NCERT Book Class 11 Maths Chapter 12 Limits and Derivatives

Chapter 12

LIMITS AND DERIVATIVES

vWith the Calculus as a key, Mathematics can be successfully applied to the
explanation of the course of Nature – WHITEHEAD v

12.1 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.
12.2 Intuitive Idea of Derivatives Sir Issac Newton
Physical experiments have confirmed that the body dropped (1642-1727)
from a tall cliff covers a distance of 4.9t2 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 = 4.9t2.
The adjoining Table 13.1 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 = 2 seconds from this
data. One way to approach this problem is to find the average velocity for various
intervals of time ending at t = 2 seconds and hope that these throw some light on the
velocity at t = 2 seconds.
Average velocity between t = t1 and t = t2 equals distance travelled between
t = t1 and t = t2 seconds divided by (t2 – t1). Hence the average velocity in the first
two seconds