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Calculus can be applied to parametric equations as well.
Use differentiation to describe the vertical and horizontal rates of change in terms of t
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 straight-forward.
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 .
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 term-wise. Thus, one can describe the velocity of a particle following such a parametrized path as:
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.