1. Calculus with Parametric Curves
Section 11.2
Calculus with Parametric Curves

2. 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

3. HORIZONTAL AND VERTICAL TANGENTS
A parametric curve will have a horizontal tangent when dy/dt = 0 (provided dx/dt ≠ 0).
A parametric curve will have a vertical tangent when dx/dt = 0 (provided dy/dt ≠ 0).

4. THE SECOND DERIVATIVE
For a set of parametric equations, the second derivative is given by
Note that this is NOT the same as

5. AREA If the parametric curve is given by x = f (t) y = g(t)
and is traversed once as t increase from α to β, then the area under the curve is given by

6. ARC LENGTH
If a curve C is described by the parametric equations x = f (t), y = g(t), α ≤ t ≤ β, where f ′ and g′ are continuous on [α, β] and C is traversed exactly once as t increasing from α to β, then the length of C is

7. SURFACE AREA
Let C be the curve given by the parametric equations x = f (t), y = g(t), α ≤ t ≤ β, is rotated about the x-axis, where f ′ and g′ are continuous, and g(t) ≥ 0, then the surface area is given by

8. SURFACE AREA (CONCLUDED)
Let C be the curve given by the parametric equations x = f (t), y = g(t), α ≤ t ≤ β, is rotated about the y-axis, where f ′ and g′ are continuous, and f (t) ≥ 0, then the surface area is given by