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ESSENTIALS OF ENGINEERING MATHEMATICS Worked Examples and Problems Alan Jeffrey SECOND EDITION CHAPMAN & HALL/CRC www.EngineeringEBooksPdf.com ESSENTIALS OF ENGINEERING MATHEMATICS Worked Examples and Problems SECOND EDITION www.EngineeringEBooksPdf.com This page intentionally left blank www.EngineeringEBooksPdf.com ESSENTIALS OF ENGINEERING MATHEMATICS Worked Examples and Problems SECOND EDITION Alan Jeffrey CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C www.EngineeringEBooksPdf.com CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2004 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20140513 International Standard Book Number-13: 978-1-4822-8604-5 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com www.EngineeringEBooksPdf.com Contents Preface ix Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section Section 10 11 12 13 14 15 16 17 18 19 20 21 Section Section Section Section Section Section Section 22 23 24 25 26 27 28 Section Section Section Section 29 30 31 32 Real numbers, inequalities and intervals Function, domain and range Basic coordinate geometry Polar coordinates Mathematical induction Binomial theorem Combination of functions Symmetry in functions and graphs Inverse functions Complex numbers: real and imaginary forms Geometry of complex numbers Modulus Á/argument form of a complex number Roots of complex numbers Limits One-sided limits: continuity Derivatives Leibniz’s formula Differentials Differentiation of inverse trigonometric functions Implicit differentiation Parametrically defined curves and parametric differentiation The exponential function The logarithmic function Hyperbolic functions Inverse hyperbolic functions Properties and applications of differentiability Functions of two variables Limits and continuity of functions of two real variables Partial differentiation The total differential The chain rule Change of variable in partial differentiation www.EngineeringEBooksPdf.com 12 17 35 40 44 50 55 60 66 75 81 87 92 101 109 123 128 133 138 141 148 156 164 169 174 192 198 204 217 224 230 vi ESSENTIALS OF ENGINEERING MATHEMATICS Section Section Section Section Section 33 34 35 36 37 Section Section 38 39 Section Section Section Section Section 40 41 42 43 44 Section Section Section Section Section Section 45 46 47 48 49 50 Section Section Section 51 52 53 Section Section Section 54 55 56 Section Section Section Section Section Section Section Section Section Section Section 57 58 59 60 61 62 63 64 65 66 67 Section 68 Section 69 Antidifferentiation (integration) Integration by substitution Some useful standard forms Integration by parts Partial fractions and integration of rational functions The definite integral The fundamental theorem of integral calculus and the evaluation of definite integrals Improper integrals Numerical integration Geometrical applications of definite integrals Centre of mass of a plane lamina (centroid) Applications of integration to he hydrostatic pressure on a plate Moments of inertia Sequences Infinite numerical series Power series Taylor and Maclaurin series Taylor’s theorem for functions of two variables: stationary points and their identification Fourier series Determinants Matrices: equality, addition, subtraction, scaling and transposition Matrix multiplication The inverse matrix Solution of a system of linear equations: Gaussian elimination The GaussÁ/Seidel iterative method The algebraic eigenvalue problem Scalars, vectors and vector addition Vectors in component form The straight line The scalar product (dot product) The plane The vector product (cross product) Applications of the vector product Differentiation and integration of vectors Dynamics of a particle and the motion of a particle in a plane Scalar and vector fields and the gradient of a scalar function Ordinary differential equations: order and degree, initial and boundary conditions www.EngineeringEBooksPdf.com 238 249 262 266 277 288 296 311 317 324 333 341 348 351 354 374 382 403 416 435 450 456 467 474 489 496 505 511 519 523 528 533 540 550 561 567 578 CONTENTS Section 70 Section Section Section Section Section Section 71 72 73 74 75 76 Section 77 Section 78 Section Section 79 80 Section Section Section 81 82 83 Section Section 84 85 Section Section 86 87 First order differential equations solvable by separation of variables The method of isoclines and Euler’s methods Homogeneous and near homogeneous equations Exact differential equations The first order linear differential equation The Bernoulli equation The structure of solutions of linear differential equations of any order Determining the complementary function for constant coefficient equations Determining particular integrals of constant coefficient equations Differential equations describing oscillations Simultaneous first order linear constant coefficient differential equations The Laplace transform and transform pairs The Laplace transform of derivatives The shift theorems and the Heaviside step function Solution of initial value problems The delta function and its use in initial value problems with the Laplace transform Enlarging the list of Laplace transform pairs Symbolic algebraic manipulation by computer software Answers Reference information Index www.EngineeringEBooksPdf.com 586 592 605 612 617 623 625 631 640 649 657 662 669 671 680 691 697 701 731 865 875 vii This page intentionally left blank www.EngineeringEBooksPdf.com Preface to the Second Edition This book evolved from lectures given in Newcastle over many years, and it presents the essentials of first year engineering mathematics as simply as possible It is intended that the book should be suitable both as a text to supplement a lecture course and also, because it contains a full set of detailed solutions to problems, as a book for private study The success with which the style and content of the first edition was received has persuaded me that these features should be preserved when preparing this second edition Accordingly, the changes made to the original material have, in the main, been confined to small amendments designed to improve the understanding of some basic concepts Typical of these amendments to the first edition is the inclusion in Section 26 of some new problems involving the mean value theorem for derivatives, an extension of the account of stationary points of functions of two variables in Section 50 to include Lagrange multipliers, and the introduction of the concept of the direction field of a first order differential equation in Section 71, now made possible by the ready availability of suitable computer software While making these changes, the opportunity has also been taken to correct some typographical errors More important, however, is the inclusion of a considerable amount of new material The first is to be found in Section 85, where the reader is introduced to the delta function and its uses with the Laplace transform when solving initial value problems for linear differential equations The second, which is far more fundamental, is the inclusion of an introductory account in Section 87 of the use of new computer software that is now widely available The purpose of this software is to enable a computer to act in some ways like a person with pencil and paper, because it allows a computer to perform symbolic operations, like differentiation, integration, and matrix algebra, and to give the results in both symbolic and numerical form The two examples of software described here are called MAPLE and MATLAB, each of which names is the registered trademark of a software company quoted in Section 87 In fact MAPLE, which provides the symbolic capabilities of MATLAB, was used when preparing the new material for this second edition When symbolic software is available the reader is encouraged to take full advantage of it by using it to explore the properties of functions, mathematical operations, and differential equations, and also by using its excellent graphical output to gain a better understanding of the geometrical implications of mathematical results Alan Jeffrey Newcastle upon Tyne www.EngineeringEBooksPdf.com m Example 49.9 Deduce the series expansion of ln (1/x) from the binomial expansion of 1/(1/x ) Solution From the binomial theorem we have 1t 1 tt t3 .( 1)n t , which we know to be absolutely convergent for / B/t B/1 Thus we may integrate this result from to x to obtain g x dt 1t g x dt g x t dt g x t dt g x t dt .( 1)n x g t dt , n so ln (1x) x x2 x3 x4 .( 1)n xn1 n1 , for / B/x B/1, in agreement with the result obtained in Example 49.3 m Example 49.10 Deduce the series expansion of arctan x from the binomial expansion of 1/(1/x2) Solution From the binomial theorem we have 1 t2 1 t t t .( 1)n t 2n , which is absolutely convergent for / 1B/t B/1 Thus we may integrate this result from to x to obtain g x dt t2 g x dt g x t dt g x t dt .( 1)n www.EngineeringEBooksPdf.com x gt 2n dt , TAYLOR AND MACLAURIN SERIES so arctan x x x3 x5 x7 .( 1)n1 x2n 2n , for / B/x B/1 m Often when working with Maclaurin series only the first few terms are required In such circumstances, when the function f(x ) involved is the product of two functions with known Maclaurin series, the simplest way of finding first few terms of the required expansion of f (x ) may be by multiplication of the two series Example 49.11 Find the first three non-zero terms of the Maclaurin series expansion of (i) x ln (1/x2); (ii) e x sin 2x Solution (i) We saw in Example 49.3 that ln (1x) x x2 x2 x4 , thus replacing x by x2 gives ln (1x2 ) x2 x4 so x ln (1x2 ) x3 x6 x8 , x5 x7 , for / 1B/x B/1 (the interval of convergence of ln (1/x) and thus of ln (1/x2)) (ii) We saw in Example 49.2 that ex 1x x2 x3 , 2! 3! for / B/x B/ , so replacing x by / x gives e x 1 x x2 2! x3 3! , for / B/x B/ In Example 49.1 we found that sin x x x3 3! x5 5! x7 7! , www.EngineeringEBooksPdf.com 393 394 ESSENTIALS OF ENGINEERING MATHEMATICS for / B/x B/ , so replacing x by 2x gives sin 2x 2x (2x)3 (2x)5 , 3! 5! or 4 sin 2x 2x x3 x5 , 15 for / B/x B/ Thus x2 x3 4 x e sin 2x 1 x 2x x x , 15 2! 3! so multiplying out the first few terms we have 4 4 x e sin 2x 2x x x 2x x x 15 15 2 2 x3 x5 x7 x4 x6 x8 , 15 45 after which collecting terms gives the required result e x sin 2x 2x 2x2 x3 : When carrying out this operation care must always be taken to see that each function is expanded far enough to ensure that all terms of the required degree are contained in the answer m Example 49.12 Use the Maclaurin series expansion of ex to find the Maclaurin series expansions of sinh x and cosh x Solution We have from Example 49.2 that ex 1x x2 2! x3 3! x4 4! for / B/x B/ , so replacing x by / x gives e x 1 x x2 x3 x4 , 2! 3! 4! for / B/x B/ www.EngineeringEBooksPdf.com .. .ESSENTIALS OF ENGINEERING MATHEMATICS Worked Examples and Problems SECOND EDITION www.EngineeringEBooksPdf.com This page intentionally left blank www.EngineeringEBooksPdf.com ESSENTIALS OF ENGINEERING. .. ESSENTIALS OF ENGINEERING MATHEMATICS Worked Examples and Problems SECOND EDITION Alan Jeffrey CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C www.EngineeringEBooksPdf.com... means that the product of a number and a sum is equal to the sum of the respective products www.EngineeringEBooksPdf.com ESSENTIALS OF ENGINEERING MATHEMATICS Real numbers and exist, called identity