Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Calculus II
Topic: Parametric, Polar, and Conic Curves
Subtopic: Calculus of Parametric Curves

Overview

Recall from a Pre-Calculus course parametric equations (see Lesson Notes - Trigonometry - Parametric Equations and Graphs but ignore any "text notes"). A parametric equation, e.g. x=t^3, y=tan(t), can be converted to a rectangular equation (the previous example would be y=tan(root3(x)), but some calculus operations may be simplified by working with the parametric form. All the calculus we have done with rectangular equations we'll redo in this lesson but using parametric form. We'll find derivatives and integrals of parametric equations, slopes of tangent lines to parametric curves, arc lengths of parametric curves, and more.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 10.1.1 Find the first and second derivative of a parametric equation (dx/dt, dy/dt, dy/dx, (d^2y)/dx^2)
• 10.1.2 Find the slope and equation of a tangent line to a parametric curve
• 10.1.3 Find the point(s) at which the tangent line to a parametric curve is horizontal or vertical
• 10.1.4 Find the critical points, inflection points, relative extrema, and concavity of a parametric curve
• 10.1.5 Set-up and evaluate the integral that gives the area under (or inside) a parametric curve
• 10.1.6 Set-up and evaluate the integral that gives the arc length of a parametric curve
• 10.1.7 Use integration to find the volume and surface area of a solid formed by revolving a parametric curve about a horizontal or vertical axis
• 10.1.8 Know some facts about the cycloid curve (hypocycloid and epicycloid) including its basic shape, how it is generated, and applications

Terminology

Terms you should be able to define: parametric equation, parameter

Mini-Lectures and Examples

Supplemental Resources (recommended)

Simple sheet with Calculus Formulas for Parametric Equations from Dr. Rahman, Univ. of N. Florida.

Supplemental Resources (optional)

These are both quite good reads. The first reviews concepts from Pre-Calculus and the second covers Calculus concepts.

This article about Ocean Waves and the Trochoid Curve provides a nice little application of a particular parametric curve.

Bezier curves are not covered in Calculus II, but they are a beautiful and truly useful in real-life application of parametric curves so worth exploring for motivation or simply intriguing fun. See Bezier Curves - Getting the Shape you Want, Dale Hoffman's Contemporary Calculus

rev. 2021-02-27