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5: Differential Calculus with Parametric Curves

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    Introduction

    While functions, given by \(y=f(x)\), and implicitly defined curves, given by \( f(x,y)=c \), provide us with a variety of curves, parametric equations expand the possibilities. For example, parametric equations are useful in describing the motion of objects in terms of time in two dimensional space, with a natural extension to three dimensions. They also give us some pretty pictures!

    Repeated floral shape given by parametric equations.
    Figure \(\PageIndex{1}\): A curve given by parametric equations of the form \( x(t) = (a-b) \cos(t)+b \cos(t((\frac{a}{b}-1))\), \(y(t) = (a-b) \sin(t) - b \sin(t(\frac{a}{b} - 1)) \) (CC BY-SA 2.0; Josep M Battle i Ferrer via Wikimedia)

    • 5.1: 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.
    • 5.2: Differential 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?
    • 5.3: Chapter 5 Review Exercises

    Thumbnail: A hypocycloid. (CC BY NC SA; Openstax via Calculus-Volume-2)

    Contributors and Attributions

    Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

    This page titled 5: Differential Calculus with Parametric Curves was last modified on Fri, 05 Sep 2025 23:29:35 GMT and is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Doli Bambhania, De Anza College.