Calculus single variable

745 Pages·2017·11.696 MB·English

by  Bretscher, Otto, Hughes-Hallett, Deborah, Sloane, David

Lines Distance andMidpoint Formulas
Slopeoflinethrough(𝑥 ,𝑦 )and(𝑥 ,𝑦 ): Distance𝐷between(𝑥 ,𝑦 )and(𝑥 ,𝑦 ):
1 1 2 2 1 1 2 2
𝑦 −𝑦
𝑚= 𝑥2−𝑥1 𝐷= (𝑥2−𝑥1)2+(𝑦2−𝑦1)2
2 1 √
Point-slope equation of line through (𝑥1,𝑦1) Midpointof(𝑥1,𝑦1)and(𝑥2,𝑦2):
withslope𝑚:
𝑥 +𝑥 𝑦 +𝑦
1 2, 1 2
𝑦−𝑦 =𝑚(𝑥−𝑥 ) ( 2 2 )
1 1
Slope-intercept equation of line with slope 𝑚 QuadraticFormula
and𝑦-intercept𝑏: If𝑎𝑥2+𝑏𝑥+𝑐 =0,then
𝑦=𝑏+𝑚𝑥 −𝑏± 𝑏2−4𝑎𝑐
𝑥=
√2𝑎
FactoringSpecial Polynomials
RulesofExponents 𝑥2−𝑦2 =(𝑥+𝑦)(𝑥−𝑦)
𝑥3+𝑦3 =(𝑥+𝑦)(𝑥2−𝑥𝑦+𝑦2)
𝑎𝑥𝑎𝑡 =𝑎𝑥+𝑡 𝑥3−𝑦3 =(𝑥−𝑦)(𝑥2+𝑥𝑦+𝑦2)
𝑎𝑥
=𝑎𝑥−𝑡
𝑎𝑡 Circles
(𝑎𝑥)𝑡 =𝑎𝑥𝑡
Center(ℎ,𝑘)andradius𝑟:
DefinitionofNatural Log
(𝑥−ℎ)2+(𝑦−𝑘)2 =𝑟2
𝑦=ln𝑥 means 𝑒𝑦 =𝑥
ex:ln1=0since𝑒0 =1 Ellipse
𝑦 𝑥2 𝑦2
𝑦=𝑒𝑥 𝑎2 + 𝑏2 =1
1 𝑦=ln𝑥
𝑥 𝑦
𝑏
1
𝑥
−𝑎 𝑎
Identities
−𝑏
ln𝑒𝑥 =𝑥 Hyperbola
𝑒ln𝑥 =𝑥
𝑥2 𝑦2
RulesofNatural Logarithms 𝑎2 − 𝑏2 =1
𝑦
ln(𝐴𝐵)=ln𝐴+ln𝐵 𝑦=𝑏𝑥∕𝑎
𝐴
ln =ln𝐴−ln𝐵
𝐵
(ln𝐴)𝑝 =𝑝ln𝐴 𝑎 𝑥
𝑦=−𝑏𝑥∕𝑎
GeometricFormulas
ConversionBetweenRadiansandDegrees: 𝜋radians =180◦
Triangle Circle SectorofCircle
𝐴= 1𝑏ℎ 𝐴=𝜋𝑟2 𝐴= 1𝑟2𝜃 (𝜃 inradians)
= 12𝑎𝑏sin𝜃 𝐶 =2𝜋𝑟 𝑠=𝑟𝜃2 (𝜃inradians)
2
𝑎 𝑟 𝑟 𝑠
ℎ
𝜃 𝜃
✛ 𝑏 ✲ 𝑟
Sphere Cylinder Cone
𝑉 = 4𝜋𝑟3 𝐴=4𝜋𝑟2 𝑉 =𝜋𝑟2ℎ 𝑉 = 1𝜋𝑟2ℎ
3 3
TrigonometricFunctions
𝑦
sin𝜃 = (𝑥,𝑦)
𝑟 ✛ ✻ sin(𝐴±𝐵)=sin𝐴cos𝐵±cos𝐴sin𝐵
𝑥
cos𝜃 =
𝑟 𝑟 cos(𝐴±𝐵)=cos𝐴cos𝐵∓sin𝐴sin𝐵
𝑦
𝑦
tan𝜃 = sin(2𝐴)=2sin𝐴cos𝐴
𝑥
✛
sin𝜃 𝜃 ❄ cos(2𝐴)=2cos2𝐴−1=1−2sin2𝐴
tan𝜃 = ✛ 𝑥 ✲
cos𝜃
cos2𝜃+sin2𝜃 =1
𝑦 𝑦 𝑦
𝑦=tan𝑥
1 𝑦=sin𝑥 1 𝑦=cos𝑥
𝑥 𝑥 𝑥
𝜋 2𝜋 𝜋 2𝜋 −𝜋 𝜋
−1 −1
The Binomial Theorem
(𝑥+𝑦)𝑛 =𝑥𝑛+𝑛𝑥𝑛−1𝑦+ 𝑛(𝑛−1)𝑥𝑛−2𝑦2+ 𝑛(𝑛−1)(𝑛−2)𝑥𝑛−3𝑦3+⋯+𝑛𝑥𝑦𝑛−1+𝑦𝑛
1⋅2 1⋅2⋅3
(𝑥−𝑦)𝑛 =𝑥𝑛−𝑛𝑥𝑛−1𝑦+ 𝑛(𝑛−1)𝑥𝑛−2𝑦2− 𝑛(𝑛−1)(𝑛−2)𝑥𝑛−3𝑦3+⋯±𝑛𝑥𝑦𝑛−1∓𝑦𝑛
1⋅2 1⋅2⋅3
CALCULUS
SeventhEdition
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CALCULUS
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PREFACE
Calculusisoneofthegreatestachievementsofthehumanintellect.Inspiredbyproblemsinastronomy,
NewtonandLeibnizdevelopedtheideasofcalculus300yearsago.Sincethen,eachcenturyhasdemonstrated
thepowerofcalculustoilluminatequestionsinmathematics,thephysicalsciences,engineering,andthesocial
andbiologicalsciences.
Calculushasbeensosuccessfulbothbecauseitscentraltheme—change—ispivotaltoananalysisofthe
naturalworldandbecauseofitsextraordinarypowertoreducecomplicatedproblemstosimpleprocedures.
Thereinliesthedangerinteachingcalculus:itispossibletoteachthesubjectasnothingbutprocedures—
therebylosingsightofboththemathematicsandofitspracticalvalue.ThiseditionofCalculuscontinuesour
efforttopromotecoursesinwhichunderstandingandcomputationreinforceeachother.Itreflectstheinput
ofusersatresearchuniversities,four-yearcolleges,communitycolleges,andsecondaryschools,aswellas
ofprofessionalsinpartnerdisciplinessuchasengineeringandthenaturalandsocialsciences.
MathematicalThinkingSupportedbyTheoryandModeling
Thefirststageinthedevelopmentofmathematicalthinkingistheacquisitionofaclearintuitivepictureofthe
centralideas.Inthenextstage,thestudentlearnstoreasonwiththeintuitiveideasinplainEnglish.Afterthis
foundationhasbeenlaid,thereisachoiceofdirection.Allstudentsbenefitfromboththeoryandmodeling,
butthebalancemaydifferfordifferentgroups.Somestudents,suchasmathematicsmajors,mayprefermore
theory,whileothersmayprefermoremodeling.Forinstructorswishingtoemphasizetheconnectionbetween
calculusandotherfields,thetextincludes:
• Avarietyofproblemsfromthephysicalsciencesandengineering.
• Examplesfromthebiologicalsciencesandeconomics.
• Modelsfromthehealthsciencesandofpopulationgrowth.
• Problemsonsustainability.
• CasestudiesonmedicinebyDavidE.Sloane,MD.
ActiveLearning:GoodProblems
As instructors ourselves, we know that interactive classrooms and well-crafted problems promote student
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problemsprobestudentunderstandinginwaysoftentakenforgranted.Praisedfortheircreativityandvariety,
theseproblemshavehadinfluencefarbeyondtheusersofourtextbook.
TheSeventhEditioncontinuesthistradition.Underourapproach,whichwecallthe“RuleofFour,”ideas
are presentedgraphically,numerically,symbolically,and verbally,therebyencouragingstudents to deepen
theirunderstanding.Graphsandtablesinthistextareassumedtoshowallnecessaryinformationaboutthe
functionstheyrepresent,includingdirectionofchange,localextrema,anddiscontinuities.
Problemsinthistextinclude:
• Strengthen Your Understanding problemsat the end of every section. These problemsask students
to reflect on what they have learned by deciding “What is wrong?” with a statement and to “Give an
example”ofanidea.
• ConcepTestspromoteactivelearningintheclassroom.Thesecanbeusedwithorwithoutpersonalre-
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• Data and Models Many examples and problemsthroughoutthe text involve data-drivenmodels. For
example,Section11.7hasaseriesofproblemsstudyingthespreadofthechikungunyavirusthatarrived
v
vi Preface
intheUSin2013.ProjectsattheendofeachchapteroftheE-Text(atwww.wiley.com/college/hughes-
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• DrillExercisesbuildstudentskillandconfidence.
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andProblems,CASChallengeProblems,andProjects
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tions
• Sectiononthe𝜖,𝛿definitionoflimit(1.10)
• Appendicesthatincludepreliminaryideasusefulinthiscourse
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FlexibilityandAdaptability:VariedApproaches
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encesregardinginclusionoftopicsandapplicationsandtheuseofcomputationaltechnology.Forthosewho
prefertheleantopiclistofearliereditions,wehavekeptclearthemainconceptualpaths.Forexample,
• TheKeyConceptchaptersonthederivativeandthedefiniteintegral(Chapters2and5)canbecovered
attheoutsetofthecourse,rightafterChapter1.