Sir Isaac Newton invented calculus and was a pioneer in the study of motion. Although he did not develop calculus specifically to study motion (as is sometimes asserted), the tools of differentiation and integration he developed prove very useful in this realm.

Above, you see the graph of a function x(t) representing the position with respect to time of an object moving along a straight line. In fact, this graph describes the motion of the red mass attached to a spring, moving over a friction-free surface like an ideal air hockey table. In an overhead view, you see the mass moving back and forth as a sheet of paper scrolls beneath it, and its position over time is graphed on the paper. Later, you will study this form of motion, called simple harmonic motion, in more depth.

The x position is the distance of the mass from its "zero" or equilibrium position, where it would be at rest if undisturbed. Its velocity (both speed and direction) changes as it moves.

The position of the mass with respect to time is described by the cosine function you see in the illustration of Example 1. In the examples, we compute the velocity and acceleration of the mass using derivatives. To do so, we use the derivative of the cosine function, which is the negative of the sine function. Since the argument to the cosine is a function of t, we use the chain rule, taking the derivative of the argument also and multiplying the two derivatives. This gives us the velocity function. Similarly, the derivative of the sine is the cosine and we again apply the chain rule to find the function for acceleration.

Notice that the velocity and acceleration functions are also trigonometric − in fact, sine and cosine − functions. The position, velocity, and acceleration of a mass attached to an ideal spring like this are all sinusoidal functions.