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KIET GROUP OF INSTITUTIONS, DELHI-NCR, GHAZIABADB.Tech (I sem), 2020-21
MATHEMATICS-I (KAS-103T)
MODULE 2 Differential Calculus- I
SYLLABUS: Introduction to limits, continuity and differentiability, Rolle’s Theorem,
Lagrange’s Mean value theorem and Cauchy mean value theorem, Successive
Differentiation (nth order derivatives), Leibnitz theorem and its application, Envelope,
Involutes and Evolutes, Curve tracing: Cartesian and Polar co-ordinates
CONTENT
Serial no. Topic Page No.
2.1 Limit 02
2.2 Continuity 07
2.3 Differentiability 10
2.4 Rolle’s theorm 16
2.5 Lagrange’s theorem 19
2.6 Cauchy’s theorem 22
2.7 Successive differentiation 25
2.8 Leibnitz theorem 29
2.9 Curve tracing 39
2.9.1 In cartesian coordinate 39
2.9.2 In polar coordinate 48
2.10 Envelopes and evolutes 55
2.11 E-link for more understanding 64
1
, KIET GROUP OF INSTITUTIONS, DELHI-NCR, GHAZIABAD
LIMIT, CONTINUITY AND DIFFERENTIABILITY
Derivatives and Integrals are the core practical aspects of Calculus. They were the first things
investigated by Archimedes and developed by Liebnitz and Newton. The process involved examining
smaller and smaller pieces to get a sense of a
progression toward a goal. This process was not formalized algebraically, though, at the time. The
theoretical underpinnings of these operations were developed and formalized later by Bolzano,
Weierstrauss and others. These core concepts in this area are Limits, Continuity and Differentiability.
Derivatives and Integrals are defined in terms of limits. Continuity and Differentiability are important
because almost every theorem in Calculus begins with the condition that the function is continuous
and differentiable. The Limit of a function is the function value (y-value) expected by the trend (or
sequence) of y-values yielded by a sequence of x-values that approach the x-value being investigated.
In other words, the Limit is what the y-value should be for a given x-value, even if the actual y-value
does not exist. The limit was created /defined as an operation that would deal with y-values that were
of an indeterminate form.
APPLICATIONS IN REAL LIFE
In engineering for building skyscrapers, bridges.
In robotics to know how robotic parts work on given command.
For system design in electrical and computer engineering.
Used in improving safety of vehicles.
Derivatives are used to calculate profit and loss in business using graphs.
Derivatives are used to check temperature variation.
In medical sciences and biology to model growth/decay, chaos, inter-dependable
relations and much more.
Edge detection in image processing.
2
, KIET GROUP OF INSTITUTIONS, DELHI-NCR, GHAZIABAD
2.1 LIMIT
Approaching….
Sometimes we can’t work something out directly ….but we can see what it should be as we get closer
and closer!
For example
x 2
1
( x 1)
Let’s work out for x=1
1 2
1
1 1 0
1 1 1 1 0
Now 0/0 is a difficulty…because we don’t know the value of 0/0 as it is indeterminate form hence we
need another way to answer this.
So instead of trying to work it out for x=1 let’s try approaching it closer and closer
x x 2
1
x 1
0.5 1.50000
0.9 1.90000
0.99 1.99000
0.999 1.99900
0.9999 1.99990
0.99999 1.99999
So with the help of this example we can see that as x get closer to 1, then
x 2
1
gets close to 2.
( x 1)
Interesting situation!!
When x=1 we don’t know the answer ( It is indeterminate)
But we can see that it is going to be 2
We want to give answer “2” but can’t , so instead mathematicians say exactly what is going on by
using the special word “limit”
3
, KIET GROUP OF INSTITUTIONS, DELHI-NCR, GHAZIABAD
The limit of
x 2
1
as x approaches 1 is 2 and it written in symbols as: lim x 1
x 2
1
2
( x 1) ( x 1)
So it is special way of saying, “ ignoring what happens when we get there, but as we get closer the
answer gets closer and closer to 2”.
As a graph it look like this
Fig.1
So, in truth, we cannot say what is the value at x=1 is but we can say that as we approach 1, the limit is
2.
In language of Mathematics we can say that “f(x) gets close to some limit as x gets close to some
value”.
When we call the limit “L”, and the value that x gets close to “a” we can say - “ f(x) gets close to L as
x gets close to a”
f(x) →L as x→a
2.1.1 Definition of a Limit
limxa f ( x) L If and only if for every 0 , there exist 0 s.t f ( x) L for all values of x for
which 0 x a .
4
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