Dave4Math / Courses / Conic Sections, Plane Curves, and Polar Coordinates

Conic Sections, Plane Curves, and Polar Coordinates

Learn all about conic sections and plane curves. You'll master how to manipulate and graph conic sections (and other plane curves) in both rectangular and polar coordinates in this course.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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So many topics, so little time. How can you keep this all organized? Is your teacher going too fast? Introducing Dave’s online math course: Conic Sections, Plane Curves, and Polar Coordinates. Dave really cares about his students. With Dave, you’ll grow into being organized and, in doing so, be inspired to study more.

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in conic sections, plane curves, and polar coordinates.
  • Students who are enrolled in calculus 2 and want to improve their grade.
  • Anyone interested in getting prepared for calculus 3.
  • Anyone wanting to learn about conic sections, plane curves, parametric equations, and polar coordinates.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Conic Sections, Plane Curves, and Polar Coordinates

  • Conic Sections
  • Parabola
  • Reflective Property of the Parabola
  • Ellipses
  • Reflective Property of the Ellipse
  • Eccentricity of an Ellipse
  • Hyperbolas
  • Shifted Conics
  • Why We Use Parametric Equations
  • Sketching Parametric Curves
  • Tangent Lines to Curves
  • Horizontal and Vertical Tangents
  • Higher Order Derivatives
  • The Length of a Smooth Curve
  • The Area of a Surface of Revolution
  • The Polar Coordinate System
  • Relationship Between Polar and Rectangular
  • Graphs of Polar Equations
  • Symmetry
  • Tangent Lines to Graphs of Polar Equations
  • Areas in Polar Coordinates
  • Area Bounded by Two Graphs
  • Arc Length in Polar Coordinates
  • Area of a Surface of Revolution
  • Points of Intersections in Polar Coordinates
  • Polar Equations
  • Eccentricity of a Conic
  • Motion of Celestial Bodies

Course Description

This course provides a detailed review of conic sections. We emphasize concepts and work through examples. After this review, we then focus on plane curves and parametric equations, including sketching curves defined by parametric equations.

Next, we begin a thorough exploration of the calculus of parametric equations. Finding derivatives and second derivatives, finding the length of a smooth curve, and finding the area of a surface of revolution are all detailed.

So far, in this course, we have concentrated on rectangular coordinates. Now here we examine conic sections and plane curves in polar coordinates. We explain symmetry, converting between coordinate systems, and sketching graphs. For example, tangent lines to graphs of polar equations, and areas and arc lengths in polar coordinates are covered. Kepler’s Laws are discussed at the end.

Recommended Prerequisites for Conic Sections, Plane Curves, and Polar Coordinates

I recommend the prerequisite course Techniques of Integration. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 2 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.