Samenvatting Introduction to Computational Thinking - Minor BFW

Aantekeningen colleges ICT
Mathematical fundamentals (ma 2-9)
Parameter = constant & make calculations more general, usually considered positive
Operations  + - / *
Opening parentheses:
Introducing parentheses (= factoring):
Parameter equation solving: | Factoring: | Solving quadratic equations:(check afterwards!)
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Multiple variables:  solve how one variable depends on the other  choose simplest option
 Substitution approach ----------------------->
 Elimination approach
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Function = formula describing how one quantity depends on other quantities (y = f(x) -> f(x) = x^2 )
 x = independent variable y = dependent variable
 Functions have a domain (x) = set of all possible input values & range (y) = set of all possible output values
 [ of ] = tot die waarde < of > = bij infinity
Limit (L): L is the limit of the function f(x) when f(x) approaches L in case x approaches a  lim
x→ ∞
f ( x ) =L

 Only defined when approached from either side is the same: lim
x ↑a
f ( x )=lim f ( x )=L
x ↓a
 Limits can be found by filling out x = a in f(x)
 Exceptions: limits at infinity and infinite ‘limits’ (vb. f(x) = b + 1 / x)
g (x)
Limits for rational functions (vb. f ( x )= )
h( x)'
 Divide by highest power in denominator
 Result depends on highest powers in nominator and denominator
 Highest power numerator = denominator  limit = constant
 Highest power numerator < denominator  limit = 0
 Highest power numerator > denominator  no limit (± ∞ )

Local slope (derivative)  differentiation = calculating deriv of a function
- ( f +g )' ( x )=f ' ( x ) + g' ( x )
- ( f −g )' ( x ) =f ' ( x )−g' ( x )
- ( Cf )' ( x )=C f ' ( x )
- Product rule: h ( x )=f ( x ) g ( x ) → h' ( x )=f ' ( x ) g ( x ) + f ( x ) g ' ( x )
- Chain rule: h ( x )=f ( g ( x ) ) =f ∙ g ( x ) →h' ( x )=f ' ( g ( x ) ) g '( x)
f (x ) g ( x ) f ' ( x ) −f ( x ) g ' (x)
- Quotient rule: h ( x )= → h' ( x )= 2
g ( x) (g ( x ))

Shapes of common functions:
o Parabolic (no asymp)
o Cubic (no asymp)
o Square root func
o Hyperbolic func
o Exponential growth(e a) / decay (e−a)
Plan to draw graphs:
I. Intersection points x-axis  solve y = f (x) = 0
II. Intersection points y-axis  fill in x = 0 in y = f (x)
III. lim f ( x )
Horizontal asymptotes  find limit x→ ±∞
p (x)
IV. Vertical asymptotes for rational functions  x values for which q (x) = 0 ?
q( x )
V. X values of maxima / minima  solve f ‘ (x) = 0
VI. Y values at maxima / minima  fill in x value(s) in f (x)
VII. Sketch all possible graphs

, Introduction to modeling (di 3-9)
Model = simplified abstraction of reality  focus on only certain aspects of study object
Types of models: Animal/disease Conceptual/verbal Cartoon Quantitative
Why quantitative models: increased precision/remove uncertainty | predicition (inter- & extrapolation) | possibility to
analyze (simulation, mathematics) | automated analysis | explain ‘complex’ system behaviour based on individual
components | integrative view on data acquired at different levels
Models based on observations cannot be proven correct (only in mathematical statements)
 Model falsification: bewijs waarom model incorrect is
 Model validation: verify predictions by experimentation  increase confidence in model
 Scope of model (beschrijft bepaald deel / specific circumstances)
Mechanistic models describe mechanism underlying observed behaviour  understanding
Descriptive/phenomenological models summarize data  powerful for prediction
Damped oscillations vs. persistent oscillations -------------------------------------------------------->
 Negative feedback can lead to oscillations
Modeling of pathway  include isoforms in model  compare model & experiment  knockout of isoforms  result


Differential equations I (di 3-9)
Cartoon network models:
 Based on verbal description | nodes: molecular species | arrows: molecular interactions (form, degr, regulation)
Mathematical network models:
 Remove uncertainty of model behaviour by becoming quantitive
 Modeling: Ordinary Differential Equations (ODEs)  describe dynamics | arrows = quantitative eaction rates
(State) variables: abundance of modeled molecular species | can vary over time
Parameters: values are fixed over studied time scale | characterizes environmental effects & interactions (vb. degr rate)
 In biology, parameters are positive
Reaction rates: predict changes over time
- Depends on: conc of reactants | environmental conditions (temp, pH)
If rate is known, reactions can be described as Ordinary Differential Equations (ODEs)
ODE assumptions: reaction rates are approximated:
I. Well-mixed environment  rates considered independent of position in space (but: spacial structure in cells)
II. Many molecules are present  continuous rather than discrete (but: some processes rely on only 1e 5 molecs)
Translation from cartoon network to a quantitative description (here: reactions)


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Law of mass-action:  reaction rate is proportional to the product of the concs of the reactants
 k, k1, k2, k3 = rate constants
k 0 A k1 da(t )
 =rate of change of [ A ] =k 0−k 1 a ( t )=rate of production−rate of decay
→ → dt
Cartoon to quantitative description: chemical reaction network  reaction rates  assumptions  ODE


Differential equations II (wo 4-9)
Analysis of ODEs: I. Analytical/symbolic solution II. Numerical simulationIII. Model analysis
Ak da
Analytical: =−ka a ( t )=D e−kt = exponential decay (D = initial conc)
→ dt
Numerical: in silico experi  how does system behave?
 Predict system behaviour over time for given conditions | use numerical simulations in software packages (vb. R)
da(t ) a ( t+ h )−a (t)
Approximation of solution: Euler’s method =f (a (t ) ) f ( a (t)) ≈ a ( t +h ) ≈ a ( t )+ hf ¿ )
dt h