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Discipline- Op tijd opstaan- ‘Combineer’ discipline, geen geheelonthouding- Gewoon doen!
Plannen- Stel haalbare doelen- Schets mogelijke scenario’s- Leg duidelijke prioriteiten wanneer dat nodig is- Plan in dagdelen (‘s morgens, ‘s middags, ‘s
avonds)- Plan resultaatgericht
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Makkelijk punten scoren- Prioriteit bij projecten
- Beter 2 zessen dan 3 vijven
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Verder- Thuis of UB, wat werkt voor jou het beste?
- Regelmaat, afleiding (toko eten etc.)
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Studeerkunde Analyse Tentamen
Analyse 1
Calculus Differentiation Integration
- Trigonometry
- Logarithms
- Complexe Numbers
- Vectors
- Limits
- Differentials
- Productrule
- Chain Rule
- Impliciet Differentiëren
- Differential Equations
- Integrals
- Substitution
- Integration by Parts
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Calculus
Appendix D: TrigonometryAppendix H: Complex numbers
H12: Vectors and the geometry of spaceH2: Limits and derivatives
APPENDIX DTrigonometry
Calculus Differentiëren Integreren
b
a
c
sin( )
cos( )
tan( )
b
ca
cb
a
What exactly is a cosine or sine?
APPENDIX DTrigonometry
Calculus Differentiëren Integreren
APPENDIX DTrigonometry
Calculus Differentiëren Integreren
1
1
2
45o
60o
30o
2
1
3
180o rad
APPENDIX HComplex numbers
Calculus Differentiëren Integreren
2 1i
Complex numbers are ‘imaginary’, but very useful in engineering situations. Especially Euler’s formula.
( ) cos( ) sin( )ixe x i x
CHAPTER 12Vectors and the geometry of space
Calculus Differentiëren Integreren
A vector is a point in space, and can be used to visualize a mathematical problem.
CHAPTER 12Vectors and the geometry of space
Calculus Differentiëren Integreren
Important formulas concerning vectors
Length of a vector
Angle between two vectors
Volume determined by three vectors
CHAPTER 12Vectors and the geometry of space
Calculus Differentiëren Integreren
Parametric equations of a line
Parametric equations of a function
Differentiëren
H3, H4, H9
Differentials
Calculus Differentiëren Integreren
1
01
( )'n n
cdxxdxx nx
cos( )' sin( )sin( )' cos( )
x xe dx ex xx x
1( )'
( ( ))' ( ( ))'
( ) ( ) ' ( )' ( )'
n nx nx
c f x c f x
f x g x f x g x
Power Rule
Constant Multiple Rule
Sum Rule
‘Core Analysis Business’, very important for engineering purposes. Lot of different notations.
Product- & Quotiëntregel
Calculus Differentiëren Integreren
Chain Rule
Calculus Differentiëren Integreren
If g is differentiable at x and f is differentiable at g(x), then the composite function F= f o g defined by F(x) = f(g(x)) is differentiable at x and F’ is given by the product:
Implicit Differentiation
Calculus Differentiëren Integreren
Occurs when functions are defined implicitly by a relation between x and y such as:
2 2
3 3
256
x yx y xy
For example, differentiate with respect to x,2 2 25x y
Implicit Differentiation
Calculus Differentiëren Integreren
!!!
Because y is a function of x, apply chain rule:
Integration
H5, H7
Integrals
Calculus Differentiëren Integreren
11
1n n ncx dx c x dx c x
n
( ) ( ) ( ) ( )bb
a a
f x dx F x F b F a
( ) ( ) ( ) ( )f x g x dx f x dx g x dx
1ln( )
x xe dx e
x dxx
The Fundamental Theorem of Calculus states that if:
( ) '( ) ( )f x dx means F x f x
Integrals
Calculus Differentiëren Integreren
There are two important techniques for integrals:
- Integration by parts
- Substitution Rule
( ( )) '( ) ( )f g x g x dx f u du
( ) '( ) ( ) ( ) ( ) '( )f x g x dx f x g x g x f x dx
Mind the Chain Rule!
Tentamen
WTB & MT, Januari 2008
Studeerkunde Analyse Tentamen
Studeerkunde Analyse Tentamen
Studeerkunde Analyse Tentamen
Studeerkunde Analyse Tentamen
Studeerkunde Analyse Tentamen
Studeerkunde Analyse Tentamen
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