Calculus of Parametric Curves

Parametric differentiation

Velocity and speed

Arc length parametrically

Surface area of revolution

For a parametric curve x = x(t), y = y(t), the slope dy/dx equals which ratio?

A parametric curve has a vertical tangent where which condition holds?

The parametric arc length integral for the unit circle x = cos t, y = sin t over [0, 2 pi] gives what integrand?

The arc length of one cycloid arch works out to 8r. Compared to the rolling circle's circumference 2 pi r, the arch is about how many times longer?

At a parametric curve's singular point, both dx/dt and dy/dt vanish. What behavior typically occurs there?

At the projectile's peak, the vertical velocity vanishes and the speed equals the horizontal component v0 cos theta. As a function name, what trig function appears? Answer in lowercase.

The arc length integrand is the speed root of (x')^2 plus (y')^2. As a single word, what physical quantity is it? Answer in lowercase.

The second derivative d^2y/dx^2 divides d/dt of dy/dx by dx/dt. As a derivative, what is the denominator? Answer like dx/dt.

Rotating a parametric curve about the y axis instead of the x axis changes the surface area formula how?

Rotating the parametric curve about the x axis gives surface area 2 pi integral of y times speed dt. As a single letter, what is the band's radius? Answer with one lowercase letter.