Parametric Equations, Polar Coordinates, and Vector-Valued Functions – Unit 9

Unit 9 includes all the topics listed in the title. These are BC only topics (CED – 2019 p. 163 – 176). These topics account for about 11 – 12% of questions on the BC exam.

Comments on Prerequisites: In BC Calculus the work with parametric, vector, and polar equations is somewhat limited. I always hoped that students had studied these topics in detail in their precalculus classes and had more precalculus knowledge and experience with them than is required for the BC exam. This will help them in calculus, so see that they are included in your precalculus classes.

Parametric/Vector topics are largely limited to motion in a plane. There are a number of connections between 1-dimensional motion (motion on a number line), and 2- and 3- dimensional motion. Also, most textbooks seem to jump directly into 3-dimensional parametric and vector equations without stopping in 2-dimensions. Some polar equation questions also concern motion along a polar path. You may have to supplement your textbook. Use questions from past exams both multiple-choice and free-response.

Topics 9.1 – 9.3 Parametric Equations

Comment: There is little difference (really only in the notation) between parametric form and vector form of curves in two dimensions and curves defining motion in the plane. Concentrate on the similarities. Other than circles, the names of various curves are not tested. The emphasis is almost totally on motion in the plane defined by parametric/vector-valued functions.

Topic 9.1: Defining and Differentiation Parametric Equations. Finding dy/dx in terms of dy/dt and dx/dt

Topic 9.2: Second Derivatives of Parametric Equations. Finding the second derivative. See Implicit Differentiation of Parametric Equations this discusses the second derivative.

Topic 9.3: Finding Arc Lengths of Curves Given by Parametric Equations.

Topics 9.4 – 9.6 Vector-Valued Functions and Motion in the plane

Comment: There is little difference (really only in the notation) between parametric form and vector form of curves in two dimensions and curves defining motion in the plane. Concentrate on the similarities. Other than circles, the names of various curves are not tested.The emphasis is almost totally on motion in the plane defined by parametric/vector-valued functions.

Topic 9.4 : Defining and Differentiating Vector-Valued Functions. Finding the second derivative. See this A Vector’s Derivatives which includes a note on second derivatives.

Topic 9.5: Integrating Vector-Valued Functions

Topic 9.6: Solving Motion Problems Using Parametric and Vector-Valued Functions. Position, Velocity, acceleration, speed, total distance traveled, and displacement extended to motion in the plane.

Topics 9.7 – 9.9 Polar Equation and Area in Polar Form.

Comment: Other than circles, students are not expected to know the names of the various polar curves (lemiçons, rose curves, etc.) Graphs will be given or available on the graphing calculator. The equations for moving from polar to rectangular form or vice versa and how to use them should be known by students

Topic 9.7: Defining Polar Coordinate and Differentiation in Polar Form. The derivatives and their meaning.

Topic 9.8: Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Topic 9.9: Finding the Area of the Region Bounded by Two Polar Curves. Students should know how to find the intersections of polar curves to use for the limits of integration.