Calculus with Parametric Curves

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Section 11.2

• Calculus with Parametric Curves

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TANGENTS TO A PARAMETRIC CURVE
If the parametric curve is given by x f (t)
y g(t). The slope of the tangent line is given
by
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HORIZONTAL AND VERTICAL TANGENTS
A parametric curve will have a horizontal tangent
when dy/dt 0. A parametric curve will have a
vertical tangent when dx/dt 0.
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THE SECOND DERIVATIVE
For a set of parametric equations, the second
derivative is given by
Note that this is NOT the same as
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AREA
If the parametric curve is given by x f (t)
y g(t) and is traversed once as t increase from
a to ß, then the area under the curve is given by
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ARC LENGTH
If a curve C is described by the parametric
equations x f (t), y g(t), a t ß, where
f ' and g' are continuous on a, ß and C is
traversed exactly once as t increasing from a to
ß, then the length of C is
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SURFACE AREA
Let C be the curve given by the parametric
equations x f (t), y g(t), a t ß, is
rotated about the x-axis, where f ' and g' are
continuous, and g(t) 0, then the surface area
is given by
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SURFACE AREA (CONCLUDED)
Let C be the curve given by the parametric
equations x f (t), y g(t), a t ß, is
rotated about the y-axis, where f ' and g' are
continuous, and f(t) 0, then the surface area
is given by