Western Calculus 2302A/B Intermediate Calculus I: Topics and Practice

Calculus 2302A/B is Intermediate Calculus I at Western University, the first course in multivariable calculus. It moves calculus into three dimensions in three stages: the geometry of vectors, lines, planes and surfaces; curves in space and how they bend; and derivatives of functions of several variables, ending with optimization. This guide maps every Calculus 2302 topic to free practice so you can learn each idea by working problems.

What Western Calculus 2302A/B covers

The Western academic calendar lists three blocks. Three dimensional analytic geometry: dot and cross product; equations for lines and planes; quadric surfaces. Vector functions and space curves: arc length; curvature; velocity; acceleration. Differential calculus of functions of several variables: level curves and surfaces; limits; continuity; partial derivatives; tangent planes; differentials; chain rule; implicit functions; extrema; Lagrange multipliers. It is a 0.50 course with three lecture hours a week.

That is the first half of a standard Calculus 3 course. The integral half (multiple integrals, line and surface integrals and the big theorems) is in Calculus 2303A/B.

Calculus 2302A/B topic map

Calculus 2302 and 2303

Before Calculus 2302 you take Calculus 1000A/B (or 1500) and then Calculus 1501A/B or 1301A/B.

Example problems at the Calculus 2302 level

1. Distance from a point to a plane. The distance from (1, 1, 1) to 2x + y - 2z = 6 is

2. Velocity and acceleration. For \mathbf{r}(t) = \langle t, t^2, t^3 \rangle: \mathbf{v} = \langle 1, 2t, 3t^2 \rangle and \mathbf{a} = \langle 0, 2, 6t \rangle.

3. A tangent plane. For z = x^2 + y^2 at (1, 2, 5): z = 5 + 2(x - 1) + 4(y - 2).

4. Implicit functions. If x^2 + y^2 + z^2 = 9 defines z near a point with z \neq 0, then \frac{\partial z}{\partial x} = -\frac{x}{z}.

How to do well in Calculus 2302A/B

• Picture the geometry. Sketch planes and quadric surfaces using their traces in the coordinate planes.
• Use the right product. The dot product gives angles and projections; the cross product gives normals and areas.
• Watch the hypotheses. Limits in several variables can depend on the path. Try two different paths before concluding a limit exists.
• Organise optimization. List the critical points, classify each with the second derivative test, and check boundaries or constraints.

Tip: a normal vector to the plane ax + by + cz = d is \langle a, b, c \rangle. You can read it straight off the equation.