explanation of the course of Nature – WHITEHEAD v . Introduction This chapter is an introduction to Calculus. Calculus is that branch of mathematics which mainly deals with the study of change in the value of a function as the points in the domain change. First, we give an intuitive idea of derivative (without actually defining it).
Then we give a naive definition of limit and study some algebra of limits. Then we come back to a definition of derivative and study some algebra of derivatives. We also obtain derivatives of certain standard functions. .
Intuitive Idea of Derivatives Physical experiments have confirmed that the body dropped from a tall cliff covers a distance of . t metres in t seconds, i.e., distance s in metres covered by the body as a function of time t in seconds is given by s = . t . The adjoining Table .
gives the distance travelled in metres at various intervals of time in seconds of a body dropped from a tall cliff. The objective is to find the veloctiy of the body at time t = seconds from this data. One way to approach this problem is to find the average velocity for various intervals of time ending at t = seconds and hope that these throw some light on the velocity at t = seconds. Average velocity between t = t and t = t equals distance travelled between t = t and t = t seconds divided by ( t – t ) .
Hence the average velocity in the first two seconds