Parametric Functions

First Derivative:

dy/dx = dy/dt
dx/dt

Example:
x = cos q
y = 3 sin q

dy/dt = 3 cos q
dx/dt = -sin q

so dy/dt = -sin q
3 cos q

Second Derivative:

d²y/dx² = d(y' ) = d(y' )/dt
dx dx/dt

Example:

Find the second derivative if
x = t - t²
y = t – t³

First Derivative: dy/dt = 1 - 3t²
                           dx/dt = 1 - 2t
                           dy/dx = 1 - 3t²
                                        1 - 2t

Second Derivative:

Length of a Parametric Curve:

If a smooth curve x = f(t), y = g(t), a < t < b, is traversed exactly once as t increases from a to b, the curve’s length is

Find length: x = 8 cos t + 8t sin t,
y = 8 sin t – 8t cos t , 0 < t < p/2

Surface Area:

Revolution about the x-axis (y > 0):

Revolution about the y-axis (x > 0):

Example:

x = ln (sec t + tan t) – sin t
y = cos t
0 < t < p/3

Find the surface area of the solid formed when this curve is rotated around the x-axis.

Answer: