Unit 9 AP Calculus

If f and g are continuous functions of t on an interval l, then the equations x=f(g) and y=g(t) are called __ equations and t is called the __.

If f and g are continuous functions of t on an interval l, then the equations x=f(g) and y=g(t) are called __ equations and t is called the __.

parametric, parameter

When sketching a curve by hand represented by parametric equations, you use increasing values of t, so the curve will be traced over a specific __.

direction

If we only wish to sketch the general shape of a plane curve, we do so by eliminating the parameter to create a rectangular equation y=f(x). How?

Solve for the parameter in one of the parametric equations, then replacing the result in the other equation.

If a smooth curve C is given by the equations x=f(t) and y=g(t), then the slope of C at (x,y) is dy/dx=

dy/dt / dx/dt

Second derivative of a parametric d^2y/dx^2=

d/dt(dy/dx) / dx/dt

Third derivative of a parametric d^3y/dx^3=

d/dx(d^2y/dx^2) / dx/dt

Arc length in parametric form

(integral from a to b) ∫√(dx/dt)^2 + (dy/dt)^2 dt

Area of a surface of revolution.

Revolution about the x-axis: g(t)≥0 → (integral from a to b) 2pi∫g(t)√(dx/dt)^2+(dy/dt)^2