142320Calculus Equations

EssentialToolsfor UnderstandingCalculus- Rules, Concepts,Variables, Equations,Examples, j)Helpful Hints&LhCommon PitfallsSTRATEGY FORSOLVING PROBLEMS EFFECTIVELY I.Understand the principle (business or scientific)required.II. j) Developa mathematicalstrategy.A.There are eight useful steps that will help youdevelopthecorrectstrategy.I.Sketch, diagram orchartthe relationships andinformationthatissubjectof theproblem.2.Identifyall relevant variables, concepts andconstants.3.Describethe problem situations usingappropriatemathematicalrelationships,functions,formulas,equationsorgraphs.4.Collectall essentialinformationanddata.S.Lh extraandunnecessarYinformationanddata.6.Derive a mathematical expression or statementfor the problem, making sure all measurementsarein'thecorrectunit.7.Complete the appropriate mathematicalmanipulationsandsolutiontechniques.8.Check the final answer by using the originalproblemand informationtomakecertainthattheanswers, units, signs, magnitudes, etc., all makesenseandarecorrect!FUNCTIONS I. DefinitionsA.Arelationisasetoforderpairs;written(x,y)or(x, B.Afunctionisarelationthathasx-valuesthatarealldifferent fordifferenty-values.Averticallinetestcan be used todeterminea function; everyverticallineintersectsthegraph,atmost,once.C.A one-to-one function is a function that has y-values that are all different for differentx-values.A horizontal linetestcan be used todetermine aone-to-onefunction;everyhorizontallineintersectsthegraph,atmost,once.D.Domainisthesetof allx-valuesof arelation.E. Rangeisthesetof ally-valuesof a relation.F. Afunction is anevenfunction iff(- x)=fix).G.Afunction is anoddfunctioniff(- x)= -f(x).H.Theone-to-onefunctionsf(x)andg(x)areinversefunctionsiff(g(x =g(f(x =x;f'(x)andg-'(x)indicate the inverse functions offix) and g(x),respectively. Inverse functions are reflections overthelinegraphofy = x .I.Dependent variable is the output variable in anequation and depends on or is determined by theinputvariable.1.Independent variable is the input variable in anequation.II. CommonFunctionSummaryA.Linear:f(x)= mx+bI.m istheslope;m=Y2 - y, = y,- Y2 = = risex2-x, x,-x2 run 2.bisthey-intercept.3.lt is a constant function when m = 0; it is ahorizontalline.B.Absolutevalue:f(x)= - hi+kI.(h, k)isthevertex.2.If a>0, thegraphopensup.3.If a0,thegraphgoestotheright.3.p If a. g(x) limg(x)'X-->. limg(x)""0.E. ;:{J(x)"=(limj(x)"),providedn isapositiveX----)Q X-)Qinteger.F. Iimj(x)=Ais equivalentto lim[j(x)-A]=O.x----). X----)QG.lfj(x) < g(x) < hex) for every x in a puncturedneighborhood of a (that is, x near a), andlimj(x)=IimhCxl=A,then limg(x)=A.X-HI x--+a X.....,IIf2whenfindinglimx3-28suchthaI0/'x-+l x-x3, x-2 x-2'x"#2 - 8 becomes (x - 2)(x2+2x +4) andthen(x'+ 2x+ 4); consequently, when x is close to 2,(xl+2x+4)iscloseto12;therefore, limxl-: =12.III. Rulesx----)2 x-A.For polynomialp(x) to the n'h power withthe lead term ofax" and polynomial Q(x) tothe m'h power with the lead term ofbX"', ifp(x)x =Q(x) and Q(x)"# 0, then when:j()l.n=m,the limj(xl=![x----)oo b2.n>m,the limj(x)= lim PxX--,)ooo X----) 0 at everypoint in an interval, thenthegraphoff{x)isconcaveupin thisinterval.d.lfj"(x), m:onh..or In)' infor:nltion .storqt and Ktricnl'YJlcm. ..... ithoul .orI( \CO (rom the:puhliJher.C2eMfbtClluts.l .e.0.409NOTE10STttOEl\.:Thi,pnOcIS inlmdtdOuclOilI6 condcmt;.Jfurm;l1. this autikroNlOtL'O'W 1,.1,.ltU6,'at qUICKStUay.com U.S.$S.95 Aut hor: Dr.S. B. KLzlik :/:Ier:ro:..:CustomerHotline# 1.800.230.9522ISBN-13: 978-142320856-3ISBN-10: 142320856-09IlfIIIlll I6