Dalhousie MATH 2001 Intermediate Calculus I: Topics and Practice

MATH 2001 is Intermediate Calculus I at Dalhousie University, the first course in multivariable calculus. It takes the curves and integrals from MATH 1010 into three dimensions: vectors and their products, curves in space, partial derivatives, and double and triple integrals. This guide maps every MATH 2001 topic to free practice so you can learn each idea by working problems.

What Dalhousie MATH 2001 covers

The Dalhousie calendar lists the topics as a review of parametric equations and polar coordinates, conic sections, coordinate systems and vectors, the dot product and cross product, vector functions, derivatives and integrals of vector functions, arc length and curvature, functions of several variables and partial derivatives, directional derivatives, and double and triple integrals. It is a 3 credit hour lecture course and the prerequisite is MATH 1010.

This covers most of a standard Calculus 3 course. The vector calculus part (line and surface integrals, Green's and Stokes' theorems) is in the next course, MATH 2002, along with change of variables in multiple integrals.

MATH 2001 topic map

MATH 2001 and MATH 2002

Before MATH 2001 you take MATH 1000 and MATH 1010.

Example problems at the MATH 2001 level

1. The angle between vectors. For \mathbf{u} = \langle 1, 2, 2 \rangle and \mathbf{v} = \langle 2, 0, 1 \rangle:

2. Curvature of a circle. For \mathbf{r}(t) = \langle a\cos t, a\sin t \rangle, the curvature is \kappa = \frac{1}{a}: smaller circles bend more.

3. A directional derivative. For f(x, y) = x^2 + y^2 at (1, 2) in the direction \mathbf{u} = \langle \tfrac{3}{5}, \tfrac{4}{5} \rangle: \nabla f = \langle 2, 4 \rangle, so D_{\mathbf{u}}f = \frac{6}{5} + \frac{16}{5} = \frac{22}{5}.

4. A double integral.

How to do well in MATH 2001

• Visualise in 3D. Sketch surfaces and regions, or use a graphing tool. Seeing the shape makes bounds and normals much easier.
• Keep the cross product formula handy. Write it as a determinant and double check the middle sign.
• Make the direction a unit vector. Directional derivatives need |\mathbf{u}| = 1, a step many students forget.
• Practise switching the order. Some double integrals are only easy in one order, so practise switching the order.

Tip: the gradient points in the direction of fastest increase, and its length is the largest possible directional derivative.