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Samenvatting - 1-Wiskunde (1001FTIWIS)
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Het van 1-wiskunde met zeer veel gegeven informatie tijdens de hoorcolleges. Deze samenvatting is een bundel van de genomen notities ui de lessen.
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1-WiskundeTable of Contents
Continuïteit van reële functies..............................................................................................................6
Reële functies...................................................................................................................................6
Soorten functies...............................................................................................................................7
De grafieken.....................................................................................................................................7
Functie vs Injectie............................................................................................................................8
Cyclometrische functies...................................................................................................................8
Continuïteit van elementaire functies...................................................................................................9
Geheelwaarde functie y=G(x)........................................................................................................10
Meervoudig voorschrift.................................................................................................................11
RC (Rechts continu).......................................................................................................................11
LC (Links continu).........................................................................................................................11
Continuïteit in een interval.............................................................................................................11
Continuïteit in een gesloten interval..............................................................................................12
Functie van Heaviside u(x).................................................................................................................12
Eigenschappen van continuïteit..........................................................................................................12
Bewerkingen met continue functies...............................................................................................12
Stelling van Weierstrass.................................................................................................................12
Stelling van tussenwaardes............................................................................................................13
De hoofdeigenschap ( van continue reële functies).......................................................................13
De stelling van bolzano (stelling van het nulpunt)........................................................................14
Elementaire grafieken....................................................................................................................15
Cyclometrische functies.....................................................................................................................18
BoogSinusFunctie..........................................................................................................................18
BoogCosinusFunctie......................................................................................................................18
BoogTangensFunctie.....................................................................................................................19
BoogCotangensFunctie..................................................................................................................19
Eigenschappen...............................................................................................................................20
Bewijzen........................................................................................................................................20
Oneindig.............................................................................................................................................22
Limieten..............................................................................................................................................23
DEF Links limiet...........................................................................................................................25
DEF Rechts limiet..........................................................................................................................25
Verband tussen limiet, linkerlimiet, rechterlimiet..........................................................................25
Rekenregels met.............................................................................................................................25
Eigenschappen van limieten..........................................................................................................26
Een limiet is uniek.....................................................................................................................26
Limiet en continuïteit (Hoofdeigenschap)................................................................................26
Stellingen over het berekenen van limieten...................................................................................26
Aparte limieten..............................................................................................................................27
Onbepaalde Vormen oplossen........................................................................................................27
Voorbeeld 1...............................................................................................................................28
Afgeleide............................................................................................................................................29
Afgeleiden bestaat niet in deze gevallen........................................................................................29
Linker en rechter afgeleiden..........................................................................................................30
1
, Verband afgeleiden in een punt en continuïteit in een punt...........................................................30
Hoge orde afgeleiden.....................................................................................................................31
Voorbeeld.......................................................................................................................................31
Afgeleiden van een inverse functies..............................................................................................31
Afgeleide functie............................................................................................................................32
Basis afgeleiden.............................................................................................................................32
Eigenschappen...............................................................................................................................32
Kettingregel...................................................................................................................................32
Differentiaal........................................................................................................................................33
Differentiaal functie.......................................................................................................................34
Basis differentiaal..........................................................................................................................34
Eigenschappen...............................................................................................................................34
verband toenamen en differentiaal.................................................................................................34
functies gegeven in parametervorm SPV (zonder expliciet voorschrift).......................................35
Samengevat....................................................................................................................................35
2de afgeleide van een functie gegeven door SPV..........................................................................35
Functies gegeven m.b.v. een impliciet voorschrift........................................................................37
Functies gegeven m.b.v. een impliciet voorschrift (verkorte vorm)..............................................38
2de afgeleide van een impliciet voorschrift...................................................................................38
Stellingen van de gemiddelde waarde................................................................................................39
Extreme waarde van een functie....................................................................................................39
De stelling van Fermat...................................................................................................................40
Berekenen min & max van een gegeven functies VB........................................................................41
De stellingen van de gemiddelde waarde...........................................................................................43
Stelling van Rolle...........................................................................................................................43
Bewijs........................................................................................................................................43
Stelling van Lagrange....................................................................................................................43
Bewijs........................................................................................................................................43
Stelling van Cauchy.......................................................................................................................44
Bewijs........................................................................................................................................44
Regel van de l’Hopital........................................................................................................................45
Verloop van functies van de eerste afgeleiden....................................................................................46
De brachistoChrone kromme..............................................................................................................47
Tautochrone kromme..........................................................................................................................47
Cycloïde..............................................................................................................................................47
SPV (Stelsel Parameter Vorm).......................................................................................................48
Hypocycloïde.................................................................................................................................48
Epicycloïde....................................................................................................................................48
Kettinglijnen.......................................................................................................................................48
Verloop van functies...........................................................................................................................49
Gebruik van de eerste afgeleiden...................................................................................................49
Gebruik van de tweede afgeleiden.................................................................................................49
Asymptoten.........................................................................................................................................50
Bepalen van asymptoten................................................................................................................50
VA x = a.....................................................................................................................................50
HA y = b....................................................................................................................................50
SA y = mx + q...........................................................................................................................51
Bepalen van asymptoten met SVP (kromme)................................................................................51
hyperbolische functies........................................................................................................................52
Soorten...........................................................................................................................................52
2
, Deel 1 eigenschappen....................................................................................................................52
Regel van Osborn......................................................................................................................52
Deel2 functies................................................................................................................................53
Deel3 inverse functies....................................................................................................................54
Deel4 afgeleide..............................................................................................................................54
Primitieve functies..............................................................................................................................55
Definities en eigenschappen..........................................................................................................55
De verzameling van alle primitieve functies van f (onbepaalde integraal)...............................56
Verband met differentiaal dx (afgeleiden).................................................................................56
Onbepaald integreren.....................................................................................................................57
Basis integralen.........................................................................................................................57
Eigenschappen..........................................................................................................................57
Integratie Methodes............................................................................................................................58
Substitutie......................................................................................................................................58
Doordachte substitutie...............................................................................................................58
Stappen.................................................................................................................................58
Wilde substitutie........................................................................................................................59
Tips.......................................................................................................................................59
Opmerking 1.........................................................................................................................60
Opmerking 2.........................................................................................................................60
Partiële integratie...........................................................................................................................61
regel...........................................................................................................................................61
bewijs........................................................................................................................................61
Opmerkingen.............................................................................................................................62
Integreren van rationale functies........................................................................................................63
splitsen in partieel breuken............................................................................................................63
Methode....................................................................................................................................63
Stelling van Jacobi....................................................................................................................64
Hoe integreren van veelterm?........................................................................................................65
H13 wortelvormen + goniometrische functies...................................................................................67
Irrationale functies (wortels)..........................................................................................................67
Tip1...........................................................................................................................................67
Tip2...........................................................................................................................................67
Goniometrische/ hyperbolische functies........................................................................................67
Belangrijk..................................................................................................................................67
Reductie (recursie) formules.....................................................................................................67
Stel integrand is en m & n zijn positief en even.......................................................................68
Stel integrand is en m & n zijn negatief en even......................................................................68
Samengevat..........................................................................................................................68
Stel integrand is en m & n zijn oneven (-2,-1,1,3,5,7,..)...........................................................68
Samengevat..........................................................................................................................68
Stel integrand is product van cos en sin met verschillend argument........................................69
H14 De bepaalde integraal.................................................................................................................70
Grondbegrippen.............................................................................................................................70
Verdeling van een interval, verfijning, onbeperkt verfijnen.....................................................70
Een keuze van punten inleid tot een verdeling van..............................................................70
Een verfijning van een verdeling..........................................................................................70
Onbeperkt verfijnen..............................................................................................................70
Riemannsom van f inbij een gegeven verdeling.......................................................................70
De bepaalde integraal van f in...................................................................................................71
3
, Integreerbare functies................................................................................................................71
Bijkomende definities...............................................................................................................72
Optelbaarheids eigenschap van de bepaalde integraal..............................................................72
De lineariteit van de bepaalde integraal....................................................................................72
Bepaalde integralen van een continue functie...............................................................................72
Middelwaarde stelling van de bepaalde integraal.....................................................................72
Eerste hoofdstelling van de integraalberekening......................................................................73
Tweede hoofdstelling van de integraalberekening....................................................................74
Berekenen van een bepaalde integraal......................................................................................74
Opmerkingen bij berekenen......................................................................................................75
Regel van Leibniz.....................................................................................................................76
H15 niet zien.......................................................................................................................................77
H16 oneigenlijke integralen...............................................................................................................77
Het begrip oneigenlijke integraal...................................................................................................77
1ste vorm...................................................................................................................................77
2de vorm...................................................................................................................................77
Probleem aanduiden.......................................................................................................................77
Convergentie en divergentie van een oneigenlijke integraal.........................................................78
Praktisch.........................................................................................................................................79
H2 Vlakke meetkunde Poolcoördinaten.............................................................................................80
Benodigdheden..............................................................................................................................80
Formules:..................................................................................................................................80
Hoe los je goniometrische vergelijking op?...................................................................................80
Hoe stel je poolcoördinaten op......................................................................................................80
Verband met carthesische (x, y) en poolcoördinaten (theta, rho)..................................................81
Poolvergelijking van een kromme.................................................................................................81
Eenvoudige vorm......................................................................................................................81
SPV (Stelsel Parameter Vergelijking) van een poolkromme....................................................81
De periode en de symmetrieën.......................................................................................................82
Besluit.......................................................................................................................................82
Raaklijn aan een poolkromme.......................................................................................................82
H17 toepassingen bepaalde integralen...............................................................................................83
Belangrijke formules......................................................................................................................83
Oppervlakte van een vlakdeel........................................................................................................83
verticaal.....................................................................................................................................83
Horizontaal................................................................................................................................84
Bij SPV.....................................................................................................................................86
Bij poolkromme........................................................................................................................86
Hoe grafische redeneren?.....................................................................................................86
Booglengte van de kromme...........................................................................................................88
Stappen:.....................................................................................................................................88
Carthesisch................................................................................................................................88
Bewijs formule.....................................................................................................................88
SPV..........................................................................................................................................88
Poolkromme..............................................................................................................................89
3 berekenen van de inhoud van een deel van de ruimte................................................................90
Oppervlakte vlakdeel................................................................................................................90
Inhoud (volume vierkant)..........................................................................................................90
Toepassing.................................................................................................................................90
H7 Functies van meer (dan 1) veranderlijken....................................................................................91
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