4: Parametric Equations
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Introduction
The chambered nautilus is a fascinating creature. This animal feeds on hermit crabs, fish, and other crustaceans. It has a hard outer shell with many chambers connected in a spiral fashion, and it can retract into its shell to avoid predators. When part of the shell is cut away, a perfect spiral is revealed, with chambers inside that are somewhat similar to growth rings in a tree.

The mathematical function that describes a spiral can be expressed using rectangular (or Cartesian) coordinates. However, if we change our coordinate system to something that works a bit better with circular patterns, the function becomes much simpler to describe. The polar coordinate system is well-suited for describing curves of this type. How can we use this coordinate system to describe spirals and other radial figures?
- 4.1: Parametric Equations
- In this section we examine parametric equations and their graphs. In the two-dimensional coordinate system, parametric equations are useful for describing curves that are not necessarily functions. The parameter is an independent variable that both x and y depend on, and as the parameter increases, the values of x and y trace out a path along a plane curve.
- 4.2: Calculus of Parametric Curves
- Now that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus. For example, if we know a parameterization of a given curve, is it possible to calculate the slope of a tangent line to the curve? How about the arc length of the curve? Or the area under the curve?
- 4.3: Chapter 4 Review Exercises
- This page offers a variety of mathematical exercises covering true or false statements, parametric curve sketching, and deriving equations. It includes tasks on finding tangent lines, computing derivatives, and determining areas and arc lengths for specified curves. Exercises involve proving/disproving statements about coordinates and equations, as well as converting parametric forms to Cartesian equations. Some final answers are also provided, detailing the results of the calculations.
Thumbnail: Mathematical Parameterization of Earth's Orbit Around the Sun. (CC BY NC SA; Openstax via Calculus-Volume-2)