The Free High School Science Texts: A Textbook for High ...

1 TheFreeHighSchool ScienceTexts:A TextbookforHighSchool Students 9, 20051 See 2003\FreeHighSchool ScienceTexts"Permissionis grantedto copy, distributeand/ormodifythisdocu-ment underthetermsof theGNUFreeDocumentationLicense, any laterversionpublishedby theFreeSoftwareFoun-dation;withnoInvaria nt Sections,noFront-Cover Texts,andnoBack-Cover copy of thelicenseis includedin thesectionentitled\GNUFreeDocumentationL icense".iContentsIMaths11 .. Numbers.. Line.. of RealNumbers.. andUnlike Surds.. Surd.. cNotation..212 Patternsin .. Equationsforsequences.. (Grade12).. niteSeries..36ii3 .. ,Constants andRelations.. nitionof a Function(grade12).. a Relation..494 .. (pleasedeletewhen nished).

2 MinimisingtheObjective Function..59 Essay1: Di erentiationin theFinancialWorld625 Di .. andlimits.. erentiatingf(x) =xn.. Di erentiation.. erentiationwithGraphs.. Lines.. Sketching..746 .. of Polygons.. Theorem.. ,Re ecting,StretchingandShrinkingGraphs:.. a Line.. Semi-circles.. transversallines..917 .. radians.. nitionof theTrigonometricFunctions.. a Right AngledTriangle.. , Cosecant, Cotangent andtheirgraphs.. ection.. `Productto Sum'and`Sumto Product'Identities.. Trigonometry.. andDepth.. theTrigonomerticRulesandIdentities.. 1288 .. thebasics.. Combiningthebasicsin a fewsteps.. quadraticfunctionin theformf(x) =a(x p)2+q.. a QuadraticEquation?.

3 QuadraticEquations.. a QuadraticInequality .. 1509 .. 154 IIOldMaths15510 .. :.. 186A GNUFreeDocumentationLicense188vPartIMath s1 Thisbookattemptsto meetthecriteriafortheSA\OutcomesBased"sy l-labusof fewnotesto authors:All\realworldexamples"shouldbe in thecontextof HIV/AIDS,labourdisputes,humanrights,soci al,economical,cultural, ,everysectionshouldhave a fordisprovingsomethingis by cationforany mathematicalgeneralisationsof appliedexamplesis ( andsurds?)A number is a way to represent quantity. Numbersarenotsomethingthatwe cantouch or hold, threeapples,threepencils, justtouch three,youcanonlytouch threeof ,youdon'tneedto seethreeapplesin frontof youto know thatif youtake oneappleaway, thattherewillbe two number represents quantity becausewe canlookat theworldaroundusandquantifyit many minutes?

4 How many kilometers?How many apples?How much money?How much medicine?Theseareallquestionswhich canonlybe answeredusingnumbersto tellus \how much"ofsomethingwe want to number canbe writtenmany di erent ways andit is always bestto choosethemostappropriateway of example,thenumber \ahalf" may be spokenaloudor writtenin words,butthatmakes mathematicsverydi betterway of writing\ahalf" is as afraction12or as a decimalnumber 0; is stillthesamenumber,nomatterwhich way ,allthenumberswhich youwillseearecalledreal num-bers(NOTE:Advanced:Thename\realnumbe rs"is usedbecausetherearedi erent andmorecomplicatednumbersknownas \imaginarynumbers", won'tbe lookingat numberswhich aren'treal,if youseea number youcanbe sureit is a realone.)

5 AndmathematiciansusethesymbolRto standforthesetof all real numbers, which simplymeansallof theserealnumberscanbe writtenin a particularway, erent ways of writingany number ,andwheneach way of writingthenumber is best.(NOTE:Thisintroneedsmoremotivationf ordi erent types of numbers, erent whenwe dothehistoryeditnearrelease.) syllabusrequires: algebraicmanipulationisgovernedbythealge braoftherealnumbers manipulateequations(rearrangefory,expand asquaredbracket)(NOTE:\algebraof theReals".why ,like changefroma ,squaredbrackets,fractions, onesideandtheother.)Whenyou add,subtract,multiplyor dividetwo numbers,you areperformingarithmetic1. Thesefourbasicoperations(+; ; ; ) canbe performedonanytwo two realnumbers,it wouldtake forever to writeouteverypossiblecombination,sinceth ereareanin nite(NOTE:Advanced:wereallyneedto de newhatin nitemeans,nicely!)

6 Amount of realnumbers!Tomake thingseasier,it is convenient to uselettersto standin forany number2,andthenwe can llin a particularnumber whenwe example,thefollowingequationx+y=z( )can ,xrepresents theamount of changeyoushouldget,zis theamount youpayed andyis thepriceof dois writetheamount youpayedinsteadofzandthepriceinsteadofy, yourchangeis thenx. Butto be ableto ndyourchangeyouwillneedtorearrangetheequ ationforx. We'll ndouthow to dothatjustafterwe ,subtracting,multiplyinganddividingareth emostbasicoperationsbetweennumbersbutthe yareverycloselyrelatedtoeach subtractingas beingtheoppositeof addingsinceaddinga number andthensubtractingthesamenumber ,if we startwithaandaddb, thensubtractb, we willjustgetback toaagaina+b b=a( )5 + 2 2 = 5(NOTE:reworkthesebitsinto theNegative needsmoreattentionthanwe initiallythought.)

7 Subtractionis actuallythesameas addinganegativenumber. A negative number is a number Inthisexample,aandbarepositive numbers,but bis a negative numbera b=a+ ( b)( )5 3 = 5 + ( 3)1 Arithmeticis theGreekwordfor\ number "2We willlookat thisin moredetailin doesn'tmatterwhich orderyouwriteadditionsandsubtractions(NO TE:Advanced:Thisis a property knownasassociativity, which meansa+b=b+a),butit looksbetterto writesubtractionsto theright. Youwillagreethata blooksneaterthan b+a, andit makes somesumseasier,forexample,mostpeople nd12 3 a loteasierto workoutthan 3 + 12, NumbersNegative numberscanbe veryconfusingto beginwith,butthereis nothingtobe negative number ,it is thesameassubtractingthatnumber if it werepositive.

8 Likewise,if yousubtracta negativenumber,it is thesameas addingthenumber if it werepositive. Numbersareeitherpositive or negative, andwe positive number haspositive sign,anda negative number hasa negative sign.(NOTE: number movingto left,addingis movingto theright. maybe somethingelseaboutnegative numbers?) to calculatethesignof theanswer whenyoumultiplytwo rstcolumnshowsthesignof oneof thenumbers,thesecondcolumngives thesignof theothernumber,andthethirdcolumnshowswha tsigntheanswer negative number byaba b++++---+---+ :Tableof signsformultiplyingtwo positive number alwaysgives youa negative number ,whereasmultiplyingnumberswhich have thesamesignalways gives a positive example,2 3 = 6 and 2 3 = 6, but 2 3 = 6 and2 3 = erently, have a lookat youaba+b++++-?

9 -+? :Tableof signsforaddingtwo positive numbersyouwillalways geta positive number ,butif youaddtwo negative numbersyouwillalways geta negative thenumbershave di erent sign,thenthesignof theanswer dependsonwhich oneis equation( )we usedbrackets3around b. Bracketsareusedto show theorderin which important as youcangetdi erentanswersdependingontheorderin which example(5 10)+ 20 = 70( )whereas5 (10+ 20)= 150( )If youdon'tseeany brackets,youshouldalways domultiplicationsanddivi-sions rstandthenadditionsandsubtractions4. Youcanalways putyourownbracketsinto equationsusingthisruleto make thingseasierforyourself,forexample:a b+c d=(a b) + (c d)( )5 10 + 20 4=(5 10)+ (20 4) additionandsubtraction,multiplicationand divisionareoppositesofeach a number andthendividingby thesamenumbergetsus back to thestartagain:a b b=a( )5 4 4 = 5 Sometimesyouwillseea multiplicationof letterswithoutthe symbol,don'tworry, towritethingsin theneatestway b c( )It is usuallyneaterto writeknownnumbersto theleft,andlettersto arethesamething(NOTE:Advanced.)

10 Thisisa property knownascommutativity, which meansab=ba), it youseea multiplicationoutsidea bracket like thisa(b+c)( )3(4 3)thenit meansyouhave to multiplyeach partinsidethebracket by thenumberoutsidea(b+c)=ab+ac( )3(4 3)=3 4 3 3 = 12 9 = 33 Sometimespeoplesay \parenthesis"insteadof \brackets".4 Multiplyinganddividingcanbe performedin any orderas it doesn' 'tmatterwhich longas youdoany beforeany + .6unlessyoucansimplifyeverythinginsideth ebracket into a fact ,in theabove example,it wouldhave beensmarterto have donethis3(4 3) = 3 (1)= 3( )It canhappenwithletterstoo3(4a 3a) = 3 (a) = 3a( )If therearetwo bracketsmultipliedby each other,thenyoucandoit onestepat a time(a+b)(c+d)=a(c+d) +b(c+d)( )=ac+ad+bc+bd(a+ 3)(4+d)=a(4 +d) + 3(4+d)=4a+ad+ 12 + to theexampleaboutchange,which we wantedto solve earlierinequation( )x+y=zTo recapyourmemory,zis theamount you (ora customer)