10.2 Calculus with Parametric Curves

For the parametric curve x = f(t), y = g(t), the derivative can be computed as

dy/dx = (dy/dt) / (dx/dt) = (g’(t)/f’(t))

To find horizontal tangent lines to this parametric curve

solve dy/dx = 0, so g’(t) = 0 and f’(t) ≠ 0 or dy = 0 and dx ≠ 0

To find vertical tangent lines to this parametric curve

solve dy/dx = undefined, so dx = 0 or f’(t) = 0

If there is a t = a such that g’(a) = 0 and f’(a) = 0, then use

L’Hospital’s rule on the limit lim (t goes to a) dy/dx

second derivative (d²y/dx²) of the parametric curve x = f(t), y = g(t) is computed by

d²y/dx² = (d/dx)(dy/dx)

second derivative (d²y/dx²) of the parametric curve x = f(t), y = g(t) is computed by (WRITE IN TERMS OF d/dt)

d²y/dx² = ( (d/dt) (dy/dt) ) / (dx/dt)

second derivative (d²y/dx²) of the parametric curve x = f(t), y = g(t) is computed by (WRITE IN TERMS OF f’(t) and g’(t))

d²y/dx² = ( ( g’(t) / f’(t) )’ / f’(t) )