applied math 2nd semester diploma

As per MSBTE’s ‘ I ’ Scheme Revised syllabus w.e.f. academic year 2017-2018
Subject Code : 22206 (AMP)
First Year Diploma
SEMESTER – II
For
Auto/Mech/Production/Chemical Engg. (AE/ME/PG/PT/CH/)
M.Sc. Ph.D. (Mathematics)
Head, Dept. of Applied Sciences
S.S.V.P.Sanstha’s. Bapusaheb Shivajirao Deore Polytechnic, Dhule.
‘I’ Scheme Committee Member
M.Sc., M.Phil. (Mathematics)
Govt. Polytechnic, Miraj
‘I’ Scheme Committee Member
M.Sc.(Mathematics)
PVPIT Budhgaon, Sangli
M.Sc. (Mathematics)
Guru Gobind Singh Polytechnic, Nashik
M.Sc. (Mathematics)
K.K. Wagh Polytechnic, Nashik
M.Sc. (Mathematics)
MVPS’s Rajashri Shahu Maharaj
Polytechnic College, Nashik
M.Sc. (Mathematics) B.Ed. (Maths)
Sanjivani K. B. P. Polytechnic, Kopargaon,
Dist. Ahmednagar
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Gigatech Publishing House
Igniting Minds

Applied Mathematics (AMP) 22206
First Year Diploma (Semester – II)
Auto/Mech/Production/Chemical Engg. (AE/ME/PG/PT/CH)
Dr. S. P. Pawar
First Edition : 2017
Published By : Gigatech Publishing House
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Shan Bramha Complex, Pune – 411 002.
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Preface
With a great pleasure and satisfaction, we present the text book of
‘Applied Mathematics’ for the new curriculum (Semester pattern) ‘ I ’
scheme with effect from the academic year 2017-18 for First Year Diploma
Course in ‘Engineering and Technology’ (Semester-II). In presenting this
First Edition, an utmost care has been taken to make the contents precise,
simple and perfect. From our long experience, we have constantly kept in
mind the requirements of the common student for understanding the
subject Mathematics, as related to the technology. Hence, the contents are
presented in very simple & easy language. The special feature is that we
have included lot of exercises at the end of each chapter with answers,
which will certainly help to understand the subject.
We are very thankful to “Gigatech Publishing House, TEAM” for
their encouragement and co-operation to write this text book.
We are also thankful to Shri. Kaustubh S. Pawar who has taken
untiring wholeheartedly efforts and gave innumerable suggestions to
make the book effective especially for common students coming from the
rural areas.
In spite of our best efforts to make the book unique and complete, it
may have some shortcomings. From bottom of our heart we earnestly and
sincerely request the Students, Professors and other Readers to inform us
any discrepancies observed in this book on the following e-mail address
which may be incorporated in the next edition.
Dr. S. P. Pawar Prof. H. D. Jadhav
sppawar17@yahoo.co.in j.hindurao@yahoo.com

Dedicated to
My beloved Father
Late
Appasaheb
Pandurang B. Pawar
&
My Mother
Late
Taisaheb
Venubai Pawar
who was the source of
my inspiration.

IMPORTANCE OF MATHEMATICS
 Mathematics is very important in our daily life. It finds application in vari-
ous types of professions.
 Mathematics is the language used in the understanding and deliverance of
scientific notions.
 Mathematics has a vital role in the engineering education.
 Mathematics equips pupils with uniquely powerful ways to describe, ana-
lyze and change the world.
 Mathematical thinking is important for all members of a modern society as
a habit of mind for its use in the workplace, business and finance; and for
personal decision-making.
 Generality and interconnection between subjects which can only be made
possible by the marriage between mathematics and engineering
knowledge.
 For the common man, knowledge of mathematics helps him in his personal
development and enhancing his mental abilities.
 Engineering is one of the most important professions for the mathematics
discipline.
 Engineering is a quantitative discipline, traditionally strongly based on
mathematics.


SYLLABUS
The following topics/subtopics should be taught and assessed in order to develop LOs in cognitive
domain for achieving the COs to attain the identified competency.
Unit
Major Cognitive domain
Learning Outcomes
Topics and
Sub-topics
Unit – I
Differential
Calculus
1a. Solve the given simple
problems based on functions.
1b. Solve the given simple
problems based on rules of
differentiation
1c. Obtain the derivatives of
logarithmic, exponential
functions.
1d. Apply the concept of
differentiation to find given
equation of tangent and normal
1e. Apply the concept of
differentiation to calculate
maxima and minima and
radius of curvature for given
function.
1.1 Functions and Limits :
a) Concept of function and
simple examples
b) Concept of limits without
examples.
1.2 Derivatives :
a) Rules of derivatives such as
sum, product, quotient of
functions.
b) Derivative of composite
functions (chain Rule),
implicit and parametric
functions.
c) Derivatives of inverse,
logarithmic and exponential
functions.
1.3 Applications of derivative :
a) Second order derivative
without examples
b) Equation of tangent and
normal
c) Maxima and minima
d) Radius of curvature
Unit – II
Integral Calculus
2a. Solve the given simple
problem(s) based on rules of
integration.
2b. Obtain the given integral(s)
using substitution method.
2c. Integrate given simple
functions using the integration
by parts.
2d. Evaluate the given simple
integral by partial fractions.
2.1 Simple Integration: Rules of
integration and integration of
standard functions.
2.2 Methods of Integration:
a. Integration by substitution.
b. Integration by parts
c. Integration by partial fractions.

Unit – III
Applications of
Definite
Integration
3a. Solve given simple problems
based on properties of definite
integration.
3b. Apply the concept of definite
integration to find the area
under the given curve(s).
3c. Utilize the concept of definite
integration to find area
between given two curves.
3d. Invoke the concept of definite
integration to find the volume
of revolution of given surface.
3.1 Definite Integration:
a) Simple examples
b) Properties of definite integral
(without proof) and simple
examples.
3.2 Applications of integration :
a) Area under the curve.
b) Area between two curves.
c) Volume of revolution
Unit-IV
First Order First
Degree Differential
Equations
4a. Find the order and degree of
given differential equations.
4b. Form simple differential
equations for given simple
engineering problem(s).
4c. Solve given differential
equations using the method of
variable separable.
4d. Solve the given simple
problem(s) based on linear
differential equations.
4.1 Concept of differential equation
4.2 Order, degree and formation of
differential equation.
4.3 Solution of differential equation
a) Variable separable form.
b) Linear differential equation.
4.4 Application of differential
equations and related engineering
problems.
Unit –V
Probability
Distribution
5a. Make use of probability
distribution to identify discrete
and continuous probability
distribution.
5b. Solve given problems based on
repeated trials using Binomial
distribution.
5c. Solve given problems when
number of trials are large and
probability is very small
5d. Utilize the concept of normal
distribution to solve related
engineering problems.
5.1 Probability distribution
a) Discrete Probability
distribution
b) Continuous Probability
distribution
5.2 Binomial distribution.
5.3 Poisson’s distribution.
5.4 Normal distribution.


Suggested Specification Table for Question Paper Design
Unit
No.
Unit Title
Teaching
Hours
Distribution of Theory Marks
R
Level
U
Level
A
Level
Total
Marks
I Differential calculus 20 04 08 12 24
II Integral calculus 14 02 06 08 16
III
Applications of Definite
Integration.
10 02 02 04 08
IV
First Order First Degree
Differential Equations
08 02 02 04 08
V Probability distribution 12 02 05 07 14
Total 64 12 23 35 70
Legends :
R=Remember, U=Understand, A=Apply and above (Bloom’s Revised taxonomy)
Note :
This specification table provides general guidelines to assist student for their learning and to teachers to
teach and assess students with respect to attainment of LOs. The actual distribution of marks at different
taxonomy levels (of R, U and A) in the question paper may vary from above table.
Recommended by MSBTE Text Books and Reference Books
Text Books :
Sr. No. Title of Book Author Publication
1 Higher Engineering
Mathematics
Grewal, B.S. Khanna publications, New Delhi ,
2013 ISBN: 8174091955
2 A Text Book of Engineer-
ing Mathematics
Dutta, D. New Age Publications, New Delhi,
2006, ISBN-978-81-224-1689-3
3 Advanced Engineering
Mathematics
Krezig, Ervin Wiley Publications, New Delhi, 2016
ISBN:978-81-265-5423-2,
4 Advanced Engineering
Mathematics
Das, H.K. S. Chand Publications, New Delhi,
2008, ISBN:9788121903455
5 Engineering Mathematics
Volume 1 (4th edition)
Sastry, S.S. PHI Learning, New Delhi, 2009
ISBN-978-81-203-3616-2,
6 Comprehensive Basic
Mathematics, Volume 2
Veena, G.R. New Age Publications, New Delhi,
2005 ISBN: 978-81-224-1684-8

7 Getting Started with
MATLAB-7
Pratap, Rudra Oxford University Press, New Delhi,
2009, ISBN: 10: 0199731241
8 Engineering Mathematics
(3rd edition).
Croft, Anthony Pearson Education, New Delhi,2010
ISBN: 978-81-317-2605-1
Software/Learning Websites :
a. www.scilab.org/ - SCI Lab
b. www.mathworks.com/products/matlab/ - MATLAB
c. Spreadsheet applications
d. www.dplot.com/ - DPlot
e. www.allmathcad.com/ - MathCAD
f. www.wolfram.com/mathematica/ - Mathematica
g. http://fossee.in/
h. https://www.khanacademy.org/math?gclid=CNqHuabCys4CFdOJaAoddHoPig
i. www.easycalculation.com
j. www.math-magic.com


Chapter 1 : Functions and Limits......................................................................... 1.1 – 1.12
1.1 Introduction
1.2 Function
1.3 Types of Functions
1.4 Other Functions
1.5 Limit of A Function
1.6 Definition of Limit
1.7 Algebra of Limits
1.8 Types of Limits
Chapter 2 : Derivatives.......................................................................................... 2.1 – 2.64
2.1 Concept of Derivative
2.2 Derivative of A Function
2.3 Derivative of Standard Functions
2.4 Rules of Differentiation
2.5 Derivative of Composite Functions
2.6 Derivative of Implicit Functions
2.7 Derivative of Parametric Functions
2.8 Derivative of Inverse Functions
2.9 Derivative of Inverse Trigonometric Functions
2.10 Derivative of Exponential & Logarithmic Functions
2.11 Logarithmic Differentiation
Chapter 3 : Applications of Derivative ................................................................ 3.1 – 3.40
3.1 Introduction
3.2 Successive Differentiation
3.3 Tangent and Normal
3.4 Maxima and Minima
3.5 Radius of Curvature

Chapter 4 : Simple Integration............................................................................. 4.1 – 4.20
4.1 Introduction
4.2 Definition of Integration ss Anti−Derivative
4.3 Constant of Integration
4.4 Rules of Integration
4.5 Integration of Standard Functions
Chapter 5 : Methods of Integration...................................................................... 5.1 – 5.54
5.1 Methods of Integration
5.2 Integration by Substitution
5.3 Integration of Rational Functions
5.4 Integrals of The Type 
Or 
Or  
5.5 Integration by Parts
5.6 Integration by Partial Fractions
Chapter 6 : Definite Integration .......................................................................... 6.1 – 6.18
6.1 Definite Integration
6.2 Definition
6.3 Fundamental Properties of Definite Integration
Chapter 7 : Applications of Integration.............................................................. 7.1 – 7.22
7.1 Applications of Definite Integrals
7.1.1 Area Under The Curve
7.2 Area Between Two Curves
7.3 Volume of Solid of Revolution
Chapter 8 : Differential Equation......................................................................... 8.1 – 8.28
8.1 Concept of Differential Equation
8.2 Order, Degree and Formation of Differential Equation
8.3 Solution of Differential Equation
8.4 Ordinary Differential Equation of First Order & First Degree

Chapter 9 : Application of Differential Equations.............................................. 9.1 – 9.14
9.1 Application of Differential Equations and Related Engineering Problems
Chapter 10 : Probability Distribution.............................................................. 10.1 – 10.54
10.1 Basic Ideas of Probability
10.2 Probability Distribution of Random Variables
10.3 Binomial Distribution (Discrete)
10.4 Poison Distribution
10.5 Normal Probability Distribution


Unit – I
Differential Calculus
Teaching Hours Total Marks
20 24
Chapter No. Chapter Name
1. Functions and Limits
2. Derivatives
3. Applications of derivative

Topics and Sub-topics
1.1 Functions and Limits :
a) Concept of function and simple examples
b) Concept of limits without examples.
1.2 Derivatives :
a) Rules of derivatives such as sum, product, quotient of functions.
b) Derivatives of inverse, logarithmic and exponential functions.
1.3 Applications of derivative :
a) Second order derivative without examples
b) Equation of tangent and normal
c) Maxima and minima
d) Radius of curvature

Chapter 1
Syllabus :
 Concept of function and simple examples.
 Concept of limits without examples.
1.1 INTRODUCTION :
In mathematics we are using the words constants, variables and functions
frequently. But we don’t know the exact meaning of these words. In this chapter
we see how these terms are to be defined.
1. Constant : A mathematical quantity whose value is fixed is called constant.
There are two types of constants namely absolute constants and arbitrary
constants.
a) Absolute constant : A constant whose value is fixed in any experiment is
called absolute constant. e.g. π, e, g, etc.
b) Arbitrary constant : A constant whose value is fixed for a particular
experiment and it changes experiment to experiment is called arbitrary
constant.
e.g. y = mx + c is the equation of the straight line. Here m & c are the constants
and we can assign different values to m & c.
Unit I

Applied Mathematics 1.2 Functions and Limits
Igniting Minds
2. Variable : A mathematical quantity whose value is not fixed is called variable. We
can assign different values to the variables. There are two types of variables
namely dependent variable and independent variable.
a) Dependent variable : A variable whose value is decided by some other
variable is called dependent variable.
b) Independent variable : A variable which is free to assign any value to itself is
called independent variable. e.g. y = 8x + 2x − 68
Here x is free to take any value to itself but y is not free to take any value. Value of
y is decided by x only.
Therefore, here x is independent variable and y is dependent variable.
1.2 FUNCTION :
The relation between dependent variable and independent variable is called a
function. e.g. y = 8x + 2x − 68
We denote a function by the letters F, f. ϕ, ψ, h, etc.
i.e. y = f(x) means y is function of x.
1.3 TYPES OF FUNCTIONS :
There are two types of functions namely algebraic functions and transcendental
functions.
1) Algebraic Functions : The functions which are made up of constants and
variables attached with algebraic signs + , − , ×, ÷ are called algebraic functions.
Following are the types of algebraic functions.
a) Explicit Function : When dependent & independent both variables are
separately expressed then the function is called explicit function.
i.e. y = f(x)
e.g. y = x + x − x − 1

Igniting Minds
b) Implicit Function : When dependent & independent both variables are
combinely expressed and we cannot separate them then the function is
called implicit function. i.e. f(x, y) = 0
e.g. x3
+ y3
− x2
y + x y2
− 8x = 0
c) Composite Function : A function of function is called composite function
i.e. y = f(u) and u = g(x) then y = h(x) = f g(x)
e.g. y = u − 3u + 4u + 8 where u = x + 2
then y = ( x + 2) − 3( x + 2) + 4( x + 2) + 8
2) Transcendental Functions : The functions other than algebraic functions are
called transcendental functions.
Following are the types of transcendental functions.
a) Trigonometric/Circular Functions : Functions in which trigonometric/
circular functions are present are called trigonometric /Circular functions.
e.g. y = sin x + 7 cos x ⎯ 2
b) Inverse trigonometric/circular Functions : The functions in which
inverse trigonometric/circular functions presents are called inverse
trigonometric/circular functions. e.g. y = sin 1
3x − 4x3
c) Exponential Functions : The functions in which exponentials are present
are called exponential functions. e.g. y = e2x
+ 8e x
− 4x
d) Logarithmic Functions : The functions in which logarithms are present are
called logarithmic functions. e.g. y = log8
(x + 4) − log2 17

Igniting Minds
1.4 OTHER FUNCTIONS: Other than algebraic and transcendental functions,
following are the other functions.
a) Parametric Function : When dependent & independent both variables are
expressed in terms of some other parameter then the function is called
parametric function. i.e. x = f( t) and y = g( t )
e.g. x = t3
+ 1 and y = t2
− 2t + 8 where t is a parameter.
b) Even and Odd Function : (April-2012; Nov-2015; April-2016; April-2017)
Let f(x) be the function of x.
If f (– x) = f(x) then the function is called an even function and
If f (– x) = – f(x) then the function is called an odd function.
e.g. f(x) = sin x
∴ f (– x) = sin(−x) = − sin(x) = – f(x)
⟹ f (– x) = – f(x)
⟹ f(x) = sin x is an odd function.
Also, f(x) = cosx
∴ f (– x) = cos(−x) = cos(x) = f(x)
⟹ f (– x) = f(x)
⟹ f(x) = cos x is an even function.
c) Rational Function : A function defined by quotient of two polynomials is
called rational function. e.g. f(x) =
x2 2x 5
x (x 3)
d) Periodic Function : A function f(x) is said to be periodic function with
period P if f (x + P) = f(x)
e.g. f(x) = sin x
∴ f (x + 2π) = sin(x + 2π) = sin(x) = f(x)
 f(x) = sin x is periodic function with period 2π .
Similarly, cos x & tan x are periodic function with period 2π & π respectively.

Igniting Minds
1) If ( ) = − + , ( ) + ( ). April – 2008
Ans: Given: f(x) = x − 3x + 5
∴ f(0) = (0) − 3(0) + 5 by substituting x = 0
⟹ f(0) = 5
Also, f(3) = (3) − 3(3) + 5 by substituting x = 3
⟹ f(3) = 27 − 27 + 5
⟹ f(3) = 5
∴ f(0) + f(3) = 5 + 5
⟹ f(0) + f(3) = 10
2) If ( ) = − + , ( ) + ( ). May – 2007; Nov-2011 ; Nov-2016
Ans: Given: f(x) = x − 2x + 7
 f(0) = (0) − 2(0) + 7 by substituting x = 0
⟹ f(0) = 7
Also, f(2) = (2) − 2(2) + 7 by substituting x = 2
⟹ f(2) = 16 − 4 + 7
⟹ f(2) = 19
∴ f(0) + f(2) = 7 + 19
⟹ f(0) + f(2) = 26
3) If ( ) = + − find ( ) − ( ) April – 2009
Ans: Given: f(x) = x3 + x2 − 2
 f(1) − f(2) = (1) + (1) − 2 − (2) + (2) − 2
⟹ f(1) − f(2) = 1 + 1 − 2 − 8 + 4 − 2
⟹ f(1) − f(2) = 0 − 10
⟹ f(1) − f(2) = −10
Illustrative Examples

Igniting Minds
4) If ( ) = ( ) , . April – 2009
Ans: Given: f(x) = log(sinx)
 f
π
2
= log sin
π
2
by substituting x =
⟹ f
π
2
= log(1)
⟹ f
π
2
= 0
5) If ( ) = , ( ).
Ans: Given: f(x) = tan
 f(1) = tan (1) by substituting x =1
⟹ f(1) =
π
4
6) If ( ) = , √ .
Ans: Given: f(x) = tan
 f √3 = tan √3 by substituting x = √3
⟹ f √3 =
π
3
7) If ( ) = , ( ).
Ans: Given: f(x) = cos
 f(1) = cos (1) by substituting x = 1
⟹ f(1) = 0
8) If ( ) = + + ( ) + (− ). Nov-2012 ; April-2015
Ans: Given: f(x) = x + 6x + 10
 f(2) + f(−2) = (2) + 6(2) + 10 + (−2) + 6(−2) + 10
⟹ f(2) + f(−2) = 4 + 12 + 10 + 4 − 12 + 10
⟹ f(2) + f(−2) = 26 + 2
⟹ f(2) + f(−2) = 28

Igniting Minds
9) State with proof whether the function ( ) = − + is even or odd.
Ans: Given: f(x) = x − 3x + sin x ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ( i )
Consider f(−x) = (−x) − 3(−x) + sin(−x)
⟹ f(−x) = − x + 3x − sin x
⟹ f(−x) = − x − 3x + sin x
⟹ f(−x) = −f(x)
⟹ f(x) is an odd function.
10) State whether the function ( ) =
+ −
is even or odd. Nov-2011; April-2015
Ans: Given: f(x) =
ax + a−x
2
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ( i )
Consider f(−x) =
a−x + a−(−x)
2
⟹ f(−x) =
a−x + ax
2
⟹ f(−x) =
ax + a− x
2
⟹ f(−x) = f(x)
⟹ f(x) is an even function.
11) , ( ) =
–
+
.
Ans: Given: f(x) = log
1 − x
1 + x
⋯ ⋯ ⋯ ⋯ ⋯ ( i )
∴ f(−x) = log
1 − (−x)
1 + (−x)
= log
1 + x
1 − x
= − log
1 − x
1 + x
∵ log
a
b
= − log
b
a
⟹ f(−x) = −f(x) from (i)

Igniting Minds
12) If ( ) = + + − + . Show that ( ) + (− ) = ( )
Ans: Given: f(x) = 3x + x + 5 − 3 cos x + 2sin x ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ( i )
Consider f(−x) = 3(−x) + (−x) + 5 − 3 cos(−x) + 2sin (−x)
= 3x + x + 5 − 3 cos x + 2sin x
 sin(−θ) = − sin θ and cos (−θ) = cos θ
⟹ f(− x) = f(x) from ( i)
∴ f(x) + f(− x) = f(x) + f(x) = 2f(x)
13) Whether the function ( ) = − + + is even or odd? April-2013
Ans: Given: f(x) = x − 3x + sin x + x cos x ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ( i )
Now f(−x) = (−x) − 3(−x) + sin(−x) + (−x) cos(−x)
⟹ f(−x) = −x + 3x − sin x − x cos x
 sin(−θ) = − sin θ and cos(−θ) = cos θ
⟹ f(−x) = − x − 3x + sin x + x cos x
⟹ f(−x) = − f(x)
14) If ( ) = + + , ( ) + (− ) .
Ans: Given: f(x) = log √1 + x + x
 f(x) + f(− x) = log √1 + x + x + log 1 + (− x) + (− x)
⟹ f(x) + f(− x) = log √1 + x + x + log √1 + x − x
⟹ f(x) + f(− x) = log √1 + x + x ∙ √1 + x − x
 log + log = log( ∙ )
⟹ f(x) + f(− x) = log 1 + x − x  (a + b)(a − b) = a − b
⟹ f(x) + f(− x) = log(1)
⟹ f(x) + f(− x) = 0

Igniting Minds
1) If f(x) = 3x – 3x + 1 find f(0) + f(1) Ans: 2
2) If f(x) = 4x + 5x + 1 find f(1) – f(0) Ans: 9
3) If f(x) = x − 5x + 2 find f(1) + f(0) Ans: 0
4) If f(x) = x + x find f(1) + f(2) April-2011 Ans: 12
5) If f(x) = x – 4x + 7 find f(0) , f(−1) & f(2) Ans: 7; 12; 3
6) If f(x) = x – 3x + 1 find f(1) + f(2) Ans: 10
7) If f(x) = log(cos x) , find f(0). Ans: 0
8) If f(x) = log(tan x) , find f 4
. Ans: 0
9) If f(x) = log(sec x) , find f(0). Ans: 0
10) If f(x) = log(cosec x) , find f 4
. Ans: : √
11) If f(x) = log(cot x) , find f 4
. Ans: 0
12) If f(x) = sin , find f(1). Ans:
13) If f(x) = cos , find f(0). Ans:
14) If f(x) = cot , find f(1). Ans:
15) If f(x) = cosec , find f √2 . Ans:
16) If f(x) = log
1 + x
1 − x
state with proof whether function is even or odd. Ans: Odd
17) State whether the function f(x) =
ex + e−x
2
is even or odd. Nov-2014 Ans: Even
18) If f(x) = 3x − 2x + cos x state whether the function is even or odd. Ans: Even
19) If f(x) = x cos x + 4 sinx − 2x show that f(x) + f(−x) = 0
20) If f(x) = x − 8x + 2sin x show that f(x) + f(−x) = 0
21) If f(x) = x + 2x − 3x ∙ cos x show that f(x) + f(−x) = 0

Exercise

Igniting Minds
1.5 LIMIT OF A FUNCTION :
The concept of limit of a function is the basic process of calculus. In the earlier article
we obtained the value of the function by direct substitution of the values of x. But in
this article this method fails. Hence the concept of ‘Limit’ came forward.
1.6 DEFINITION OF LIMIT :
“A function f(x) is said to tends to the limit ‘ ’ as x tends to ‘ a ’ if for ϵ > 0, there
exists δ > 0 such that | f(x) − | < for all values of x other than ‘ a ’ for which
|x − a| < ” Mathematically we write it as
lim
x → a
f(x) =
Note: ϵ – epsilon & δ – delta are the smallest positive numbers not equal to zero
but very close to zero.
 Meaning of
→
( ) = :
Let f(x) be a function of x. If x assumes values nearer and nearer to ‘a’ except ‘a’, f(x)
assumes values nearer and nearer to ‘ ’, then we say that “f(x) tends to the limit as
x tends to a “and mathematically, it can be written as
lim
x → a
f(x) =
 Meaning of
→ ∞
( ) = :
Let f(x) be a function of x. If x assumes that larger and larger values, f(x) assumes
values nearer and nearer to ‘ ’ ,then we say that “ f(x) tends to the limit as x tends
infinity” and mathematically, it can be written as
lim
x → ∞
f(x) =
1.7 ALGEBRA OF LIMITS :
We shall assume, without proof, the following theorems which are helpful in finding
the limits of the functions. Let f(x) and g(x) be the functions of x.
Also, let
lim
x → a
f(x) = f(a) and
lim
x → a
g(x) = g(a) then
i)
lim
x → a
f(x) ± g(x) =
lim
x → a
f(x) ±
lim
x → a
g(x) = f(a) ± g(a)

Igniting Minds
ii)
lim
x → a
f(x) × g(x) =
lim
x → a
f(x) ×
lim
x → a
g(x) = f(a) × g(a)
iii)
lim
x → a
k ∙ f(x) = k ∙
lim
x → a
f(x) = k ∙ f(a) where k is a constant.
iv)
lim
x → a
f(x)
g(x)
= →
( )
→
( )
=
( )
( )
; g(a) ≠ 0
If g(a) = 0 then (x – a) is common factor in numerator & denominator, cancel it and
proceed.
1.8 TYPES OF LIMITS :
Following are the different types of limits.
A) Limits of Algebraic functions: In this article we will deal with limits of algebraic
functions which involve the process of factorization, synthetic division,
simplification and rationalisation.
Note: i) If the given expression is irrational then we use rationalisation of
numerator or denominator whichever is suitable.
ii) For finding the limit of a ratio of two functions in x as x→ ∞, we divide
all the terms of the ratio by the highest degree of the functions so that
the function will be in terms of powers of and we know very well
that as x → ∞,
1
x
→0 which will be convenient for evaluation of the
limit.
B) Limits of Trigonometric functions :
In practice, while solving the examples on the limits of trigonometric functions,
we use the fundamental identities, concept of allied angles, multiple angles,
compound angles and inverse trigonometric functions which we have already
studied in trigonometry.
To evaluate limits of trigonometric functions, we remember the following
formulae without proof.

Igniting Minds
1.
lim
x → 0
= 1 
lim
x → 0
x
sinx
= 1 ; x in radian measure.
2.
lim
x → 0
cos x = 1 
lim
x → 0
1
cosx
3.
lim
x → 0
= 1 
lim
x → 0
x
tanx
C) Limits of Exponential & logarithmic functions :
While solving examples on the limits of exponential and logarithmic functions we
shall remember the following laws of indices .
i) a × a = a ii) = a iii) (a ) = a
iv) (ab) = a b v) a = vi) a =1
Now we shall assume the following formulae for evaluating the limits of
exponential and logarithmic functions without proof.
i)
lim
n → ∞
1 ±
1
n
± n
= e ;
lim
n → ∞
1 ±
a
n
± n/a
= e
ii)
lim
x → 0
(1 ± x)± 1/x = e ;
lim
x → 0
1 ±
x
a
± a/x
= e
iii)
lim
x → 0
= log a ;
lim
x → 0
= 1
iv)
lim
x → 0
( )
= 1


Chapter 2
Syllabus :
 Rules of derivatives such as sum, product, quotient of functions.
 Derivative of composite functions (chain Rule), implicit and parametric functions.
 Derivatives of inverse, logarithmic and exponential functions.
2.1 CONCEPT OF DERIVATIVE :
Consider the demand of shares changes with change in its price OR profit of a
company changes with change in the goods sold. Here the word ‘change’ gives the idea
of derivative. Derivative provides us the method of investigating the rate of change of
some particular quantity with respect to the corresponding change in the other.
Basically, there are two ways of thinking about the derivative of a function.
The first way is as an instantaneous rate of change. We use this concept all the time,
for example, when talking about velocity. Velocity is the instantaneous rate of
change of position.
There is also a geometric interpretation of the concept of derivative. The derivative
can be thought of as the slope of a curve. Derivative means the limiting position of the
increments.
We are very much interested in finding out the relation between these changes
which is termed as differentiation in calculus.
Unit I

Applied Mathematics 2.2 Derivatives
Igniting Minds
2.2 DERIVATIVE OF A FUNCTION : The derivative of y = f(x) w. r. t. x is denoted by
or
d
dx
(y) or f(x) or f (x) or D f(x) and is defined as
= →
( ) ( )
where h is the increment in x.
2.3 DERIVATIVE OF STANDARD FUNCTIONS:
A) Derivative of Algebraic functions:
1) (xn
) = n x 2) =
3) (constant) = 0 4) √x =
√
B) Derivative of Logarithmic & Exponential functions:
5) (loga
x) =
∙
6) (log x) =
7) (ax
) = ax
log a ; a > 0 8) (ex
) = ex
C) Derivative of Trigonometric functions:
9) (sin x) = cos x 10) (cos x) = − sin x
11) (tan x) = sec2
x 12) (cot ) = −cosec2
x
13) (sec x) = sec x tan x 14) (cosec x) = − cosec x cot x
D) Derivative of Inverse Trigonometric functions:
15) (sin−1
x) =
√
16) (cos−1
x) =
√
17) (tan−1
x) = 18) (cot−1
x) =
19) (sec−1
x) =
√
20) (cosec−1
x) =
√

Igniting Minds
2.4 RULES OF DIFFERENTIATION :
1) Addition Rule: If u and v are the differentiable functions of x & = + then
= {u + v} = +
2) Subtraction Rule: If u & v are the differentiable functions of x & = − then
= {u − v} = −
3) Product Rule: If u and v are the differentiable functions of x and = ∙ then
= {u ∙ v} = u ∙ + v ∙
Let u ,v and w be the three differentiable functions of x and = u ∙ v ∙ w then
= {u. v. w} = u ∙ v + u ∙ w + v ∙ w
(k ∙ u) = k ∙ ; k – constant
4) Quotient Rule: If u and v are the differentiable functions of x and =
u
v
then
= =
∙ ∙

Igniting Minds
1. Differentiate w. r. t. x
Ans: Let y = log sin
π
3
Diff. w. r. t. x
=
d
dx
log sin
π
3
⟹ = 0  log sin
π
3
is constant & (constant) = 0
2. Differentiate − w. r. t. x
Ans: Let y = cos −
π
6
Diff. w. r. t. x
 = cos −
⟹ = 0  (constant) = 0
3. Differentiate + w. r. t. x
Ans: Let y = tan x + cot x
⟹ y =
π
2
 tan x + cot x =
π
2
Diff. w. r. t. x
⟹ = 0  (constant) = 0
4. Differentiate √ − w. r. t. x
Ans: Let y = √sec x − tan x
⟹ y = 1  sec θ − tan θ = 1
Diff. w. r. t. x
⟹ = 0  (constant) = 0
Illustrative Examples

Igniting Minds
5. Find if = − + −
Ans: Given: y = 4 sin x − 8 cos x + tan x − sec x
Diff. w. r. t. x
 = 4 (sin x) – 8 ( cos x) + (tan x) − (sec x)
⟹ = 4 cos x − 8(− sin x) + sec x − sec x tan x
⟹ = 4 cos x + 8 sin x + sec x − secx tan x
6. Find if = − +
Ans: Given: y = 3x − 4 cos x + 2 cos x
Diff. w. r. t. x
 = 3 (x2
) – 4 (cos x) +2 (cos−1
x)
⟹ = 3 × 2x − 4 × (− sin x) + 2
−1
1 − x2
⟹ = 6x + 4 sin x −
2
1 − x2
7. Find if = + + + −
Ans: Given: y = x + 7x + 2x + 6x − 3
Diff. w. r. t. x
 = (x5
) + (7x3
) + (2x2
) + (6x) – (3)
⟹ = (x5
) + 7 (x3
) + 2 (x2
) + 6 (x) – 0
⟹ = 5 × x + 7 × (3 x ) + 2 × (2x) + 6 × 1
⟹ = 5x + 21x + 4x + 6

Igniting Minds
8. Find if = + + Dec.-2008 ; Nov-2011
Ans: Given: y = x + 10 + e
Diff. w. r. t. x
 = (x10
) + ( 10x
) + (ex
)
⟹ = 10 ∙ x + 10 ∙ log 10 + e
9. Find if = − + + √
Ans: Given: y = 3x − 2e + 4 sec x + 2√x
Diff. w. r. t. x
 = 3 (x3
) – 2 ( ex
) + 4 (sec x) + 2 √x
⟹ = 3 × 3x − 2 × (e ) + 4 (sec x tan x) + 2
1
2√x
⟹ = 9x − 2e + 4 secx tan x +
√
10. Find if = − + +
Ans: Given: y =
7
x6 −
6
x5 +
5
x3 +
1
x
Diff. w. r. t. x
 = 7 − 6 + 5 +
⟹ = 7 − 6 + 5 +  =
⟹ = + − −

Igniting Minds
11. Find if = √ +
√
April -2012
Ans: Given: y = √x +
1
√x
= √x + 2 ∙ √x ∙
1
√x
+
1
√x
⟹ y = x + 2 +
1
x
Diff. w. r. t. x
 = (x) + ( 2) +
⟹ = 1 + 0 +
⟹ = 1 −
12. Find if =
−
+
Nov-2013
Ans: Given: y =
1 − cos2x
1 + cos2x
⟹ y =
2 sin2
x
2cos2 x
 1 + cos2θ = 2 cos θ & 1 − cos2θ = 2 sin θ
⟹ y = √tan x
⟹ y = tan x
 = sec x
13. Find if =
−
Nov-2013
Ans: Given: y =
1 − cos2x
sin2
⟹ y =
2 sin2
x
2 sin . cosx
 sin2θ = 2 sin θ cos θ & 1 − cos2θ = 2 sin θ
⟹ y =
sinx
cosx
⟹ y = tan
 = sec x

Igniting Minds
14. Find if = ( + ) ∙ ( + ) May-2010
Ans: Given: y = (x + 1) ∙ (x + 2)
Diff. w. r. t. x
 = (x + 1) (x + 2) + (x + 2)
d
dx
(x + 1)

d
dx
(u × v) = u ×
dv
dx
+ v ×
du
dx
⟹ = (x + 1) (1) + (x + 2) (1) = x + 1 + x + 2
⟹ = 2x + 3
15. Find if = ( − ) ∙ + May-2010
Ans: Given: y = (x − 1) ∙ x2 + 2
Diff. w. r. t. x
 = (x − 1) (x2
+ 2) + x2 + 2
d
dx
(x − 1)
⟹ = (x − 1) (2x) + x2 + 2 (1)
⟹ = 2x2 − 2 + x2 + 2
⟹ = 3x2 − 2 + 2
16. Find if = ∙
Ans: Given: y = x ∙ tan x
Diff. w. r. t. x
 = x ∙
d
dx
(tan x) + tan x
d
dx
( x)
⟹ = x ∙ sec x + tan x ∙ 1
⟹ = x ∙ sec x + tan x

Igniting Minds
17. Find if = ∙
Ans: Given: y = x ∙ cos x
Diff. w. r. t. x
 = x (cos x) + cos x
d
dx
( x )
⟹ = x(− sin x) - + cos x 1
⟹ = cos x − x sin x
18. Find if = ∙ April–2009 ; Nov--2012
Ans: Given: y = e ∙ tan x
Diff. w. r. t. x
 = e ∙
d
dx
(tan x) + tan x
d
dx
( e )
⟹ = e ∙ sec x + tan x ∙ e
⟹ = e (sec x + tan x)
19. Find if = .
Ans: Given: y = x ∙ log
Diff. w. r. t. x
 = x (log ) + log
d
dx
( x )
⟹ = x ∙ + log ∙ 1
⟹ = 1 + log

Igniting Minds
20. Find if = + ∙ May – 2007
Ans: Given: y = (1 + x ) ∙ tan x
Diff. w. r. t. x
 = (1 + x ) (tan−1
x) + tan x
d
dx
( 1 + x )
⟹ = (1 + x ) ∙ + tan x ∙ 2x
⟹ = 1 + 2x ∙ tan x
21. Find if = ( + ). ( )
Ans: Given: y = (x + 1). log(x)
Diff. w. r. t. x
 = (x + 1) log(x) + log(x)
d
dx
(x + 1)
⟹ = (x + 1) ∙ + log(x) ∙ 1
⟹ = + log(x)
22. Find if =
+
−
April-2011
Ans: Given: y =
x + 1
x − 1
Diff. w. r. t. x
 =
( ) ( ) ( ) ( )
( )
∵ =
∙ ∙
⟹ =
( ) ∙ ( ) ∙
( )
=
( )
⟹ =
( )

Igniting Minds
23. Find if =
+ √
− √
April – 2008
Ans: Given: y =
1 + √x
1 − √x
Diff. w. r. t. x
 =
√ √ ( √ ) √
√
⟹ =
√
√
( √ )
√
√
⟹ =
√
√ √
√
⟹ =
√ √
⟹ =
√ √
24. Find if =
+
Ans: Given: y =
tanx
1 + x2
Diff. w. r. t. x
 =
( ) ( ) ( )
( )
⟹ =
( ) ∙ ∙
( )
⟹ =
( ) ∙ ∙
( )

Igniting Minds
25. Find if =
Ans: Given: y =
logx
x
Diff. w. r. t. x
 =
( ) ( )
⟹ =
∙ ∙
⟹ =
26. Find if =
Ans: Given: y =
ex
Diff. w. r. t. x
 =
( ) ( )
⟹ =
∙ ∙
⟹ =
( )
27. Find if =
+
−
Dec.-2007; Nov-2012
Ans: Given: y =
ex + 1
ex − 1
Diff. w. r. t. x
 =
( ) ( ) ( ) ( )
( )
⟹ =
( ) ( )
( )
⟹ =
{ }
( )
⟹ =
( )

Applied Mathematics (Automobile and
Mechanical Branch Only)
Publisher : Gigatech Publishing
House
ISBN : 9788193505748 Author : Dr S P Pawar
Type the URL : http://www.kopykitab.com/product/13488
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