A TEXTBOOK OF

ENGINEERING MATHEMATICS

A TEXTBOOK OF

ENGINEERINGMATHEMATICS

For

B.Tech. I Year (I Semester)

ForMAHAMAYA TECHNICAL UNIVERSITY (M.T.U.), NOIDA

(Strictly as per the Latest Revised Syllabus)

By

N.P. BALI Dr. MANISH GOYALFormer Principal M.Sc. (Mathematics), Ph.D., CSIR-NET

S.B. College, Gurgaon Associate Professor

Haryana Department of Mathematics

Institute of Applied Sciences & Humanities

G.L.A. University, Mathura,

U.P.

UNIVERSITY SCIENCE PRESS(An Imprint of Laxmi Publications (P) Ltd.)

BANGALORE CHENNAI COCHIN GUWAHATI HYDERABAD

JALANDHAR KOLKATA LUCKNOW MUMBAI RANCHI

NEW DELHI BOSTON, USA

Copyright © 2013 by Laxmi Publications Pvt. Ltd. All rights reserved. Nopart of this publication may be reproduced, stored in a retrieval system, ortransmitted in any form or by any means, electronic, mechanical, photocopying,recording or otherwise without the prior written permission of the publisher.

Published by :

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First Edition : 2002 ; Second Edition : 2003 ;Third Edition : 2006 ; Fourth Edition : 2008 ;

Fifth Edition : 2009 ; Sixth Edition : 2012 ;Seventh Edition : 2013

OFFICES

Bangalore 080-26 75 69 30 Jalandhar 0181-222 12 72Chennai 044-24 34 47 26 Kolkata 033-22 27 43 84Cochin 0484-237 70 04, 405 13 03 Lucknow 0522-220 99 16Guwahati 0361-251 36 69, 251 38 81 Mumbai 022-24 91 54 15, 24 92 78 69Hyderabad 040-24 65 23 33 Ranchi 0651-220 44 64

UEM-9675-395-ENGG MATH I (MTU)-BAL C—Typeset at : Excellent Graphics, Delhi. Printed at :

ToMy Sweet Little ChildMaster Yash Goyal

— Dr. Manish Goyal(Author)

CONTENTS

Units Pages

Preface ... (ix)Scheme of Evaluation ... (x)Syllabus ... (xi)Standard Results ... (xii)–(xvi)

1. Differential Calculus—I ... 12. Differential Calculus—II ... 1533. Multiple Integrals ... 2884. Vector Calculus ... 3765. Matrices ... 463

Examination Papers ... 567

(vii)

PREFACE TO THE SEVENTH EDITION

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We are glad to present the Seventh edition of ‘‘A Textbook of EngineeringMathematics’’ for the students of B.Tech. I year I semester of Mahamaya TechnicalUniversity (M.T.U.), Noida.

The speciality of this textbook is its lucid presentation and addition or submission of allthe questions asked in the examination papers till date.

The university has implemented new revised syllabus of Engineering Mathematics-Ifor B. Tech. I semester students. In view of this, the textbook has been renovated and revisedstrictly as per it. The new topics have been added in units I and II. Separate assignmentscontaining 2 marks short answer type questions for Section-A have been given at the end ofeach unit to familiarize the students with the recent trend of questions.

We have no words to convey our very special gratitude to Dr. Hari Kishan, a renownedauthor, the great Mathematician and the father of Dr. Manish Goyal, one of the authors, forproviding necessary instructions and help round the clock during the entire revision.

The authors are also thankful to the faculty members of the department of Mathematics,G.L.A. University, Mathura particularly Mr. Amit Saraswat, Mr. Ashish Sharma, Mr. UmeshSharma, Ms. Shweta, Mr. Mukesh Kumar and Mr. Ambuj Mishra for giving necessarysuggestions to make this book ‘THE FIRST CHOICE’ of readers all over the country.

We are also obliged to all those teachers and authors, whom we consulted while givingfinal shape to the matter. We are indebted to GOD for shower of blessings.

The suggestions for the welfare of students regarding the book are always welcome.

—AUTHORS

(ix)

MAHAMAYA TECHNICAL UNIVERSITY, NOIDAScheme of evaluation B.Tech. First Year

SEMESTER I

Periods Evaluation SchemeS. Code Subjects Total Credit

No.L T P Sessional End Semester

CT TA TOT P Th P

1. AS101* Engineering 3 1 0 30 20 50 – 100 – 150 4Mathematics–I

2. AS102 Engg. Physics–I 3 0 2 15 10 25 15 80 30 150 4

3. CS101/ Computer 3 1 2 20 10 30 15 100 30 175 5ME101* Programming/

Engg. Mechanics

4. EE101/ Electrical Engg. / 3 1 2 20 10 30 15 100 30 175 5EC101 Electronics Engg.

5. AS103/ Engg. Chemistry / 3 0 2 15 10 25 15 80 30 150 4ME102 Manufacturing

Practices 2 1 2 15 10 25 25 50 50

6. **/ Branch Elective / 3 0 0 10 10 20 – 80 – 100 3CE101 Energy,

Environment andEcology 3 0 0 10 10 20 – 80 –

7. AS105/ Professional 0 1 2 – – 20 30 50 2CE102 Communication/

Computer AidedEngg. Graphics 0 1 2

8. GP101 General 50 – – 50 –Proficiency

18/17 4/5 10 1000 27

*Syllabi of these courses are different for Biotechnology and Agricultural Engineering.**LIST OF BRANCH ELECTIVES

AS104/AS204 Introduction to Bio Sciences (CS/EC/EE/IC/EI/IT/BT)ME103/ME203 Manufacturing Science (ME/MT/Chemical)CE103/CE203 Geological Sciences (Civil Engineering)AG102/AG202 Material Science (Agriculture Engineering)

L: Lecture T: Tutorial P: Practical/Project CT: Class Test TA: Teacher’sAssessment and Attendance Th: Theory TOT: TotalTA = 10 (5 for teachers assessment plus 5 for attendance),TA = 15 (10 for teachers assessment plus 5 for attendance),TA = 20 (10 for teachers assessment plus 10 for attendance)

Note: Grouping of batches will be done in a way that groups select either all subjects given innumerator or denominator, choice of mix of numerator and denominator is not permitted.

(x)

SYLLABUSMAHAMAYA TECHNICAL UNIVERSITY, NOIDA

ENGINEERING MATHEMATICS—I L T P Credit

(B. Tech. Semester-I) 3 1 0 4

UNIT-I: DIFFERENTIAL CALCULUS-IDetermination of nth derivative of standard functions-illustrative examples.* Leibnitz’s theorem(without proof) and problems. Taylor’s and Maclaurin’s series for one variable (without proof).Differential coefficient of length of arc (concept and formulae without proof). Asymptotes forcartesian coordinates only. Curvature, cartesian formula for radius of curvature, centre ofcurvature. Curve tracing (cartesian and polar coordinates), simple problems. [10 hours]*Note: In the case of illustrative examples, questions are not to be set.

UNIT-II: DIFFERENTIAL CALCULUS-IIPartial differentiation. Euler’s theorem. Change of variables. Expansion of functions of severalvariables (without proof). Jacobians. Approximation of errors. Extrema of function of severalvariables. Lagrange’s method of multipliers (simple problems only). Envelopes. Evolutes.

[9 hours]

UNIT-III: MULTIPLE INTEGRALSDouble and triple integrals. Change of order of integration. Change of variables. Applicationof double and triple integrals to area and volume. Beta and Gamma functions. Dirichlet’sintegral and applications. [9 hours]

UNIT-IV: VECTOR CALCULUSVector differentiation: Vector point function. Gradient, divergence and curl of a vector pointfunction and their physical interpretation.Vector integration: Line, surface and volume integrals. Statement of Green’s, Stoke’s andGauss divergence theorems (without proof) and problems. [9 hours]

UNIT-V: MATRICESElementary row and column transformations. Rank of matrix. Linear dependence. Consistencyof linear system of equations and their solution. Characteristic equation. Cayley-Hamiltontheorem. Eigen values and Eigen vectors. Diagonalisation - Complex and unitary matrices.Application of matrices to engineering problems. [10 hours]

(xi)

STANDARD RESULTS

1.ddx

(xn) = nxn–1 2. ddx

(ax) = ax loge a

3.ddx

(ex) = ex 4.ddx

(loge x) = 1x

5.ddx

(log10 x) = 1x

log10 e 6. ddx

(sin x) = cos x

7. ddx

(cos x) = – sin x 8. ddx

(tan x) = sec2 x

9. ddx

(cosec x) = – cosec x cot x 10. ddx

(cot x) = – cosec2 x

11.ddx

(sec x) = sec x tan x 12.ddx

(sin–1 x) = 1

1 2− x

13.ddx

(cos–1 x) = −

−

1

1 2x14. d

dx (tan–1 x) =

1

1 2+ x

15. ddx

(sec–1 x) = 1

12x x −16.

ddx

(cot–1 x) = −+

1

1 2x

17. ddx

(cosec–1 x) = – 1

12x x −18. sinh x = e ex x− −

2

19. cosh x = e ex x+ −

220. tanh x =

e e

e e

x x

x x−+

−

−

21. cosh2 x – sinh2 x = 1, sech2 x + tanh2 x = 1, coth2 x = 1 + cosech2 x22. cosh2 x + sinh2 x = cosh 2x

23. sinh–1 x = log (x + x2 1+ ), cosh–1 x = log (x x+ −2 1)

24. ddx

(sinh x) = cosh x 25. ddx

(cosh x) = sinh x

26. ddx

(tanh x) = sech2 x 27. ddx

(coth x) = – cosech2 x

28.ddx

(sech x) = – sech x tanh x 29.ddx

(cosech x) = – cosech x coth x

30. Product rule: ddx

(uv) = udvdx

+ vdudx

31. Quotient rule: ddx

uvFHGIKJ =

vdudx

udvdx

v

−2

32. dydx

= dydt

dtdx

. if y = f1(t) and x = f2(t)

(xii)

33. sin–1 x + cos–1 x = π2

, tan–1 x + cot–1 x = π2

, sec–1 x + cosec–1 x = π2

34. tan–1 a bab

−+FHG

IKJ1

= tan–1 a – tan–1 b, tan–1 a bab

+−FHG

IKJ1

= tan–1 a + tan–1 b

35. tan–1 2

1 2x

x−

FHG

IKJ = sin–1

2

1 2x

x+

FHG

IKJ = 2 tan–1 x

36. sin 3x = 3 sin x – 4 sin3 x, cos 3x = 4 cos3 x – 3 cos x, tan 3x = 3

1 3

3

2tan tan

tan

x x

x

−−

sin 2x = 2 sin x cos x, tan 2x = 2

1 2tan

tan

x

x−,

cos 2x = 2 cos2 x – 1 = 1 – 2 sin2 x = cos2 x – sin2 x = 1

1

2

2−+

tan

tan

x

x

37. sin x = x – x3

3 ! + x5

5 ! –

x7

7 ! + ..., cos x = 1 – x2

2 ! +

x4

4 ! – x6

6 ! + ...

ex = 1 + x + x2

2 ! +

x3

3 ! + ...

(1 – x)–1 = 1 + x + x2 + x3 + ... ; | x | < 1 (1 – x)–2 = 1 + 2x + 3x2 + 4x3 + ...

(1 + x)–1 = 1 – x + x2 – x3 + ... (1 + x)–2 = 1 – 2x + 3x2 – 4x3 + ...

38. sin C + sin D = 2 sin C D2+ cos C D

2− , sin C – sin D = 2 cos C D

2+ sin C D

2−

cos C + cos D = 2 cos C D2+ cos C D

2− , cos C – cos D = 2 sin C D

2+ sin D C

2−

39. 2 cos A cos B = cos (A + B) + cos (A – B), 2 sin A sin B = cos (A – B) – cos (A + B)

2 sin A cos B = sin (A + B) + sin (A – B), 2 cos A sin B = sin (A + B) – sin (A – B)

40. sin (A + B) = sin A cos B + cos A sin B, sin (A – B) = sin A cos B – cos A sin B

cos (A + B) = cos A cos B – sin A sin B, cos (A – B) = cos A cos B + sin A sin B

41. ddx

(sinh–1 x) = 1

1 2+ x,

ddx

(cosh–1 x) = 1

12x −

ddx

(tanh–1 x) = 1

1 2− x, where | x | < 1,

ddx

(coth–1 x) = 1

12x −, where | x | > 1

ddx

(sech–1 x) = – 1

1 2x x−,

ddx

(cosech–1 x) = – 1

12x x +

42. (cos θ + i sin θ)n = cos nθ + i sin nθ, (cos θ + i sin θ)–n = cos nθ – i sin nθ43. sin2 θ + cos2 θ = 1, sec2 θ – tan2 θ = 1, 1 + cot2 θ = cosec2 θ

(xiii)

A Textbook Of EngineeringMathematics Sem I (MTU) Noida

Publisher : Laxmi Publications ISBN : 9789381159477Author : N P Bali, Dr.Manish Goyal

Type the URL : http://www.kopykitab.com/product/12181

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