Now that we know how to compute the derivatives of many common functions, we can give a few examples of why the derivative is such a useful tool. In this chapter, we will look at four different applications of the derivative.

The first application, is to use the derivative to find the velocity and acceleration of a particle moving in a straight line. When we are given a function f (t) describing the position of a particle at time t, the velocity of the particle at time t is the derivative f'(t) and the acceleration is the second derivate f''(t).

The second application is the analysis of graphs of functions . We can use the derivative to find critical points and inflection points on graphs, from which a reasonably good sketch of a function can be constructed.

The second applications is related to the third, optimization of functions . For example, one may encounter a function in the business world that gives the total profit of producing a certain number of goods. It would then be natural to try to maximize such a function.

The fourth and final application concerns related rates . Suppose water is flowing into a gigantic ice cream cone at a fixed rate (for some strange reason). Through a clever application of differentiation, it is possible to determine how quickly the water level will be rising when it reaches any particular height in the cone.

Next section Velocity and Acceleration

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