vWith the Calculus as a key, Mathematics can be successfully applied to the

explanation of the course of Nature – WHITEHEAD v

13.1 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.

13.2 Intuitive Idea of Derivatives

Physical experiments have confirmed that the body dropped

from a tall cliff covers a distance of 4.9t

2

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 = 4.9t

2

.

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 = t

1

and t = t

2

equals distance travelled between

t = t

1

and t = t

2

seconds divided by (t

2

– t

1

). Hence the average velocity in the first

two seconds

13Chapter

LIMITS AND DERIVATIVES

Sir Issac Newton

(1642-1727)

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282 MATHEMATICS

=

2 1

2 1

Distance travelled between 2 0

Time interval ( )

t and t

t t

= =

−

=

(

)

( )

19.6 0

9.8 /

2 0

m

m s

s

−

=

−

.

Similarly, the average velocity between t = 1

and t = 2 is

(

)

( )

19.6 – 4.9

2 1

m

s

−

= 14.7 m/s

Likewise we compute the average velocitiy

between t = t

1

and t = 2 for various t

1

. The following

Table 13.2 gives the average velocity (v), t = t

1

seconds and t = 2 seconds.

Table 13.2

t

1

0 1 1.5 1.8 1.9 1.95 1.99

v 9.8 14.7 17.15 18.62 19.11 19.355 19.551

From Table 13.2, we observe that the average velocity is gradually increasing.

As we make the time intervals ending at t = 2 smaller, we see that we get a better idea

of the velocity at t = 2. Hoping that nothing really dramatic happens between 1.99

seconds and 2 seconds, we conclude that the average velocity at t = 2 seconds is just

above 19.551m/s.

This conclusion is somewhat strengthened by the following set of computation.

Compute the average velocities for various time intervals starting at t = 2 seconds. As

before the average velocity v between t = 2 seconds and t = t

2

seconds is

=

2

2

Distance travelled between 2 seconds and

seconds

2

t

t −

=

2

2

Distance travelled in seconds Distance

travelled in 2 seconds

2

t

t

−

−

t s

0

0

1 4.9

1.5 11.025

1.8 15.876

1.9 17.689

1.95 18.63225

2 19.6

2.05 20.59225

2.1 21.609

2.2 23.716

2.5 30.625

3 44.1

4 78.4

Table 13.1

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LIMITS AND DERIVATIVES 283

=

2

2

Distance travelled in seconds 19.6

2

t

t

−

−

The following Table 13.3 gives the average velocity v in metres per second

between t = 2 seconds and t

2

seconds.

Table 13.3

t

2

4 3 2.5 2.2 2.1 2.05

2.01

v 29.4 24.5 22.05 20.58 20.09 19.845 19.649

Here again we note that if we take smaller time intervals starting at t = 2, we get

better idea of the velocity at t = 2.

In the first set of computations, what we have done is to find average velocities

in increasing time intervals ending at t = 2 and then hope that nothing dramatic happens

just before t = 2. In the second set of computations, we have found the average velocities

decreasing in time intervals ending at t = 2 and then hope that nothing dramatic happens

just after t = 2. Purely on the physical grounds, both these sequences of average

velocities must approach a common limit. We can safely conclude that the velocity of

the body at t = 2 is between 19.551m/s and 19.649 m/s. Technically, we say that the

instantaneous velocity at t = 2 is between 19.551 m/s and 19.649 m/s. As is

well-known, velocity is the rate of change of displacement. Hence what we have

accomplished is the following. From the given data of distance covered at various time

instants we have estimated the rate of

change of the distance at a given instant

of time. We say that the derivative of

the distance function s = 4.9t

2

at t = 2

is between 19.551 and 19.649.

An alternate way of viewing this

limiting process is shown in Fig 13.1.

This is a plot of distance s of the body

from the top of the cliff versus the time

t elapsed. In the limit as the sequence

of time intervals h

1

, h

2

, ..., approaches

zero, the sequence of average velocities

approaches the same limit as does the

sequence of ratios

Fig 13.1

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284 MATHEMATICS

3 3

1 1 2 2

1 2 3

C BC B C B

, ,

AC AC AC

, ...

where C

1

B

1

= s

1

– s

0

is the distance travelled by the body in the time interval h

1

= AC

1

,

etc. From the Fig 13.1 it is safe to conclude that this latter sequence approaches the

slope of the tangent to the curve at point A. In other words, the instantaneous velocity

v(t) of a body at time t = 2 is equal to the slope of the tangent of the curve s = 4.9t

2

at

t = 2.

13.3 Limits

The above discussion clearly points towards the fact that we need to understand limiting

process in greater clarity. We study a few illustrative examples to gain some familiarity

with the concept of limits.

Consider the function f(x) = x

2

. Observe that as x takes values very close to 0,

the value of f(x) also moves towards 0 (See Fig 2.10 Chapter 2). We say

(

)

0

lim 0

x

f x

→

=

(to be read as limit of f (x) as x tends to zero equals zero). The limit of f (x) as x tends

to zero is to be thought of as the value f (x) should assume at x = 0.

In general as x → a, f (x) → l, then l is called limit of the function f (x) which is

symbolically written as

(

)

lim

x a

f x l

→

=

.

Consider the following function g(x) = |x|, x

≠

0. Observe that g(0) is not defined.

Computing the value of g(x) for values of x very

near to 0, we see that the value of g(x) moves

towards 0. So,

0

lim

x

→

g(x) = 0. This is intuitively

clear from the graph of y = |x| for x

≠

0.

(See Fig 2.13, Chapter 2).

Consider the following function.

( )

2

4

, 2

2

x

h x x

x

−

= ≠

−

.

Compute the value of h(x) for values of

x very near to 2 (but not at 2). Convince yourself

that all these values are near to 4. This is

somewhat strengthened by considering the graph

of the function y = h(x) given here (Fig 13.2).

Fig 13.2

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LIMITS AND DERIVATIVES 285

In all these illustrations the value which the function should assume at a given

point x = a did not really depend on how is x tending to a. Note that there are essentially

two ways x could approach a number a either from left or from right, i.e., all the

values of x near a could be less than a or could be greater than a. This naturally leads

to two limits – the right hand limit and the left hand limit. Right hand limit

of a

function f(x) is that value of f(x) which is dictated by the values of f(x) when x tends

to a from the right. Similarly, the left hand limit. To illustrate this, consider the function

( )

1, 0

2, 0

x

f x

x

≤

=

>

Graph of this function is shown in the Fig 13.3. It is

clear that the value of f at 0 dictated by values of f(x) with

x ≤ 0 equals 1, i.e., the left hand limit of f (x) at 0 is

0

lim ( ) 1

x

f x

→

=

.

Similarly, the value of f at 0 dictated by values of

f (x) with x > 0 equals 2, i.e., the right hand limit of f (x)

at 0 is

0

lim ( ) 2

x

f x

+

→

=

.

In this case the right and left hand limits are different, and hence we say that the

limit of f (x) as x tends to zero does not exist (even though the function is defined at 0).

Summary

We say

lim

x a

→

–

f(x) is the expected value of f at x = a given the values of f near

x to the left of a. This value is called the left hand limit of f at a.

We say

lim ( )

x a

f x

+

→

is the expected value of f at x = a given the values of

f near x to the right of a. This value is called the right hand limit of f(x) at a.

If the right and left hand limits coincide, we call that common value as the limit

of f(x) at x = a and denote it by

lim

x a

→

f(x).

Illustration 1 Consider the function f(x) = x + 10. We want to find the limit of this

function at x = 5. Let us compute the value of the function f(x) for x very near to 5.

Some of the points near and to the left of 5 are 4.9, 4.95, 4.99, 4.995. . ., etc. Values

of the function at these points are tabulated below. Similarly, the real number 5.001,

Fig 13.3

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286 MATHEMATICS

5.01, 5.1 are also points near and to the right of 5. Values of the function at these points

are also given in the Table 13.4.

Table 13.4

From the Table 13.4, we deduce that value of f(x)

at x = 5 should be greater than

14.995 and less than 15.001 assuming nothing dramatic happens between x = 4.995

and 5.001. It is reasonable to assume that the value of the f(x) at x = 5 as dictated by

the numbers to the left of 5 is 15, i.e.,

(

)

–

5

lim 15

x

f x

→

=

.

Similarly, when x approaches 5 from the right, f(x) should be taking value 15, i.e.,

(

)

5

lim 15

x

f x

+

→

=

.

Hence, it is likely that the left hand limit of f(x) and the right hand limit of f(x) are

both equal to 15. Thus,

(

)

(

)

(

)

5

5 5

lim lim lim 15

x

x x

f x f x f x

− +

→

→ →

= = =

.

This conclusion about the limit being equal to 15 is somewhat strengthened by

seeing the graph of this function which is given in Fig 2.16, Chapter 2. In this figure, we

note that as x approaches 5 from either right or left, the graph of the function

f(x) = x +10 approaches the point (5, 15).

We observe that the value of the function at x = 5 also happens to be equal to 15.

Illustration 2 Consider the function f(x) = x

3

. Let us try to find the limit of this

function at x = 1. Proceeding as in the previous case, we tabulate the value of f(x) at

x near 1. This is given in the Table 13.5.

Table 13.5

From this table, we deduce that value of f(x) at x = 1 should be greater than

0.997002999 and less than 1.003003001 assuming nothing dramatic happens between

x 0.9 0.99 0.999 1.001 1.01 1.1

f(x) 0.729 0.970299 0.997002999 1.003003001 1.030301 1.331

x 4.9 4.95 4.99 4.995 5.001 5.01 5.1

f(x) 14.9 14.95 14.99 14.995 15.001 15.01 15.1

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