Question

Define, describe and solve for speed, velocity and acceleration

Answer 1

Velocity = dy/dx = first derivative of f(x) evaluated at given point. It is the change in position over time.

speed = absolute value of velocity (square your velocity and then square root it)

acceleration = d^2y/dx^2 = second derivative of f(x) evaluated at a given point. It is also the first derivative of your velocity function, or the change in velocity over time.

Both velocity and acceleration are vectors, that is, they are either positive or negative in a direction. Speed is the magnitude of your velocity and has no direction: it is simply the absolute value of your velocity.

2) For multi-variable functions: f(x,y)

velocity = first partial derivative with respect to x + first partial derivative with respect to y. It is the vector sum of these two vectors.

speed = the same as it is for single variable, that is, speed is the absolute value of the velocity. However, in multi-variable functions you must square all your components and then square root it (ie: Pythagorean Theorem: sqrt(i^2+j^2+k^2), with (i,j,k) being the x,y,z, respectively).

acceleration is analogous to single variable acceleration. It is now the 2nd partial with respect to x + 2nd partial with respect to y.