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Cheatsheet and summary lineair algebra
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Cheatsheet and summary lineair algebra conversion class of maths 1.
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Rekenvolgorde: Rekenregels machten: Stappenplan vergelijkingen oplossen: Functions:1. Haakjes Ø a0 ∙ a1 = a021 1. Haakjes wegwerken
2. Machtsverheffen & 2. Vereenvoudigen h=g∘f
3!
Worteltrekken Ø 3" = a041 3. Alle x-en naar 1 kant werken (links) ( )
3. Vermenigvuldigen & Delen 4. Alle getallen naar de andere kant 𝑥 →𝑦→𝑧
0 0 0 0
4. Optellen & Aftrekken Ø (a ∙ b) = a ∙ b = (ab) (recht)
5. Vereenvoudigen VB:
! Ø (a0 )1 = a0∙ 5
x= numerator y= denominator !
6. Oplossen 𝑓: 𝑥 → 𝑥 $ + 3
" "
Øa = √
" a0 𝑔: 𝑥 → sin 𝑥
abc-formule
6 𝑧 = 𝑔(𝑓(𝑥)) = sin(𝑥 $ + 3)
Ø a40 = 3# $%± (
𝑥!,# = #) =
$%± √+ & $,)-
" #)
Ø a0 = (a0 )5 = (a1 )7 = a01 D = 3(b); − 4ac
7 0 7 6 6$
Ø ( )8 = ∙ ∙ … = 𝑝8 ∙ = 𝑝8 ∙ = X=…… V X=……
1 1 5 1$ 1$
6 8 D < 0 à geen oplossing
𝑝 8 +1 , D = 0 à 1 oplossing
D > 0 à 2 oplossingen
1 8 1$
Ø +1 , = 1$
= 𝑞8 𝑞48 = 𝑞8± 8 = 𝑞: = 1
! % % 1 Stappenplan gebroken vergelijkingen
0 1. Werk eventuele haakjes weg.
Ø𝑏 =𝑏" " = 0𝑏 1 = 𝑏
"
2. Werk de breuk weg d.m.v. de
balansmethode
3. Los x op m.b.v. de abc-formule of door
Rekenen met wortels: ontbinden in factoren.
Ø √a ∙ √b = √a ∙ b
*+,-.%+
Ø
√"
='
" Snijpunt berekenen: Stretching: ↕ : f(x) :⎯⎯⎯⎯< a ∙ f(x)
√# # Snijpunt met x-as: door vergelijking gelijk te 𝑟𝑒𝑝𝑙𝑎𝑐𝑒 𝑥
$ stellen aan y=0. ⟷ : f(𝑥) :⎯⎯⎯⎯< 𝑎
Rekenen met logaritmen: Ø √a ∙ √a = (√a) = a Snijpunt met y-as: door vergelijking in te
"#$ % "
Ø g log (a) = !
vullen met x=0. CDEFG
"#$ $ Ø √b = b ! Shifting: 𝑥 ;⎯⎯= 𝑥 − 𝑎
g g g
Ø log (a ∙ b) = log (a) + log HIJ
Hogere machtwortel:
(b) !
√A → derde machtswortel van A
Stappenplan Wortelvergelijkingen: 𝑥 ;= 𝑥 + 𝑎
%
Ø g
log , -= g log (a) – g log (b) ! 1. Beide zijden kwadrateren. ↑: f(𝑥) àf(𝑥) + 𝑎
& √8 = 2 , want 2# = 2 ∙ 2 ∙ 2 = 8
g ' g 2. Sorteren.
log (a )= p ∙ log (a)
"
Ø √−243 = −3, want (−3)$ = −243
3. Indien nodig helen eruit halen Mirroring: f(𝑥) à−f(𝑥)
Ø log wegwerken door 10de
macht, van de rest vd (vb 2√𝑥 à 2 eruit halen). f(𝑥) à−𝑥
vergelijking Ontbinden in factoren: 4. Oplossen.
x ! + (a + b)x + ab = (x + a)(x + b)
Ezelsbruggetje: Analyze functions: Example Analyze functions:
Voorbeeld: Rekenregel: som 1) Calculate intersection with axes (f(0) 𝑥+1
(plus) 𝑓(𝑥) =
10
log (100) = 2 a
log (b) = c and f(x) =0). 𝑥−2
product
102 = 100 ac = b 2) Max & minima: derivative = 0 #1 f(x)= 0 & f(0):
(keer)
à if f’(x)¹ 0 à no max or min.
! ()* ,)* *
√100 = 10 √b = a 𝑓(𝑥) = = 0 𝑓(0) = =
(+! ,+! +!
Rekenregels ontbinden in factoren 3)Inflection point: f’’(x)=0. 𝑥+1=0 y = -½
x ! + (a + b)x + ab = (x + a)(x + b)
x ! + (a + a)x + aa = (x + a)(x + a)
à if f’’(x)¹0 à no inflectionpoint à 𝑥 = −1
Intersect x=-1 & y = ½
Ezelsbruggetje Machten: E ax ! − a(b + c)x + abc = a(x ! − bx + c) turnover graph from incr ⟷ decr.
" % # & = a(x + b)(x + c) #2 f’(x)=0 à no max or min
Ø Omklappen: =&→"=
#
" "
' x ! − a! = (x + a)(x − a) 𝑓 "(𝑥) =
1(𝑥 − 2) − 1(𝑥 + 1) 𝑥 − 2 − 𝑥 − 1
= =
−3
Ø Noemerwissel: # = d → & = b (𝑥 + 2)# (𝑥 + 2)# (𝑥 + 2)#
VB: ≠0
Ø Kruislingsvermenigvuldigen:
" ' 5x2- 4x – 1 = 0
= & → ad = bc #3 f’’(x)= 0 à no inflection point
# (5x +1)(x -1)= 0 à eerst starten met (5x…)(x..). −6
"
x = -1/5 V x =1 𝑓 " (𝑥) = −3(𝑥 + 2)# = −6(𝑥 + 2) = ≠0
4) Draw graph. 𝑥+2
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