We have one independent variable, usually denoted, t for time, and f(t) is a vector in three-dimensional space. Thus f(t)=(f1(t),f2(t),f3(t))---we call it the coordinates of a point at time t. We call the equations x=f1(t), y=f2(t), z=f3(t) parametric, where t is the parameter.
The velocity of the point is the derivative of f, i,e, the function f'(t) = (f1'(t),f2'(t),f3'(t)). The speed is the absolute value of the velocity. The acceleration is the second derivative f''(t) = (f1''(t),f2''(t),f3''(t)) of f.
Let T=f'(t)/|f'(t)| denote the unit tangent vector for time t .