Samenvatting Introduction to Computational Thinking - Minor BFW
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Course
Introduction to Computational Thinking 2019
Institution
Universiteit Leiden (UL)
Samenvatting voor het tentamen van de colleges van het minor vak 'Introduction to Computational Thinking'. Deze samenvatting is in het Engels. Tip: oefen ook veel. Oefententamen was representatief voor het tentamen.
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
------------------------------------------->
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)
------------------>
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
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