BMTC - 131
CALCULUS
Indira Gandhi National Open University
School of Sciences
Block
4
APPLICATIONS OF DIFFERENTIAL CALCULUS
Block Introduction 3
Notations and Symbols 4
UNIT 12
Indeterminate Forms 5
UNIT 13
The Ups and Downs 37
UNIT 14
Curvature 79
UNIT 15
Asymptotes 117
UNIT 16
Curve Tracing 137
Miscellaneous Examples and Exercises 177
, Course Design Committee*
Prof. Rashmi Bhardwaj Prof. Meena Sahai
G.G.S. Indraprastha University, Delhi University of Lucknow
Dr. Sunita Gupta Dr. Sachi Srivastava
L.S.R. College, University of Delhi University of Delhi
Prof. Amber Habib
Shiv Nadar University Faculty Members
Gautam Buddha Nagar, U.P School of Sciences, IGNOU
Prof. M. S. Nathawat (Director)
Prof. S. A. Katre
University of Pune, Pune Dr. Deepika
Mr. Pawan Kumar
Prof. V. Krishna Kumar Prof. Poornima Mital
NISER, Bhubaneswar Prof. Parvin Sinclair
Dr. Amit Kulshreshtha Prof. Sujatha Varma
IISER, Mohali Dr. S. Venkataraman
Dr. Aparna Mehra
I.I.T. Delhi
Prof. Rahul Roy
Indian Statistical Institute, Delhi
* The course design is based on the recommendations of the Programme Expert Committee and the UGC-
CBCS template
Block Preparation Team
Prof. Amber Habib (Editor) Dr. Deepika
Shiv Nadar University School of Sciences
Gautam Buddha Nagar, U.P IGNOU, New Delhi
Dr. Malathy A. (Language Editor)
School of Humanities
IGNOU, New Delhi
Course Coordinators: Prof. Parvin Sinclair and Dr. Deepika
Acknowledgement: To Prof. Parvin Sinclair for comments on the manuscript. Also, to
Sh. Santosh Kumar Pal for the word processing and to Sh. S. S. Chauhan for preparing CRC of
this block. Parts of this block are based on the course material of the previous course Calculus
(MTE-01).
July, 2019
© Indira Gandhi National Open University, 2019
ISBN-978-93-89200-41-6
All right reserved. No part of this work may be reproduced in any form, by mimeograph or any other means,
without permission in writing from the Indira Gandhi National Open University.
Further information on the Indira Gandhi National Open University courses, may be obtained from the
University’s office at Maidan Garhi, New Delhi-110 068 and IGNOU website www.ignou.ac.in.
Printed and published on behalf of the Indira Gandhi National Open University, New Delhi by
Prof. M. S. Nathawat, Director, School of Sciences.
2
,BLOCK 4 APPLICATIONS OF DIFFERENTIAL
CALCULUS
In Block3, you have learnt some techniques of differentiation, and have differentiated a
wide variety of functions. In this block, we shall use the derivative to explore various
geometrical features of a curve, like maxima/minima, concavity/convexity, tangents,
normals, asymptotes and so on. For this we have to make use of not only the first
derivative, but also some higher order derivatives.
In Unit 12, we shall use the first derivative to find the limits involving an algebraic
combination of functions in an independent variable, in which evaluation of limit gives form
0 ∞
like , , 0 × ∞, 0 ∞ , etc. Such a form is called indeterminate form.
0 ∞
In the next three units, Unit 13, Unit 14 and Unit 15, we shall illustrate how we can find the
exact shape of a curve, when its equation is given to us. You will be surprised at the
amount of information which is revealed by the first and second derivatives. We shall use
this information to trace various standard curves in Unit 16. In Unit 16, we shall also tell
you how the properties of some remarkable curves are put to use. We shall also ask you
to trace some curves yourself. Do try and trace them by systematically following the
procedure which we have outlined in Unit 16. We are sure, that after reading this block
you will be aware of the presence of many of these curves in the objects around you, as
also in nature.
We have also made a video programme, “Curves”, which you can watch after going
through this block. This programme is available at your study center.
A word about some signs used in the unit! Throughout each unit, you will find theorems,
examples and exercises. To signify the end of the proof of a theorem, we use the sign .
To show the end of an example, we use ***. Further, equations that need to be referred to
are numbered sequentially within a unit, as are exercises and figures. E1, E2 etc. denote
the exercises and Fig. 1, Fig. 2, etc. denote the figures.
3
, NOTATIONS AND SYMBOLS (used in Block 4)
See the list of notations and symbols in Block 1, Block 2 and Block 3.
4
CALCULUS
Indira Gandhi National Open University
School of Sciences
Block
4
APPLICATIONS OF DIFFERENTIAL CALCULUS
Block Introduction 3
Notations and Symbols 4
UNIT 12
Indeterminate Forms 5
UNIT 13
The Ups and Downs 37
UNIT 14
Curvature 79
UNIT 15
Asymptotes 117
UNIT 16
Curve Tracing 137
Miscellaneous Examples and Exercises 177
, Course Design Committee*
Prof. Rashmi Bhardwaj Prof. Meena Sahai
G.G.S. Indraprastha University, Delhi University of Lucknow
Dr. Sunita Gupta Dr. Sachi Srivastava
L.S.R. College, University of Delhi University of Delhi
Prof. Amber Habib
Shiv Nadar University Faculty Members
Gautam Buddha Nagar, U.P School of Sciences, IGNOU
Prof. M. S. Nathawat (Director)
Prof. S. A. Katre
University of Pune, Pune Dr. Deepika
Mr. Pawan Kumar
Prof. V. Krishna Kumar Prof. Poornima Mital
NISER, Bhubaneswar Prof. Parvin Sinclair
Dr. Amit Kulshreshtha Prof. Sujatha Varma
IISER, Mohali Dr. S. Venkataraman
Dr. Aparna Mehra
I.I.T. Delhi
Prof. Rahul Roy
Indian Statistical Institute, Delhi
* The course design is based on the recommendations of the Programme Expert Committee and the UGC-
CBCS template
Block Preparation Team
Prof. Amber Habib (Editor) Dr. Deepika
Shiv Nadar University School of Sciences
Gautam Buddha Nagar, U.P IGNOU, New Delhi
Dr. Malathy A. (Language Editor)
School of Humanities
IGNOU, New Delhi
Course Coordinators: Prof. Parvin Sinclair and Dr. Deepika
Acknowledgement: To Prof. Parvin Sinclair for comments on the manuscript. Also, to
Sh. Santosh Kumar Pal for the word processing and to Sh. S. S. Chauhan for preparing CRC of
this block. Parts of this block are based on the course material of the previous course Calculus
(MTE-01).
July, 2019
© Indira Gandhi National Open University, 2019
ISBN-978-93-89200-41-6
All right reserved. No part of this work may be reproduced in any form, by mimeograph or any other means,
without permission in writing from the Indira Gandhi National Open University.
Further information on the Indira Gandhi National Open University courses, may be obtained from the
University’s office at Maidan Garhi, New Delhi-110 068 and IGNOU website www.ignou.ac.in.
Printed and published on behalf of the Indira Gandhi National Open University, New Delhi by
Prof. M. S. Nathawat, Director, School of Sciences.
2
,BLOCK 4 APPLICATIONS OF DIFFERENTIAL
CALCULUS
In Block3, you have learnt some techniques of differentiation, and have differentiated a
wide variety of functions. In this block, we shall use the derivative to explore various
geometrical features of a curve, like maxima/minima, concavity/convexity, tangents,
normals, asymptotes and so on. For this we have to make use of not only the first
derivative, but also some higher order derivatives.
In Unit 12, we shall use the first derivative to find the limits involving an algebraic
combination of functions in an independent variable, in which evaluation of limit gives form
0 ∞
like , , 0 × ∞, 0 ∞ , etc. Such a form is called indeterminate form.
0 ∞
In the next three units, Unit 13, Unit 14 and Unit 15, we shall illustrate how we can find the
exact shape of a curve, when its equation is given to us. You will be surprised at the
amount of information which is revealed by the first and second derivatives. We shall use
this information to trace various standard curves in Unit 16. In Unit 16, we shall also tell
you how the properties of some remarkable curves are put to use. We shall also ask you
to trace some curves yourself. Do try and trace them by systematically following the
procedure which we have outlined in Unit 16. We are sure, that after reading this block
you will be aware of the presence of many of these curves in the objects around you, as
also in nature.
We have also made a video programme, “Curves”, which you can watch after going
through this block. This programme is available at your study center.
A word about some signs used in the unit! Throughout each unit, you will find theorems,
examples and exercises. To signify the end of the proof of a theorem, we use the sign .
To show the end of an example, we use ***. Further, equations that need to be referred to
are numbered sequentially within a unit, as are exercises and figures. E1, E2 etc. denote
the exercises and Fig. 1, Fig. 2, etc. denote the figures.
3
, NOTATIONS AND SYMBOLS (used in Block 4)
See the list of notations and symbols in Block 1, Block 2 and Block 3.
4
