Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
SlideShare a Scribd company logo

applied math 2nd semester diploma

•
•
sakhi pathak
sakhi pathak

The document discusses the syllabus for Unit I of Applied Mathematics, which covers differential calculus. The topics covered in Chapter 1 include the concepts of functions and limits. Functions are mathematical relations that assign unique outputs to inputs. The chapter discusses types of functions and introduces the concept of a limit, which defines the behavior of a function at a certain input, without providing examples of limits. The syllabus also covers derivatives and their applications in subsequent chapters, including rules of differentiation, derivatives of standard functions, and using derivatives to find equations of tangents, normals, maxima, minima, and radii of curvature.

1 of 41
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) Head, Dept. of Applied Sciences Govt. Polytechnic, Miraj ‘I’ Scheme Committee Member M.Sc.(Mathematics) Head, Dept. of Applied Sciences PVPIT Budhgaon, Sangli M.Sc. (Mathematics) Guru Gobind Singh Polytechnic, Nashik M.Sc. (Mathematics) Head, Dept. of Applied Sciences 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 Download your free e-Book here 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 631/32, Budhwar Peth, Office No. 105, First Floor, Shan Bramha Complex, Pune – 411 002. Phone No. 7757042853 Copyright  Dr. S. P. Pawar All rights reserved. No part of this publication can be stored in any retrieval system or reproduced in and form or by any means without prior permission of the Author. LIMITS OF LIABILITY AND DISCLAIMER OF WARRANTY : The Authors and Publisher of this book have tried their best to ensure that the program, pro- cedure and function described in the book are correct. However the author and publisher make no warranty with regard to the program and documentation contained in the book. ISBN : 978-81-935057-4-8 Price : `300/- Available at All Leading Book Stalls Distributor PRADEEP BOOK DISTRIBUTORS 102/3/4 1 ST Floor, Shan Bramha Complex, Pune. Telephone No. : 020 – 24458333, 942 208 7031 / 844 699 5959 Email : pbd0909@gmail.com Note : For any queries, doubts, suggestions and complaints regarding the subject, please feel free to contact on the below e-mail or phone number. We will be more than thankful for your feedback. Ph: 7757042853 E-mail : info.gigatech1@gmail.com
What is an e-Book? Download your free e-Book here
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 Gigatech Publishing House 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
Applied Mathematics 1.3 Functions and Limits Gigatech Publishing House 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
Applied Mathematics 1.4 Functions and Limits Gigatech Publishing House 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.
Applied Mathematics 1.5 Functions and Limits Gigatech Publishing House 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
Applied Mathematics 1.6 Functions and Limits Gigatech Publishing House 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
Applied Mathematics 1.7 Functions and Limits Gigatech Publishing House 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) ⟹ f(x) is an odd function.
Applied Mathematics 1.8 Functions and Limits Gigatech Publishing House 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) ⟹ f(x) is an odd function. 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
Applied Mathematics 1.9 Functions and Limits Gigatech Publishing House 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
Applied Mathematics 1.10 Functions and Limits Gigatech Publishing House 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)
Applied Mathematics 1.11 Functions and Limits Gigatech Publishing House 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.
Applied Mathematics 1.12 Functions and Limits Gigatech Publishing House 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 = 1 ; x in radian measure. 3. lim x → 0 = 1  lim x → 0 x tanx = 1 ; x in radian measure. 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 Gigatech Publishing House 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) = √
Applied Mathematics 2.3 Derivatives Gigatech Publishing House 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 = = ∙ ∙
Applied Mathematics 2.4 Derivatives Gigatech Publishing House 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
Applied Mathematics 2.5 Derivatives Gigatech Publishing House 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
Applied Mathematics 2.6 Derivatives Gigatech Publishing House 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 +  = ⟹ = + − −
Applied Mathematics 2.7 Derivatives Gigatech Publishing House 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
Applied Mathematics 2.8 Derivatives Gigatech Publishing House 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
Applied Mathematics 2.9 Derivatives Gigatech Publishing House 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
Applied Mathematics 2.10 Derivatives Gigatech Publishing House 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  = ( ) ( ) ( ) ( ) ( ) ∵ = ∙ ∙ ⟹ = ( ) ∙ ( ) ∙ ( ) = ( ) ⟹ = ( )
Applied Mathematics 2.11 Derivatives Gigatech Publishing House 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  = ( ) ( ) ( ) ( ) ⟹ = ( ) ∙ ∙ ( ) ⟹ = ( ) ∙ ∙ ( )
Applied Mathematics 2.12 Derivatives Gigatech Publishing House 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 Get this eBook

More Related Content

applied math 2nd semester diploma

  • 2. 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) Head, Dept. of Applied Sciences Govt. Polytechnic, Miraj ‘I’ Scheme Committee Member M.Sc.(Mathematics) Head, Dept. of Applied Sciences PVPIT Budhgaon, Sangli M.Sc. (Mathematics) Guru Gobind Singh Polytechnic, Nashik M.Sc. (Mathematics) Head, Dept. of Applied Sciences 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 Download your free e-Book here Gigatech Publishing House Igniting Minds
  • 3. 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 631/32, Budhwar Peth, Office No. 105, First Floor, Shan Bramha Complex, Pune – 411 002. Phone No. 7757042853 Copyright  Dr. S. P. Pawar All rights reserved. No part of this publication can be stored in any retrieval system or reproduced in and form or by any means without prior permission of the Author. LIMITS OF LIABILITY AND DISCLAIMER OF WARRANTY : The Authors and Publisher of this book have tried their best to ensure that the program, pro- cedure and function described in the book are correct. However the author and publisher make no warranty with regard to the program and documentation contained in the book. ISBN : 978-81-935057-4-8 Price : `300/- Available at All Leading Book Stalls Distributor PRADEEP BOOK DISTRIBUTORS 102/3/4 1 ST Floor, Shan Bramha Complex, Pune. Telephone No. : 020 – 24458333, 942 208 7031 / 844 699 5959 Email : pbd0909@gmail.com Note : For any queries, doubts, suggestions and complaints regarding the subject, please feel free to contact on the below e-mail or phone number. We will be more than thankful for your feedback. Ph: 7757042853 E-mail : info.gigatech1@gmail.com
  • 4. What is an e-Book? Download your free e-Book here
  • 5. 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
  • 6. Dedicated to My beloved Father Late Appasaheb Pandurang B. Pawar & My Mother Late Taisaheb Venubai Pawar who was the source of my inspiration.
  • 7. 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. 
  • 8. 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.
  • 9. 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. 
  • 10. 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
  • 11. 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 
  • 12. 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
  • 13. 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
  • 14. 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 
  • 15. Unit – I Differential Calculus Teaching Hours Total Marks 20 24 Chapter No. Chapter Name 1. Functions and Limits 2. Derivatives 3. Applications of derivative
  • 16. 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
  • 17. 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
  • 18. Applied Mathematics 1.2 Functions and Limits Gigatech Publishing House 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
  • 19. Applied Mathematics 1.3 Functions and Limits Gigatech Publishing House 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
  • 20. Applied Mathematics 1.4 Functions and Limits Gigatech Publishing House 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.
  • 21. Applied Mathematics 1.5 Functions and Limits Gigatech Publishing House 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
  • 22. Applied Mathematics 1.6 Functions and Limits Gigatech Publishing House 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
  • 23. Applied Mathematics 1.7 Functions and Limits Gigatech Publishing House 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) ⟹ f(x) is an odd function.
  • 24. Applied Mathematics 1.8 Functions and Limits Gigatech Publishing House 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) ⟹ f(x) is an odd function. 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
  • 25. Applied Mathematics 1.9 Functions and Limits Gigatech Publishing House 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
  • 26. Applied Mathematics 1.10 Functions and Limits Gigatech Publishing House 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)
  • 27. Applied Mathematics 1.11 Functions and Limits Gigatech Publishing House 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.
  • 28. Applied Mathematics 1.12 Functions and Limits Gigatech Publishing House 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 = 1 ; x in radian measure. 3. lim x → 0 = 1  lim x → 0 x tanx = 1 ; x in radian measure. 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 
  • 29. 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
  • 30. Applied Mathematics 2.2 Derivatives Gigatech Publishing House 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) = √
  • 31. Applied Mathematics 2.3 Derivatives Gigatech Publishing House 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 = = ∙ ∙
  • 32. Applied Mathematics 2.4 Derivatives Gigatech Publishing House 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
  • 33. Applied Mathematics 2.5 Derivatives Gigatech Publishing House 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
  • 34. Applied Mathematics 2.6 Derivatives Gigatech Publishing House 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 +  = ⟹ = + − −
  • 35. Applied Mathematics 2.7 Derivatives Gigatech Publishing House 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
  • 36. Applied Mathematics 2.8 Derivatives Gigatech Publishing House 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
  • 37. Applied Mathematics 2.9 Derivatives Gigatech Publishing House 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
  • 38. Applied Mathematics 2.10 Derivatives Gigatech Publishing House 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  = ( ) ( ) ( ) ( ) ( ) ∵ = ∙ ∙ ⟹ = ( ) ∙ ( ) ∙ ( ) = ( ) ⟹ = ( )
  • 39. Applied Mathematics 2.11 Derivatives Gigatech Publishing House 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  = ( ) ( ) ( ) ( ) ⟹ = ( ) ∙ ∙ ( ) ⟹ = ( ) ∙ ∙ ( )
  • 40. Applied Mathematics 2.12 Derivatives Gigatech Publishing House 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  = ( ) ( ) ( ) ( ) ( ) ⟹ = ( ) ( ) ( ) ⟹ = { } ( ) ⟹ = ( )
  • 41. 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 Get this eBook