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These notes coven prerequisitesof Single Variable CalculusNotes
are based on Apostol's calculus Volume 1 Ed 2
TWOCOWCEPTSOFCALC.lkTSome problems are intrinsic components in
many fieldsof
science Calculus is a Technical Tool used to solve twoparticular
problems
1 Area unclear curve2 Sleepinessof slope
ARCHIMEDESMETHODOFEXHAustiohft.tninscribe polygon with shape
2 Increase numberofshapes sides3 Create upper and lower bound4
Calculate area
Problems in order to understand a theory musthave definitions
for symbols and wordsA deductevesystein have undefined conceptsall
other concepts are defined by thoseconcepts
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Statements aboutthese undefinedconcepts arecalled axioms
postulates Deductions about thesystem made from the axioms are
calledtheorems
Calculus could be defined Archimedesmethodofreal Numbers
exhaustionIR undefinedobjects
made claims about aconcept which wasn'tdefined
Integralassumedeveryregion
Derivative has airedProperties assumedproperty
theorems that nestedregionhas smaller airedAurchimedes method is
valuable for assumedpropertyofadditivityofthe sensible way to
define naked rid regions arearegions not because it is a
particularlyusefultechniqueforcalculating area
To learn the technique and some theoryofcalculusone doesn't need
to formallydevelop all the propertiesand theorems Howeverunderstand
that they have beenand onlybecauseof that Caculus and its concepts
can beconsidered valid
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AxioMSFoRTHEREALNUMBERSYSTEMfFormally Inteougans can be used to
construct national
numbers national numbers can be used to construct
immationalnumbers from these constructions the
theoremsofcalculusmust be deduced
Apostol'sbook takes a nonconstructiveapproach Insteadtaking Real
Numbers as undefined objects andassuming propertiesabout them as
axioms
These axioms can be divided into three1 Field Axioms about t x
to2 Order Axioms Cabout E z3 Least Upperbound Axiom i e
axiomofcontinuity
axiomofcompleteness
Fieldroxionislit Chdeerraxion likeCommutativity 1 x y E Rt ay
and ay CRt2 Associativity
B Distributivity2 K 0 A KEIRN x CIRT
4 NegativeNumbers B O IEIR'sExistenceofReciprocals
dppenBound_A number B is the least upperbound of a nonempty sets
if B hasthe followingproperties
1 B is a upperboundfor S Z alls Es2 Nonumberless than B is an
upperboundforS
1 1 41 1 4 11 O111LCcccec l nT R LOB TT A WB when T has no
largestnd
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Axiom Every non empty sets ofruralnumbers which isboundedabove
was a supremum's theme is a realnumber B such that B sup s
AochonidesisPropertyIf x 0 and if g is an arbitrary real number
there exists apositiveinteger in such that ax y
A small enough ruler can be used to measure
arbitrarily longdistancemeans no infinitely large or
smallmembers
Decimal expansion is a exampleofnestedintervals
MATHEMAT1CALlNDUCTloTExample
Assume formula has been proven for a k a forek's 1A k 12 t22t t
K 1 2 s KB
JDeduce result of Ktt Show ifholdsfor aninteger it alsothere
start with Acid and add k for next integer
12 t22t t K 1 2 t K2 2k K2To obtain ACKt 1 show74 s k 5
This can beshownbyexpanding 3 7 Kt 1B
KEIBKGB K2t k t I
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Show k 1 holds
Act 0 133
Principal of Mathematical InductionMethodofproofbyinduction
Let Acn be an assertion involving an integer nConclude Acn is
there for every n E ne if we can do
9 Prove Acne is true2 Let k be an arbitrary but fixed intergen z
n s Assumethat ACID is true and prove that ACKt 1 is also true
In practice he isusually 1The justification for this proof is
thetheorem
THEOREM PrincipleofMathematicalInductionLets be a set
ofpositiveintegers whichhas thefollowing two properties1 the number
1 is in the set s2 if the integer K is in S then so is k t 1
then every positive integer is in the sets
Proof 1 and 2 show S is an inductive setPositiveIntegers are
defined to exactlybethe meal numbers
whichbelongtoeveryinductiveset
The Well OrderingPruinciple
THEOREM WellOrderingPrinciple Everynonempty set
ofpositiveintegers contains a smallestmember
a A consequenceof the principle of induction