A Text book of MATHEMATICS-I. Career Institute of Technology and Management, Faridabad. Manav Rachna Publishing House Pvt. Ltd.

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2 A Tet book of ENGINEERING MATHEMATICS-I by Prof. R.S. Goel E. Principal, Aggarwal College, Ballabhgarh Senior Faculty of Mathematics Career Institute of Technology and Management, Faridabad Dr. Y.K. Sharma Assistant Professor in Mathematics Career Institute of Technology and Management, Faridabad Manav Rachna Publishing House Pvt. Ltd. Delhi110019

3 Published by : Manav Rachna Publishing House Pvt. Ltd. 117-F, Chitranjan Park, Delhi Administrative Office: 5E/1-A, N.I.T. Faridabad Ph.: ; Fa: ALL RIGHTS STRICTLY RESERVED No matter in full or part may be reproduced (ecept for review or criticism) without the written permission of the Author IMPORTANT Author and Publishers would welcome constructive suggestions from the readers for the improvement of the book and pointing out the errors and printing mistakes, if any First Edition: July, 008 Second Edition: July, 009 Price : Rs. 300/- Laser Type Setting Plus Computers, Delhi Printed by Sagar Printers, Delhi

4 Preface to the First Edition The present book on Mathematics has been written for the benefit of the students pursuing studies of first semester B.E./B.Tech in various Engineering Colleges and Universities. The aim of publishing the book is to provide easy and better understanding of the subjects to all concerned. The subject matter has been dealt with comprehensively and discussed lucidly for the easy understanding of the students. The distinguishing feature of the book is that a large number of typical problems selected from recent question papers of various Universities and Engineering Colleges have been solved to make the students familiar with the recent style of the papers set in the University Eaminations. We are grateful to our friends, faculty members and well wishers for encourangement, help and co-operation given for the publication of this book. We are specially indebted to our most respected Dr. O.P. Bhalla, the Hon ble President of MREI, Faridabad for all his elderly graces and blessings for writing the present book. The authors take this opportunity to epress their deep sense of gratitude to Dr. D.S. Kumar, Eecutive Director, MREI for his able guidance and invaluable suggestions for the publication of the book. The authors are also grateful to the members of M/s Manav Rachna Publishing House Pvt. Ltd., Delhi for the efforts they have put in to bring out the book in a short time. All efforts have been made to make the book useful to the students and keep the book free from errors, even then we shall be grateful if you bring to our notice any mistake you come across. All suggestions regarding the improvement of the book will be highly appreciated and gratefully acknowledged. Authors

5 Dedicated to Our Parents

6 Contents 1. Partial Differentiation Dependent and independent variables 1 1. Partial derivatives Which variable is to be treated as constant? Homogeneous functions Euler s theorem on homogeneous function of two variables Total derivatives Composite functions Differentiation of composite functions Differentiation of implicit function Concept of Jacobian Properties of Jacobians 50. Epansions of Functions Finite and infinite series 60. Taylor s infinite series 60.3 Working method for application of Taylor s infinite series 61.4 Failure of Taylor s series 61.5 Maclaurin s infinite series 70.6 Working method of epansion as Maclaurin s series 70.7 Failure of Maclaurin s series 70.8 Some useful epansions 75.9 Successive differentiation Leibnitz s rule Taylor s series for a function of two variables Maima and Minima Concept of maima and minima Stationary and etreme points Necessary conditions for eistence of a maima or minima of a function Sufficient condition for maima or minima Working rule for maima and minima Lagrange s method of undetermined multipliers Differentiation under the integral sign Leibnitz s rule of differentiation under the integral sign 14

7 4. Asymptotes and Curvature Branch of a curve Asymptotes Asymptotes parallel to coordinate ais Asymptotes parallel to coordinate ais for f (, y) 0, i.e, algebraic curve Oblique asymptotes To find the oblique asymptotes of the general algebraic curve Working rule for finding oblique asymptote of an algebraic curve of nth degree Intersection of a curve and its asymptotes Asymptotes in polar coordinates Working method for finding polar asymptotes Mathematical definition of a curvature Curvature of a circle Radius of curvature of cartesian curve Radius of curvature of polar curve Radius of curvature of pedal equation Radius of curvature at the origin Centre of curvature, circle of curvature and evolute Infinite Series Sequence 0 5. Limit Real sequence Range of sequence Constant sequence Bounded and unbounded sequence Convergent, divergent and oscillatory sequence Monotonic sequence Some standard limits Series Convergent, divergent and oscillatory series Properties of infinite series Partial sums Geometric series Positive term series Necessary condition for convergent series Cauchy s fundamental test for divergence Hyper harmonic series test, p-series test Comparison test D -Alembert s ratio test Raabe s test (higher ratio test) 30

8 5. Logarithmic test Gauss s test Cauchy s root test Integral test Alternating series Alternating convergent series Curve Tracing Curve and its forms Procedure of curve tracing for cartesian curves Procedure of curve tracing for polar curves Procedure of curve tracing for parametric curves Integral Calculus-I Solid of revolution and surface of revolution (A) Volume of solid of revolution (for cartesian curves) 30 (B) Volume of solid of revolution (for parametric curves) 303 (C) Volume of solid of revolution (for the polar curves) 303 (D) Surface area of solid of revolution Multiple integrals and their applications Evaluation of double integrals Double integration in polar co-ordinates Change of order of integration Volume as a double integral Integral Calculus-II Triple integral Change of variables Volume as triple integral Gamma function Reduction formula for (n) Beta function Some important deductions Vector Calculus Scalar and vector quantities Scalar and vector product, angular velocities Differentiation of vectors Formulae of differentiation Scalar and vector point function Vector differential operator 373

9 9.7 Geometrical interpretation of gradient Directional derivative of in the direction of PQ Del applied to vector point functions Physical interpretation of divergence Physical interpretation of curl Repeated operation by Properties of divergence and curl Some basic concepts A line integral Surface integral Volume integral Green s theorem Stoke s theorem (relation between line and surface integral) Gauss s divergence theorem Ordinary and Linear Differential Equations Eact differential equations Equations reducible to eact differential equation Linear differential equation The operator Theorems Auiliary equation (A.E.) Rules for finding the complementary function The inverse operator f( D) Rules for finding the particular integral Working procedure to solve the equation Method of variation of parameters to find particular integral Cauchy s homogeneous linear equation Legender s Linear equation Simultaneous Linear Equation with constant Coefficients 507 Inde

10 Chapter 1 Partial Differentiation This chapter deals with partial derivatives, homogeneous functions, Euler s theorem for homogeneous functions, composite functions, total derivatives, Jacobians and it s properties. 1.1 DEPENDENT AND INDEPENDENT VARIABLES The area of a rectangle depends upon its length and breadth and accordingly it may be stated that area is the function of two variables, i.e., length and breadth. If z be the area of rectangle and and y be the length and breadth respectively, then z is called a function of two variables and y. Symbolically, it is written as z f (, y) The variables and y are called independent variables, while z is called the dependent variable. Similarly, z can be defined as a function of more than two variables. 1. PARTIAL DERIVATIVES Let z f (, y) be a function of two independent variables and y. If y is taken as constant and changes, then z becomes a function of only. The derivative of z with respect to taking y as constant is called partial derivative of z with respect to and is denoted by z f or or f (, y) z f(, y) f(, y) and lim 0 Similarly if is kept constant and y changes, then z becomes a function of y only. The derivative of z with respect to y taking as constant is called partial derivative of z with respect to y and is denoted by and z y z f or or fy (, y) y y lim y 0 f(, y y) f(, y) y 1

11 Engineering Mathematics z z In general, or f (, y) and y or f y (, y) are also functions of and y and so these can be differentiated again partially with respect to and y. That is z z f or or f (, ) y z y y z f or or f (, ) yy y y y z y z f or or fy (, y) y y z z f y or or fy (, y) y y Corrollary : In general z z y y Taking an eample : z a + hy + by and differentiating it partially with respect to and y, one obtains z a + hy...(i) z and y h + by Again differentiating (i) with respect to y and (ii) with respect to, we get z y h or...(ii) z h...(a) y z and y z h or h...(b) y From (A) and (B), it is observed that z y z y EXAMPLE 1.1 Evaluate z and z y, if (i) z y sin y (ii) z log ( + y ) Solution: (i) z y sin y z y (sin y) sin y y cos y y sin y 1 ()

12 Partial Differentiation 3 and z y y y cos y sin y ( y) ( sin y) y y cos y cos y (1 cos y) (ii) z log ( + y ) z 1 ( ) y y z 1 y and ( y) y y y EXAMPLE 1. If u e r cos cos (r sin ), then find u r and u Solution : u e r cos cos (r sin ) Similarly u r e r cos [ sin (r sin ) sin ] + [cos e r cos ] cos (r sin ) e r cos [ sin (r sin ) sin + cos (r sin ) cos ] e r cos [cos (r sin + )] u er cos [ sin (r sin ) r cos ] + [ r sin e r cos ] cos (r sin ) re r cos [sin (r sin ) cos + sin cos (r sin )] re r cos [sin (r sin + )] EXAMPLE 1.3 If u sin 1 y y + tan 1, then find the value of u u y y Solution : u sin y tan y 1 1 u y y y 1 1 y 1 y y y or u y y y...(i) Similarly u y y y 1 1 y y y y

13 4 Engineering Mathematics u or y y On adding (i) and (ii), we get u u y y 0 y y y...(ii) EXAMPLE 1.4 Solution : Similarly If z e a + by f (a by), prove that z a a + by f (a by) z z b z y z z b a y abz a e a + by f (a by) + e a + by f (a by) a ab e a + by {f (a by) + f (a by)}...(i) be a + by f (a by) + e a + by f (a by) ( b) z a y ab e a + by {f (a by) f (a by)}...(ii) On adding (i) and (ii), we get z z b a y z z b a y ab e a + by f (a by) abz EXAMPLE 1.5 If u tan 1 a log ( + y ) + b tan 1 y, prove that u u 0 y y Solution : u tan 1 a log ( + y ) + b tan 1 Again 1 u tan a b y tan a by ( y ) y y y 1 u 1 [( y ) ] by tan a ( y ) ( y ) 1

14 Partial Differentiation 5 1 y 4 yb tan a ( y ) ( y ) 1 ( y ) yb tan a ( y ) ( y ) u 1 y b 1 Similarly y tan a y y 1 1 y tan a b y y u 1 [( y ) y y] b y Again tan a y ( y ) ( y ) 1 ( y ) by tan a ( y ) ( y ) On adding (i) and (ii), we get u y 0...(i)...(ii) EXAMPLE 1.6 If u (1 y y ), show that 1 u u (1 ) y y y 0 1 Solution : u (1 y y )...(i) Differentiating equation (i) partially with respect to, we get u 3 u (1 ) y and (1 ) 3/ (1 y y ) Again differentiating with respect to, we get 1 y (1 y y ) ( y ) 3/ (1 y y ) (1 y y ) ( y) ( y y ) (1 y y ) ( y) u (1 ) 3 (1 y y )

15 6 Engineering Mathematics 1 (1 y y ) (1 y y )( y) 3( y y ) 3 (1 y y ) 3 y y y 3y 5/ (1 y y ) Similarly differentiating (i) partially with respect to y, we get and y u y u y 3 1 ( y) (1 y y ) ( y ) 3/ (1 y y ) y 3 y 3/ (1 y y ) Again differentiating partially with respect to y, we get u y y y...(ii) 3/ 3 3 1/ (1 y y ) (y 3 y ) ( y y ) (1 y y ) ( y) 3 (1 y y ) 1 y y [(1 y y ) (y 3 y ) 3( y y ) ( y)] (1 y y ) 3 y y y 3y 5/ (1 y y ) On adding (ii) and (iii), we get u u (1 ) y y y 0 5/ (1 y y ) 3 3 y y y 3y y y y 3y 5/ 5/ (1 y y ) (1 y y ) 0...(iii) 1 EXAMPLE 1.7 If u t Solution : u 1 e t 4a t u t e 4a t, prove that u t u a at at t e e ( 1) t t 4a

16 A Tetbook of Engineering Mathematics- I Publisher : Manav Rachna Publishing House Pvt Ltd Author : R S Goel and Y K Sharma Type the URL : 70 Get this ebook

MEAN VALUE THEOREMS FUNCTIONS OF SINGLE & SEVERAL VARIABLES

MATHEMATICS-I MEAN VALUE THEOREMS FUNCTIONS OF SINGLE & SEVERAL VARIABLES I YEAR B.TECH By Y. Prabhaker Reddy Asst. Professor of Mathematics Guru Nanak Engineering College Ibrahimpatnam, Hyderabad. Name