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Instithion Decr:Ptors DOCUOSIT BESU r" ED 176 987 Sk 028 914 AUTHOR Cross, Judson B.; And Others TIILE Applied Mathematics: An Introductory Course. INSTITHION Boston Univ., Mass. Physical ScienceGroup. SPONS AGENCY National Science Foundation,Washingtcn, D.C. PUB DATE 75 GRANT NSF-GZ-2892 NOTE 364.; Contains occasicnal light type EDRS PiICE MF01/PC15 Plur Postage. DECR:PTORS *Calculation; Collegp Curriculum; *College Mathematics; Higher tducation; LinearPrograming; *Mathematical Applications; *Mathematical Conceyts; Mathematical Enrichment; Mt.thematical Vocabulary; *Mathematics; *Mat14matics Instruction; Science Education; SPt Theory; Trigoncmetry IDLNTIFIERS *Functions (Mathematic9' AcliTSACT The intent ofthis text is to prQvide students in a variety of science andtechnology disciplines with a tasio understanding of watLematics commcnlyused in introductory texts in such disciplines. Tae first fivechapters develop skills needed for efficient numerical calculation. Ihe lastfive chapters examine the properties.of elementary functions.Special emphasis is placed or, finding analyticalexpressions from graphical representationcf data. (Author/RE) *********************************************************************** Reproductions supplied by EDRS are thebest that can be made from the original document. *********************************************************************** -11v9S pf ITGEY 1, -6c:11:sot: U S DEPARTMENT OF HEALTH 'PERMISSION TO REPRODUCE THIS , EDUCATION I WELFARE MATERIAL HAS BEEN GRANTED BY NATtONAL iNSTITUTE OF 110vCATION ^Aar Gr1 /.17:7-7ty TH.5DuCuMENT HAS pEIN RIPRC L1 DUCED EXACTLY AS IRE 7EIVED FROM U C FII=ATION THE PERSON OR OFIC,ANI/A7ION rH% NATICAL ATING IT POInaTS Of. viE.A 0$4 ()PINIONS sTTED LX.) NOT NECESSAiiii. V REPRE- SEMTOf FICTAL NiCHONALF.4S1,.1.).1 (IF tDj(ATO POSITiOPo OP POl .( Y TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC)" 13sarit-,3ti- OriJ-fethisv5c,44;4 MEL_ AN INTR UCT RY C RSE PHYSICAL SCIENCEGROUP BOSTON! UN IVEASITY Copyright 1974, 1975 by The Trustees of BostonUniversity Printed in the United States of America IL PREFACE Are there basic needs inapplied mathematics that are sharedby beginning college students in thesocial sciences, the naturalsciences, and technology? Several topics come to mind thatpeint to an affirmative answer: presenting and interpretingdata, finding analyticalexpressions for functions from graphs, beingfamiliar with the properties ofelementary functions, and being conversantwith the language ofcalculus. By pro- viding for these needs, weenable the students to overcomeserious obstacles to the understandingof introductory texts in all1.hese fields. Thi2 intention of this textis to serve just this purpose. The first five chapters developthe skills needed for efficient numerical calculations, emphasizingthe consequences of theinherent uncertainties of most numbersused in applications. Thetopics discussed of range fromorder-of-magnitude estimatesthrough the theory and the use the slide rule to thefundamentals of the use of computers.(Although the importance of the slide rule isdeclining because of the growing useof calculators, an understr4ndngof the logarithmic scale is asimportant as ever.) The last five chapters examinethe basic properties of theelemen- tary functions,including their derivatives andintegrals.Special emphasis is placed on findinganalytic expressions fromgraphical representation of data. The book has been writtenwith an interactive modeof learning in of mind.It is suitable for section workwhere short lectures, discussion textand problems can be carriedout as needed. Whenever webelieved 4 that certain points are best made by having the students tackle thein, these points were included in the questions at the end of the section. Thus, the questions form an integral part of the course. Many of the questions can be approached in different ways and thereby present the opportunity for constructive discussion and a means for improving the communicative skills of the students. There are relatively few drill problems. Extra problems of this kind can easily be provided by the instructor. Because questions are placed after each section, the text may also be used for individual study. This book has its origin as the freshman mathematics course in our Uniergraduate Program forPhysics-Chemistry Teachers that started in 1970. However, since then it has also been usedextensively by students in other fieldi. The principal contlibutors to the preliminary edition wereJudsbn B. Cross, Thomas J. Dillon, Jo Rita Jordan, George Lukas, LeonardT. Nelson, Poul Thomsen, David B. Teague, and myself. This book constitutes a far-reaching revision of the preliminary edition, including much new mEterial. The revision was done byJudson B. Cross, Robin Esch, Romualdas Skvatcius, and myself. The revision benefited from the feedback of the following professors' who piloted the course:Leonard T. Nelson and Joseph Van Wie at South- west Minnesota State College, Henry P. Guillotte at RhodeIsland College, and Albert G. 51.arling and David B.teague at Western Carolina University. The work was illustrated by George Filgulietti and Myrna S.Goldblat, and produced by Benjamin T. R.chards.The bulk of the camera copy slas typed by Caroline E.. Russell; the typing wascompleted by Lorraine Perrotta. The development of this book is E ,Ipported by a grant from theNational Science Foundation. This financial support is gratefully acknowledged. Uri Haber-Schaim July 1975 CONTENTS Chapter 1 PHYSICAL NUMBERS 1 Mathematical and Physical Numbers;Uncertainty ... Significant Digits...Addition and Subtraction of PhysicalNu, Lbers ... Com- putations with PhysicalNumbers Chapter 2 COMPARING NUMBERS AND SETSOF NUMBERS 14 Comparing Numbers by Ordering and byDifference Comparing Numbers by Ratio ... The FractionaiDifference:Per Cent .. Specific Quantities ... Comparing Sets ofNumbers; Central Tendency...Histograms arid Frequency Distributions ... Measure of the Spread of Data Chapter 3 NUMERICAL CALCULATIONS 53 Larrje and Small-NurnherCalculations ... Estimation ...First- Order Approximations ... AnExtension of First-Order Approxi- mations .. RelativeUncertainties in Multiplir.:ation and Division...Finding Square Roots by an IterativeProcess ... The rirst Order Approximation forNil ( Chapter 4 SLIDE RULES 67 Multiplication and Division ofPowers of Two ... Non-Integral Poweis of Two ... A Power-of-TenSlide Rule ... Divis'an and Multiplication Using Only aOne-Decade Slide Rule ... Com-I mercial Slide-Rule Scales ...Multiple Multiplicaticn and Division Constant Factors, Ratios, andUncertainty ChaptEr 5 AUTOMATIC COMPUTATION 85 Programs...Loops and Branches; Flow Charts ...Basic BASIC Running BASIC Programs Chapter 6 GRAPHS 115 Functions; Independent and DeperLdentVariables ... Choosing Scales for Axes ... SmoothCurves and Uncertainty ...Interpo- lation and Extrapolation - iv- Chapter 7 LINEAR AND POWER FUNCTIONS 139 Notation...Homomorphic Curves...Direct Proportions... The Linear Function...Finding the Equation of a Straight Line...The Quadratic Function...Recognizing Quadratic Functions from Graphs..Inverse Proportions...The inverse- Square Function...Recognizing Hyperbolas and Inverse-Square Functions Chapter 8 DERIVATIVES AND ANTIDERIVATIVES 179 Function and Slope The Slope Function...The Delta Pro- cess ... The Derivative ofI and qi-c...Some Properties of Derivatives...Antiderivatives...The Constant of Integration: The Initial Condition...Short-Range Predictions...A General Way of Calculating Integrals...The Area Under a Curve... The Area Function...Numerical Integration Chapter 9 THE DOONENTIAL AND LOGARITHMIC FUNCTIONS 217 The Exponential Function y = 10x ... The Exponential Functions y = bx and y = kbx ... Inverse Functions ...g(x) = log10x; The Inverse Function of f(x) = lox The Functions ex and lnx The Derivative of y = ex; Exponential Growth and Decay ... Recognizing Functions of the Form y = Cbx ... Recognizing Func- tions of the rorm y = mlogx + b ... Recognizing Functions of the Form y = ax"...Scale Stretchinby Logarithmic Plotting Chapter 10 THE SINE AND COSINE FUNCHONS 267 Sectors and Radians ... Definitions ... Symmetries and Anti- symmetries Periodicity...Homomorphic Trigonometric Functions...The Functions y = A sin (kx) and y = cos (kx) y = A sink (x -x0) and y = A cos k(x -x0)...Recognizing Trigo- nometric Functions from a Graph ... Qualitative Observations on the Derivatives of sinx and cos x ...The Addition Formula for the Sine Function ... The Derivative of siax ... The Derivc- tive of cos x APPENDICES Appendix 1 ALGEBRAIC MANIPULATIONS 305 some Properties of Numbers ... Addition and Subtraction of Algebraic Expressions ... Multiplication of Algebraic Expres- sions ... Some Special Products ... Factoring ...Division of Polynomials in One Variable ... Algebraic Fractions ... Multiplication and Division of Algebraic Fractions Appendix 2 WHAT CAN WE DO TO EQUATIONS? 324 Adding a Well-Chosen Zero ...Multiplying by a Well-Chosen One What Can Be Done to an Equation byWorking With Both Sides ?. .Solving the Quadratic Equation ... Substitu- tion ... What Can Be Done to Two orMore Equations? ... Graphical Solution of Two Equations ...Some Precautions Appendix 3 INEQUALITIES 346 Notation...Properties of Inequalities Appendix 4 TABLES 349 Logarithms of Numbers ... TrigonometricFunctions in De- grees ... TrigonometricFunctions in Radians Chapter 1.PHYSICAL NUMBERS 1.1 Mathematical and Physical Numbers; Uncertainty Numbers mean different things in different contexts.In mathematics a number is ordinarily considered
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