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Samenvatting Wiskunde I(A) (F000135A)

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Samenvatting Wiskunde I(A) door Prof. dr. J. Van der Jeugt H1: Inleidende begrippen H2: Functies: algemene begrippen en voorbeelden H3: Limieten en continuïteit H4: Afgeleiden en differentiaalrekening ()

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  • January 7, 2021
  • 63
  • 2020/2021
  • Summary
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,Verzamelingen
-




=
geheel van
objecten ,
elementen


✓ opsomming k =
{ a. e. i. 0 ,
u }
-
Beschrijving v. =
G klinken van
alfabet }


* Eindige verzameling V
Cardioliterte Nl of # : U =




Vve H1 =
5
,
1 Al =
26


→ Element van : X E K ( ←
× ¢ k )

Gelijkheid :
U = W a. s . a × € U ⇐ × EW



Deelverzameling : w E V ⇐ Vx C- W :
x E V


[ Minstens 1 element niet : we v


→ Verschil v -
w : Vrw = { x 1 x E V en x ¢ w
}

Ure { a. G. c. d. e } Ik =
{ b. C ,
d }

Doorsnede : V NW =
{ x 1 X EU en × En W }

Vis { a. b. c. d
, e) NK =
{ a ,
e }

Unie { }
of
→ W
:
V v w =
X E V × €




Vvd { a. b
,
C
,
die 3 v K e
{ dier , Cid ,
e
,
i. 0 ,
u
}
→ Carters product : v × W =
{ v.v 1 er € V en w ⇐ W
}
In deelverzamelingen :




Vr X Vz X Vs × . . × Vn =
{ ( vr , vz ers , . . .
. om ) 1 Vi C- Vi ,
I E ft . ? ..
.
))



*
e-
Oneindige verzamelingen :




Getallen
-




Natuurlijke getallen : IN =
{ 0 . 1. 2,3 ,
. .
_


}

Gehele getallen :
Z =
{ . . .


,
-
2
,
-
1
.
0
,
1
,
2 . 3 ,
e- .


}
→ Rationale getallen :
④ =
onvermoeibare breuken
'

with : U =
VI ( o )
-




h =
{ × C- U 1 × zo }
V _
= { x E V 1 X E 03

,Etc
:
geldig voor m
C- en →
Bewijs door inductie




Ë = n t 2 t 3 + . . .
e-
~ =
min
1- 2

:
LL =
1 RL =
-2 = 1 Ok




②n=→ :
Ë i =
Ntn ( Inductiehypothese)

Na 1



③m-N I I = u ← 2 t 3 t . . - 1- N t (Nts )
in
-




=


Ntn + (Nts )


Ndt 1) t 2. ( Nts )
=
-




Z




=
(Nts ) ( Nt 2 )
-

2


D


Reële getallen
-




% De getallen x
stelling van
plantageras
⇒ TE =/ ④ →
Bewijs uit het
ongerijmde

, Rationale getallen
Irrationale getallen
} ⇒ Reële getallen IR

Vvd 52 =
1,144 . . . 1T =
3 14159 . . .



,




→ Teken van X E IR



zien × = t 1 als × > 0




riem × = -

en als × en 0




signs × =
0 als × = 0




Absolute waarde

van ×


1X 1 = X als × zo → 1×1 = x. zien ×



1×1 = -

X als X E 0




Intervallen in R
-



*
Eindige intervallen [ a. er] : [ a.
er [ i Ja .
er [ .
. . .




*
Oneindige intervallen [ •
,
te [ ; J - *
,
a ]




f)-
Begrenzing

is de V
Er bovengrens t

van ⇐ x ← u : x ← er


Er is de V 78
benedengrens ⇐ t

van x E V
:
x




Stellig : Als een
verzameling V ER naar leven begrensd is ,




kleinste
dan heeft de
verzameling van de bovengrenzen een




element .




kleinste
Supreme =
bovengrens

Inferieur grootste ondergrens

, '




N Ja er [ V Er v
eng
:
we *
sap
= = =
a
,




{ In }
*
* V
v = C-
en :
sap
= 2
inz v = 1




( v =
{ 2. 7,7 .
. . . 3
'




2- ⇒ ~
ondergrens

d
en
= t



is kleinste
x e
ondergrens
( is
1,001 ) geen ondergrens


* v =
{ 1
,
f. 7,7 .
. . .


} soep V = 1 inf v = 0

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