DOCUMENT RESUME ED 173 AEI SE 027 905 AUTHOR Herriot, Satan T.; And Others. TITLE Calculus of-EleMentary Functons, Part Student Text. Preliminry -Edition. INSTITUTION StanfordUn iv. Calif...School Mathematics Study- GIQ up. ,' SPONS AGENCY National Science Foundation, Washington, D.C. PUB DATE,- '68 . NOTE 33Bp.;.For related documents, see SE-027-906 and. ED* 143 514; Contains :occasional light and broken type -EDES PRICE MF01/P014 Plus PostSq. DESCRIPTORS *Caicu1l s;Curriculum; Graphs; *InstructiOn; Mathematics Edd?ation;! Secondary Education; '*Secondary chool Mathematics; *Textbooks; *Trigonome IDENTIFIERS Functions 10ematics);. *Polynomials; *School Math..7matits Study Group. ABtTEACT . This is part one of *a thre-part SMSG.calculus text for high scholi kdents; Th aim O_theof tr-xt is to develop. some. of th:concepts an chnigues which will ,,.?.abl..,4111. student to. obtain. important graphs oi *1--men--_ary functions. Chapter (1) alynomial fwictions; (2) the derivative of a topicsinclude: . polynomial furctio, nd ( cular functions. .("SAP) ... *************************************************************** ** * -ReproduceOns supplied by EbRS are the best that canbe made ( * 1 . -' . , 4 * . from the originaldocuwent. ...--- . CO 0' C.) SCHOOL PC\ ti MATHEMATICS t-4 STUDY;GROUP L=1 La r'_PARTMENT OF HEALTH. 4 EDUCATION 6 WEi..FARE TERMVION TO REPRODUCE THIS 14 AT icurtst INSTITUTE OF MATERIAL HAS BEEN GRATED. BY EDUCATION *4,5 DOCUMENT HAS BEEN REPRO - DUC E7,7F JD ACT i YpAG,S C E 1:./ ED ROE OR ON POINTS Of viE STATED DO NOT .1ECISSARII V REPRE- SENT nEFIci:4'NATor., INSTITUTE OF .4 O. FOi ACV TOTHE EDUCATIONAL RESOURCES INFORMA,TION CENTER (E7C) . CALCULUS OF ELEMENTARY' FUNCTIONS Part I' 1 .A Student,Text. (PrOirnimtry Edition) 4 La- . CALCULUS OF LEMENTARNA FUNCTIONS:..4 Part I Student Te5ct'' (Pieliminary'Edition)7 The following is a list of all those who participated in the preparation of this volume: Sarah T. Herriot, Gunn High School, Palo Alto, C Desmond T. Jenkins, PaloAlto Senior High Sch C. W. Leeds, III, Berk.hire School, Sheffield, Mas " PaulC. Shields, Wayne State University, Detroit . t .e . 4 . ; .4 Financial supportfor the .SchoolMuhematii Study Group has been provided by the National ScienceFoundation N Permission to make verbatim use-"ofmaterial in this bomust be secured from the Director of ch permission W be'ranted' except in unusual circumstances. Publicatz ns ,orating SMSG materials must include.both an acknOwledgment f the MSG copright (Yale Univer- sclaimer of SMSG ' 4it y or Stanford Unipersity;asthe case may be) and a endorsement. Exclusive license will notbe granted sa e. in exceptional circumstances, and then only byspecific action ?f dvisory Board of SMSG. ". 41° .0 1968 by The gpard of Trustees.of theLeland Stanford Junior University 411 rights reserved PrinteNn the .United States of America / FOREWORD The correspondellee.betw6ei, graphs in the xy-Iblane andfelationsbetv:reen x and y was one of the profound discoveries of thematics. In particular,. .ifno Itertical line can meet 'a graPh in more thanon point, then the graph that of a function f*: x that is, for eiChfir. cnrdinate x of'a 3point on the graph' there is a UniqUe. number 3ititsuchhat (x,y) lies onthe graph. A.centrel-purpose of. this.text is. to Au the relatiOnship between' graphs of functions anethe expressions which define iglese functions.We- shall concentrate our atte 'ntion on funClions defined by polyndtiq,strigonoTetric, exponential and logarithmic expressions, or. by combinations of such expressirs These functions are. usually referred to as elementary,functiips. - We ;shot:1d expedi t. erties'of the graph of-s>function are related to the expreRsion-lhiehdefines the functiolks. For exampleloyanalyzing the functional 'expression we .should beAble to determine the location high and'' lr pointS on'the graph, and in addition answer questions abou-hthe sbape.of the graph (sueh intervals of rise or fall and how the graph benda).. F. her- more, if the function is related to some physical problem involving:moti it : 9 seeMs reesonablela-hat analysis of its expression shOuld enable us4 to deter- aspectS of the' motion as velocity and acceleration. Iikewtse:we might ecitbbe able ts determine from' the functional express ion such prd7 r :/ ' ertie as ,the alt,gyae.value of the function and the area of a region bon9dedLti by the.functiOn.c , . °.Ouraim i-A to develop some of the conceptb and techni.queswhich will enable us to obtain4kPortant information about .graphs ofelementary The primary concept. 4hich We d'ezelpih Volumed s tha f the ent line at a point on .the graph of afuncti8h. his tangent .1 isd as the . - straight 1pie which best fits'the gr h near that point: .Id formulas Will be developed for findinthe slope of the tangent line toithe. graphs of various elementary function. For 'a given fanction,-. .f, such a formula defines a new function called 'the derivative of f: This derivative \ function is Often easy to obtain and its values give a measure Of the rate of 4,Z --- change of they graph. In p rtd.cular, such aspects of7ffiotion as velocity and . ., : acceleratioh are described by)' the de iiive, .Furthermore, high and,ow points -vs of ttle.graph of ,f' can be given by z k of .the derivative awhile rise or fall can be determined by the sign of the derive oe.") .4- 4 . r . ' ( N functions, Our fiat tasit is to analyzethe simplest kind's of elemeAary namely polyilomials,,the sine andcosine functions,- and simple power, expszen- These are disTassed in Volume One,whilethe tial andlogarithiniefunctions. more difficulttgihniques associated with various algebraic combinations (siltns, Troducts, quotients, composition)of't14se basic functions" are left . concept of antiderivative is to Chapter 8 (in Volume'Two).In addition, the introduced' in Volume Two', thq antiriyative of fbeing a function whose antiderivatives can be used tocalculate derivative .f. We shall show ho acceleration. areas and to -solve suchproblems as determining veloCity given Furthermore, this relationship betweenantiderivatives and areaRrovides us functions. witlya powerful .geometric tool foranalyzing and approximating bhapters l and 2- discuss polynomial functions, withChapter 1 concentrat- algebraic and .geometricpropertles. The concept . ingon definitionl.and simple of tangent line at a point on thegraph of= polynomial function is introduced in Chhter 2 and formulas for thederivative of a-polynomial function, are q> high Applications of the.derivative tographing (such as finding velocity and low,.points andintervals of rise or. fall); its interpretation as approximationarealpo discussed in Chapter 2.. or aCcele'ration, and its use in This same pattern is followed in theremaining gour chapters of this volume. Deffhittons and simple properties61.. the sine and cOsine functions and the 5; reSpee- 'power exponential and.logarihmfunctions appear in Chapters 3 and for these respeC- tively, While .Chapters 4 and6 discs derivative formula A tive classes, as well as applicationsto graphing and approximations. , . For their thoughtful suggestions.and'contributions we are grateful to 0 . Frank B. LyOna'fTownship High School . Leonard Gillman, University, of Rochester; David W. Jonah, Wayne State-University; Donald E.-Richmond, Williams College; Albert W. Tucker, Princeton UniVersity. We are also indebted to previbus SMSG writing teams, whose materials had considerable influence on this text. Many ideas and exercises were taken directly or adapted from two earlier SMSG texts: Elementary- Functions and 'Calculus. In the first part of this text we borrow heaviiifrOm SMSG Elemen- tary-Functions; Appendix 2 of this text contains Sections 2-12, 2-6, 2-8, and 2-9 of Elementary Functions. The sections indicated in the folloWing table were taken (with minor editing) from SMSGtCalculus for the appendices to this text. Contains the Following Sections of This T t SMSG CalCulus i l 1 A2-1, A2-2, A2-3, A2-4,A276 . A3 -1; A3 -2 .3. 4 10-3, 10-4', 10-5, 10-6 5c 6-1,. 6-2, A6-3:),-, 6-3, 6-4 0 .,q 6 3-2, 3-3, 3-4 7lz A4-1, A24-2,Tarts 5-3 and A1-5 . 8 l'A6-2 . r TABLE OF' CONTENTS . A Page Chapter 1. POLYNOMIAL FUNCTIONS 1-1., 'Introduction and Notation. 2 1-2. Constant and Linear Functions 7 1-3. Quadratic Functiops ' 1-4. Polynomial Functions of Degree Higher Than 'One 36 ' 1 -5. .Remainder and Factor Theorems' . .... 46 1 -6. .Locating Zeros:of Po omial Functions . 53 1.-7. Approximating Zeros . ).. 1..... 62 1-8. 'Degree an Zeros 65 ,Chapter 2. THE DERIVATIVE OF A POLYNOMIAL FUNCTION' '79 2-1. The Tangent at the y-,Intercept of A Graph , . ... 1 . 81 272. The Tangent at the y- Intercept . 1 ....... 88 2-3. The Behavior of the Graph Near anArbiriry Point .°. '96 2-4. ,,Agplicatioa to Graphing .... ........104 2-5. The Slope 'as Limit of Difference Quotients '109 :2-6. The Derivative g ..... 117, 2-7. Applications of the Derivative to Graphing 126 2-8°: Optimization ProbleMs' f3C 2-9. Rate of Change and Velocity 148' 2=10. The Second Derivative . ... .. 159 2-11. NeWton'8 Method , ...... I . 168 2-12. Translation 176 2-13..Hig11er Derivatives 181 General Polynomial PUnctions t 188 . Chapter 3. CIRCULAR FUNCTIONS .195 3 -1. The Sine and Cosine Functions ...:.. %. .- 196 3-2% Properties of the:Circular Functions 215 3-3 Graphs 'of the Circular Functions 25 3-4. Uniform Circular Motion 232 3-5-.The AdditiOn Formulas' 240 . 3-6: Pure Waves 247. 3 -7. Anaiysiswof General Waves. Period 256 4 .; fa 4 Page . 263 APPENDIX 1. FUNCTIONS AND)THEIR REPRESENTATIONS . All. Functions 263 Al -2. Composite. Functions 79 Al-B. Inverse FunctifOns. .* .; ..... 284 A1-4. MOnotone Functions ..292 297 ° Polar Coordinates . APPENDIX 2. POLYNOMIALS.4 309 ..- A2,-1. Significance of Polynomials,, .1'49 , A2-2. Rational Zeros , ? 315 , . 322 A2-3. Number of Zeros . ....... A2J-4. Complex Zeros 329 1' r O Chapter 1 POLYNOMIAL FUNCTIONS This is the first of two chapters on polynomial functions.Here we con- .