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

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

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

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

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