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=
