The derivative of a function is the instantaneous rate of change. For example, the (first) derivative of [b]position [/b]is [b]velocity [/b]and describes how quickly (or slowly) the position is changing at any given time. But, velocity itself is a function and has its own rate of change, which we call [b]acceleration[/b]. In other words, acceleration describes how the velocity changes, which in turn describes how the position changes. So, acceleration also tells us something about position. [br][br]More generally, a function has a derivative, and the derivative has a derivative (and that derivative has a derivative), and so on. The [b]order [/b]of a derivative is essentially how many times we differentiated the "original" function. So, the second derivative is the derivative of the (first) derivative. The third derivative is the derivative of the second derivative, and so on. [br][br][b]Simple Harmonic Oscillators: [/b]The standard example of this type of motion is a mass attached to the end of a spring with no other forces (e.g., friction). If you move the mass and stretch the spring, it will pull the mass back toward equilibrium but because of its momentum it will move past equilibrium to the other side until the spring compresses enough and begins pushing the mass back the other way. Because of their periodic nature, these problems are naturally modeled by trig functions.

The applet shows the motion of a point along a vertical (one-dimensional) line. [br][list][*]Notice that the (instantaneous) position, velocity, and acceleration of the point are given.[/*][*]Use the slider tool or input box to move the point P to a specific time value. [/*][*]Use the check boxes to show/hide the tangent line, graph of [math]f'(x)[/math], and graph of [math]f''(x)[/math]. [br][/*][/list]