12.1 Parametric Equations

Math 6B

Calculus II

Parametrizations and Plane Curves

Path traced by a particle moving alone the xy plane. Sometimes the graph cannot be expressed as a function of x or y.

Definition If x and y are continuous functions

x = f (t) , y = g(t)over an interval of t – values , then the set of points (x , y) = ( f (t) , g(t)) defined by these equations is a curve in the coordinate plane.

Definition The equations are parametric equations. The variable t is a parameter for the curve and its domain I is the parameter interval. If I is a closed interval, , the pt. ( f (a) , g(a)) is the initial point of the curve and ( f (b) , g(b)) is called the terminal point of the curve.

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Definition When we give parametric equations

and a parameter interval for a curve in the plane, we say that we have parameterized the curve. The equations and interval constitute a parameterization of the curve.

Tangents To find the slope of the tangent dy/dx

from the parametric equations x = f (t) and y = g (t), let us use the chain rule of dy/dt

dy dy dx

dt dx dt

Tangents We can get dy/dx by itself and

therefore get the slope of the tangent line.

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dy dy dt

dx dx dt