2 vs 3 help you see how the pieces connect.

Tip: after setting up a double integral, check it by computing the area of the region with the integrand replaced by 1. If the area is wrong, so are your limits.