Inhoudsopgave
Basic math: (chapter 1-3) .............................................................................................................. 3
Basic 1: Logic ..................................................................................................................................3
Properties of power......................................................................................................................3
Rules of algebra ...........................................................................................................................4
Fractions .....................................................................................................................................5
Summations ...............................................................................................................................8
Square roots ............................................................................................................................. 10
Basic 2: Finding critical points: .....................................................................................................11
Solving equations ...................................................................................................................... 11
Solving inequalities .................................................................................................................... 11
Factorization!!!!! ............................................................................................................................ 14
2 equations -> 2 unknowns ............................................................................................................. 15
1. Functions of 1 variable (H4) ..................................................................................................16
Find the domain. ............................................................................................................................ 17
Find the range. (R=Range) ............................................................................................................... 20
Graph of a function ........................................................................................................................ 22
Working with polynomials .............................................................................................................. 24
Cubic function:(EXAM) ............................................................................................................... 24
Rational function: ...................................................................................................................... 26
Factor for polynomials. .............................................................................................................. 27
Polynomials division!!! OPGAVE WEEKLY ASSIGNMENT boven EN MIDTERM onder! ....................... 28
Master the EXP() function !!!, Master the LN Function!!!! (EXAM) ........................................................ 30
Inverse functions! .......................................................................................................................... 33
Know how the standard graphs look like! ......................................................................................... 37
2. Differentiation of a Function F(X) ..........................................................................................38
Definition (EXAM) ........................................................................................................................... 38
Learn from looking at the first interval. ............................................................................................. 40
First derivative & second derivative: (EXAM) ..................................................................................... 42
Quotient rule: ............................................................................................................................ 44
Product rule: F(X)=F(X) X G(X) -> F’X=F’(X) X G(X) + F(X) X G’(X)...................................................... 45
Chain rule: F(x)-F(G(X))-> F’(x)=F’(GX(x)) X G’(X) ........................................................................... 45
Convex or concave? (EXAM) ....................................................................................................... 47
Rules for differentiation .................................................................................................................. 50
EXP(x) and LN (x) ............................................................................................................................ 53
Chain rule ..................................................................................................................................... 54
Limits! .........................................................................................................................................58
Limits at infinity: EXAMEN !!!!!! ........................................................................................................ 59
3. Optimization .......................................................................................................................66
pag. 1
, Single variable optimization: ........................................................................................................... 69
Def min/max .................................................................................................................................. 70
Ness FOC (First order condition) ..................................................................................................... 71
First derivative test (EXAM) ............................................................................................................. 71
Exterme value theorem (EXAM) ....................................................................................................... 74
Definition (EXAM) ....................................................................................................................... 75
Extrema for Concave/Convex functions ........................................................................................... 75
Local extrema ................................................................................................................................ 76
Concave or convex functions .......................................................................................................... 78
Inflection points (EXAM) ................................................................................................................. 81
4. Functions with more variables .............................................................................................82
Find the Domain ............................................................................................................................ 82
Partial Derivatives .......................................................................................................................... 83
Convexity/concavity ....................................................................................................................... 84
Young’s Theorem ........................................................................................................................... 84
Chain rule in 2nd dimension (EXAM) ................................................................................................. 84
Nec. FOC ...................................................................................................................................... 95
Sufficient condition!!!!! ................................................................................................................... 95
Extreme Value Theorem (EXAM) ...................................................................................................... 96
Saddle points (and other) ............................................................................................................... 97
5. Matrices (EXAM) ..................................................................................................................98
Matrix Addition:............................................................................................................................ 103
Matrix multiplication: ................................................................................................................... 104
Identity matrice !!! ........................................................................................................................ 107
Transparant matrice ..................................................................................................................... 108
Inverse matrix -> when ................................................................................................................. 110
2x2 case ...................................................................................................................................... 112
pag. 2
,Basic math: (chapter 1-3)
Basic 1: Logic
➔ Nes condition and sufficient condition
Overige regels:
Properties of power
- To the power of 0 -> Equals 1: -> 1/1=2
- 2x to the power 4: -> 2 to the power 4 and x to the power 4.
-
-
-
-
-
pag. 3
, Rules of algebra
Simpele rekenregels:
Voorbeelden:
Show that : f(x)=F(-x) -> LETTERLIJK doen wat er gevraagd wordt: voorbeeld van miderm:
Show that G(-x) = -g(x) -> G(-x) = -g(x) (formule was 3x^3 – 1/5x^5
- Voor de -g(x) -> zet een – voor de formule
- Voor de g(-x) -> zet een – voor de x en.
- Is nu gelijk.
What does this mean geometrically?
- What does this mean on the graps (= the same so (a,b) and (-a,-b) are on the
graph.
- Just try some point and give answer
Just an easy warm up question.
- 5+3=8, but 4+4 is also 8
- 4x4=16, but -4*-4 is also 16
- Something times 0 or – turns the equation negative so true, other way round not cause x
could be 3, so the y does not make a difference.
- True and True, cause: -2 X -2 X -2 = -8, so x needs to be 2 both ways round.
30 pm vast, 0.16 per minute, Cost=30+0.16x. plug in 102 and 126 on the ends, you
will get both answers.
pag. 4
Basic math: (chapter 1-3) .............................................................................................................. 3
Basic 1: Logic ..................................................................................................................................3
Properties of power......................................................................................................................3
Rules of algebra ...........................................................................................................................4
Fractions .....................................................................................................................................5
Summations ...............................................................................................................................8
Square roots ............................................................................................................................. 10
Basic 2: Finding critical points: .....................................................................................................11
Solving equations ...................................................................................................................... 11
Solving inequalities .................................................................................................................... 11
Factorization!!!!! ............................................................................................................................ 14
2 equations -> 2 unknowns ............................................................................................................. 15
1. Functions of 1 variable (H4) ..................................................................................................16
Find the domain. ............................................................................................................................ 17
Find the range. (R=Range) ............................................................................................................... 20
Graph of a function ........................................................................................................................ 22
Working with polynomials .............................................................................................................. 24
Cubic function:(EXAM) ............................................................................................................... 24
Rational function: ...................................................................................................................... 26
Factor for polynomials. .............................................................................................................. 27
Polynomials division!!! OPGAVE WEEKLY ASSIGNMENT boven EN MIDTERM onder! ....................... 28
Master the EXP() function !!!, Master the LN Function!!!! (EXAM) ........................................................ 30
Inverse functions! .......................................................................................................................... 33
Know how the standard graphs look like! ......................................................................................... 37
2. Differentiation of a Function F(X) ..........................................................................................38
Definition (EXAM) ........................................................................................................................... 38
Learn from looking at the first interval. ............................................................................................. 40
First derivative & second derivative: (EXAM) ..................................................................................... 42
Quotient rule: ............................................................................................................................ 44
Product rule: F(X)=F(X) X G(X) -> F’X=F’(X) X G(X) + F(X) X G’(X)...................................................... 45
Chain rule: F(x)-F(G(X))-> F’(x)=F’(GX(x)) X G’(X) ........................................................................... 45
Convex or concave? (EXAM) ....................................................................................................... 47
Rules for differentiation .................................................................................................................. 50
EXP(x) and LN (x) ............................................................................................................................ 53
Chain rule ..................................................................................................................................... 54
Limits! .........................................................................................................................................58
Limits at infinity: EXAMEN !!!!!! ........................................................................................................ 59
3. Optimization .......................................................................................................................66
pag. 1
, Single variable optimization: ........................................................................................................... 69
Def min/max .................................................................................................................................. 70
Ness FOC (First order condition) ..................................................................................................... 71
First derivative test (EXAM) ............................................................................................................. 71
Exterme value theorem (EXAM) ....................................................................................................... 74
Definition (EXAM) ....................................................................................................................... 75
Extrema for Concave/Convex functions ........................................................................................... 75
Local extrema ................................................................................................................................ 76
Concave or convex functions .......................................................................................................... 78
Inflection points (EXAM) ................................................................................................................. 81
4. Functions with more variables .............................................................................................82
Find the Domain ............................................................................................................................ 82
Partial Derivatives .......................................................................................................................... 83
Convexity/concavity ....................................................................................................................... 84
Young’s Theorem ........................................................................................................................... 84
Chain rule in 2nd dimension (EXAM) ................................................................................................. 84
Nec. FOC ...................................................................................................................................... 95
Sufficient condition!!!!! ................................................................................................................... 95
Extreme Value Theorem (EXAM) ...................................................................................................... 96
Saddle points (and other) ............................................................................................................... 97
5. Matrices (EXAM) ..................................................................................................................98
Matrix Addition:............................................................................................................................ 103
Matrix multiplication: ................................................................................................................... 104
Identity matrice !!! ........................................................................................................................ 107
Transparant matrice ..................................................................................................................... 108
Inverse matrix -> when ................................................................................................................. 110
2x2 case ...................................................................................................................................... 112
pag. 2
,Basic math: (chapter 1-3)
Basic 1: Logic
➔ Nes condition and sufficient condition
Overige regels:
Properties of power
- To the power of 0 -> Equals 1: -> 1/1=2
- 2x to the power 4: -> 2 to the power 4 and x to the power 4.
-
-
-
-
-
pag. 3
, Rules of algebra
Simpele rekenregels:
Voorbeelden:
Show that : f(x)=F(-x) -> LETTERLIJK doen wat er gevraagd wordt: voorbeeld van miderm:
Show that G(-x) = -g(x) -> G(-x) = -g(x) (formule was 3x^3 – 1/5x^5
- Voor de -g(x) -> zet een – voor de formule
- Voor de g(-x) -> zet een – voor de x en.
- Is nu gelijk.
What does this mean geometrically?
- What does this mean on the graps (= the same so (a,b) and (-a,-b) are on the
graph.
- Just try some point and give answer
Just an easy warm up question.
- 5+3=8, but 4+4 is also 8
- 4x4=16, but -4*-4 is also 16
- Something times 0 or – turns the equation negative so true, other way round not cause x
could be 3, so the y does not make a difference.
- True and True, cause: -2 X -2 X -2 = -8, so x needs to be 2 both ways round.
30 pm vast, 0.16 per minute, Cost=30+0.16x. plug in 102 and 126 on the ends, you
will get both answers.
pag. 4
