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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)

displacement
a vector quantity which denotes distance with a directional component

trajectory
the path of a body as it travels through space
Full Text
Parametric equations are equations which depend on a single parameter. You can rewrite y=x such that x=t and y=t where t is the parameter.
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 x and y, signifying horizontal and vertical displacement, respectively. As time goes on the object flies through its path and x and y change. Therefore, we can say that both x and y depend on a parameter t, which is 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 v is the velocity, r is the distance, and x, y, and z are the coordinates. The apostrophe represents the derivative with respect to the parameter.
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 x and y explicitly in terms of t enables one to differentiate and integrate with respect to t. The horizontal velocity is the time rate of change of the x value, and the vertical velocity is the time rate of change of the y value. Writing in parametric form makes this easier to do.
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Key Term Reference
 coordinate
 Appears in this related concepts: Parametric Equations, Calculus of VectorValued Functions, and Triple Integrals in Cylindrical Coordinates
 curve
 Appears in this related concepts: Arc Length and Surface Area, Arc Length and Speed, and Area Between Curves
 derivative
 Appears in this related concepts: Difference Quotients, Overview of Derivatives, and Financial Leverage
 differentiation
 Appears in this related concepts: The Many Goals of Persuasion, Defining Monopoly, and Socioemotional Development in Adolescence
 integration
 Appears in this related concepts: Area and Distances, Volumes of Revolution, and Consumer Purchasing Behavior
 parametric
 Appears in this related concepts: DistributionFree Tests, Integration By Parts, and Surfaces in Space
 variable
 Appears in this related concepts: Related Rates, The Linear Function f(x) = mx + b and Slope, and Math Review
 velocity
 Appears in this related concepts: Force, Average Velocity: A Graphical Interpretation, and Applications and ProblemSolving
Sources
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Cite This Source
Source: Boundless. “Calculus with Parametric Curves.” Boundless Calculus. Boundless, 02 Jul. 2014. Retrieved 21 May. 2015 from https://www.boundless.com/calculus/textbooks/boundlesscalculustextbook/differentialequationsparametricequationsandsequencesandseries4/parametricequationsandpolarcoordinates17/calculuswithparametriccurves1122828/