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: Polar Equations and Curves

Overview

Recall from a Pre-Calculus course polar equations (see Lesson Notes - Trigonometry - Polar Equations and Graphs but ignore any "text notes"). A polar equation, e.g. r=sin(3theta), can be converted to a rectangular equation, but some calculus operations may be simplified by working with the polar form. This lesson merely reviews and practices skills working with polar coordinates, polar equations, and polar curves from a Pre-Calculus course in preparation for doing calculus in polar coordinates in the next lesson.

Objectives

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

• 10.2.1 Convert polar equations to rectangular equations and visa versa using x=rcostheta, y=rsintheta, r=sqrt(x^2+y^2), theta=tan^-1(y/x)
• 10.2.2 Evaluate polar equations at specific angles
• 10.2.3 Find axis intercept points of polar curves
• 10.2.4 Graph a polar equation manually and/or electronically
• 10.2.5 Find collision points and self-intersection points on a polar curve manually and/or electronically
• 10.2.6 Find the intersection of two polar curves manually and/or electronically

Terminology

Terms you should be able to define: polar equation, pole, collision point, self-intersection point

Mini-Lectures and Examples

Supplemental Resources (recommended)

Common Polar Curves is a simple sheet from Prof. Kate Niedzielski, Univ. of Minnesota. However, this sheet is weak on limicons, so for those I would refer instead to MathWords >> Limacon which includes a limacon with a "loop", a limacon with a "dimple", and a convex ("flattened") limacon. Watch out though, MathWords swaps the a and b in the equations of a limacon compared to other sources. Explore the various limacons via DESMOS interactively and watch for the relationships between |a/b|<1, |a/b|=1, 1<|a/b|<2, |a/b|>=2.

Supplemental Resources (optional)

The material below reviews concepts from Pre-Calculus.

rev. 2021-02-28