We begin the course with Chapter 10.1: Curves Defined by Parametric Equations. In this section, we will broaden our understanding of what a function is to include functions which can be described by a parameter, such as time. The output of this new type of function can be thought of as a pair of numbers, in contrast to the usual functions we saw in single-variable calculus. We will also learn how to sketch parametric curves from equations which depend on a parameter.
10.1: #1, 3, 9, 11, 13, 17, 19, 31, 33, 47
After finishing up 10.1, we will work through Chapter 10.2: Calculus with Parametric Curves. Here, we will see how to extend some of the basic geometric ideas from single-variable calculus to the study of parametric curves. Toward the end of the week, we start Chapter 10.3: Polar Coordinates. This section consists of an introduction to the polar coordinate system, which allows us to describe points in terms of a distance r and an angle θ. We then study polar curves, which are curves given by an equation relating r and θ.
10.2: #1, 3, 5, 9, 11, 15, 17, 19, 21, 23, 33, 35, 37, 47, 49, 71, 75
10.3: #1, 3, 5, 7, 9, 11, 15, 17, 29, 33, 37, 39, 47
After finishing up our discussion of polar coordinates and polar curves introduced in 10.3, we will work through Chapter 10.4: Calculus in Polar Coordinates. In this section, we will see how to apply some of the formulas, results and techniques we saw in Chapter 10.2 to polar curves and polar representations of curves. For example, we will learn the formulas for computing tangents, areas and arc lengths of curves given in polar representation, and we will see that these formulas differ from what was obtained in section 10.2.
10.4: #3, 5, 7, 9, 11, 17, 21, 23, 37, 39, 49, 51, 63, 67, 69, 71
We begin the week by combining Chapter 10.5: Conic Sections and Chapter 12.6: Cylinders and Quadric Surfaces. Here, we will review parabolas and ellipses (two curves which will show up many times in our study of multivariable calculus) and we will study their three-dimensional counterparts and other common surfaces. We then move on to Chapter 12.1: Three-Dimensional Coordinate Systems and Chapter 12.2: Vectors, where we give a quick recap of the concept of a vector and its relation to three-dimensional coordinate systems. We finish the week by introducing two important operations on vectors in Chapter 12.3: The Dot Product and Chapter 12.4: The Cross Product.