5: Differential Calculus with Parametric Curves
- Page ID
- 168910
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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 dimensiona. They also give us some pretty pictures!
- 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?
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.