Problem Set 12—Parametric Curves and Equations

Purpose

This problem set reinforces your understanding of parametric equations.

Background

This problem set is based on material in sections 11.1 and 11.2 of our textbook. We will discuss this material in classes between April 22 and April 29.

Activity

Solve each of the following problems:

Problem 1

Give a pair of parametric equations that define the curve y = 3x² - 2.

Problem 2

Section 11.1, exercise 2 (find a Cartesian equation corresponding to the parametric equations x = -√t and y = t).

Problem 3

Section 11.1, exercise 20a (find parametric equations and a parameter interval for a certain ellipse; see the textbook for the definition of the ellipse).

Problem 4

Section 11.1, exercise 38 (find parametric equations for a general trochoid). In addition to finding the equations for trochoids, plot at least two example trochoids, one with b < a, and one with b > a; I suggest that you use a calculator or programming language to do this. (Note that solutions to this exercise are easy to look up; be sure you can explain the thinking behind your equations.)

Problem 5

Find the slope of the curve defined by x = sin(2t), y = t² + 1 at t = π/2.

Problem 6

Section 11.2, exercise 26 (find the length of the curve defined by x = t³, y = 3t²/2, 0 ≤ t ≤ √3).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as reading through it will help me know what to focus on in the rest of the meeting.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.

My basic expectation in grading this exercise is that your solution will show that you understand how to solve each problem, although there may be arithmetic or copying mistakes, inefficient solution methods, incorrect or irrelevant statements incidental to the solution, or similar minor mistakes. If you understand how to solve all the problems and have no minor errors, I will consider the solution to be in between “what I expect” and “surprisingly beyond expectations.” I will consider solutions to be 3/4, 1/2, 1/4, or none of what I expect according to what (rough) fraction of the problems your solution shows understanding of, although I will raise grades slightly if it is clear by the end of your grading meeting that you have come to understand things you didn’t understand when you arrived at the meeting.