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Calculus with Parametric Curves
Calculus can be applied to parametric equations as well.
Learning Objective

Use differentiation to describe the vertical and horizontal rates of change in terms of
$t$
Key Points
 Parametric equations are equations that depend on a single parameter.
 A common example comes from physics. The trajectory of an object is well represented by parametric equations.
 Writing the horizontal and vertical displacement in terms of the time parameter makes finding the velocity a simple matter of differentiating by the parameter time. Parameterizing makes this kind of analysis straightforward.
Terms

acceleration
the change of velocity with respect to time (can include deceleration or changing direction)

trajectory
the path of a body as it travels through space

displacement
a vector quantity which denotes distance with a directional component
Full Text
Parametric equations are equations which depend on a single parameter. You can rewrite
A common example occurs in physics, where it is necessary to follow the trajectory of a moving object. The position of the object is given by
Trajectories
A trajectory is a useful place to use parametric equations because it relates the horizontal and vertical distance to the time.
This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves termwise. Thus, one can describe the velocity of a particle following such a parametrized path as:
where
The acceleration can be written as follows with the double apostrophe signifying the second derivative:
Writing these equations in parametric form gives a common parameter for both equations to depend on. This makes integration and differentiation easier to carry out as they rely on the same variable. Writing
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Key Term Reference
 coordinate
 Appears in these related concepts: Parametric Equations, Calculus of VectorValued Functions, and Triple Integrals in Cylindrical Coordinates
 curve
 Appears in these related concepts: Arc Length and Surface Area, Arc Length and Speed, and Area Between Curves
 derivative
 Appears in these related concepts: Types of Financial Markets, Defining Financial Leverage, and Difference Quotients
 differentiation
 Appears in these related concepts: Development of Nervous Tissue, Caulobacter Differentiation, and The Challenge of Competition
 integration
 Appears in these related concepts: Area and Distances, The Definite Integral, and Volumes of Revolution
 parametric
 Appears in these related concepts: DistributionFree Tests, Integration By Parts, and Surfaces in Space
 variable
 Appears in these related concepts: Related Rates, Controlling for a Variable, and Math Review
 velocity
 Appears in these related concepts: RootMeanSquare Speed, Force, and Average Velocity: A Graphical Interpretation
Sources
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Cite This Source
Source: Boundless. “Calculus with Parametric Curves.” Boundless Calculus. Boundless, 26 May. 2016. Retrieved 30 May. 2016 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/differentialequationsparametricequationsandsequencesandseries4/parametricequationsandpolarcoordinates17/calculuswithparametriccurves1122828/