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The derivative

Lecture 5 Handling a changing world. The derivative. The derivative. y 2. y 2 -y 1. y 2 -y 1. y 1. x 2 -x 1. x 2 -x 1. x 1. x 2. The derivative describes the change in the slope of functions. The first Indian satellite. Bhaskara II (1114-1185). Aryabhata (476-550). b.

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The derivative

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  1. Lecture5 Handling a changingworld Thederivative Thederivative y2 y2-y1 y2-y1 y1 x2-x1 x2-x1 x1 x2 Thederivativedescribesthechangeintheslope of functions The first Indian satellite Bhaskara II (1114-1185) Aryabhata (476-550)

  2. b Local minimum u Stationary point, point of equilibrium Meanvaluetheorem Fourbasicrules to calculatederivatives

  3. Dy=30-10 Dy=0 Dx=15-5 Thederivative of a linearfunctiony=axequalsitsslope a Thederivative of a constanty=bisalways zero. A constantdoesn’tchange.

  4. Theimportance of e dy dx

  5. Theapproximation of a smallincrease How much largeris a ball of 100 cm radius if we extendits radius to 105 cm? ThetruevalueisDV = 0.66m3.

  6. Rule of l’Hospital Thevalue of a functionat a point x can be approximated by its tangent at x.

  7. Stationarypoints Maximum Minimum How to find minima and maxima of functions? f(x) f’=0 f’(x) f’<0 f’<0 f’>0 f’=0 f’’(x)

  8. Populations of bacteriacansometimes be modelled by a general trigonometricfunction: b a a: amplitude b: wavelength; 1/b: frequency c: shift on x-axis d: shift on y-axis d c Thetime seriesof population growth of a bacteriumismodelled by Atwhattimes t doesthispopulationhavemaximumsizes?

  9. Maximum and minimum change Positivesense f’=0 f’=0 Negativesense 4/3 Point of maximumchange Point of inflection Atthe point of inflectionthe first derivativehas a maximumor minimum. To findthe point of inflectionthesecondderivativehas to be zero.

  10. The most important growth processisthelogistic growth (PearlVerhulst model) The growth of Saccharomycescerevisiae(Carlson 1913) K Logistic growth Thefunctionissymmetricaroundthe point of fastest growth.

  11. The most important growth processisthelogistic growth (PearlVerhulst model) Theprocessconverges to an upper limit defined by thecarryingcapacity K K/2 t0 Differentialequation Maximumpopulationsizeisat Saccharomycescerevisiae Thepopulationgrowthsfastestat

  12. Thechange of populationsin time TheNicholson – Baileyapproachto fluctuations of animalpopulationsin time First order recursivefunction A simpledeterministicprocess (function) isable to generate a quasi random (pseudochaotic) pattern. K=2. 0 a=1.2 b=3.9 K=0.95 a=0.05 b=2.0 K=3. 0 a=3.0 b=6.0 Hence, seeminglycomplicatedfluctuations of populationsin time might be driven by verysimpleecologicalprocesses K=1. 5 a=0.01 b=0.5

  13. Recursivefunctions of nth order First order recursivefunction Exponential model of population growth Howfastdoes a populationincreasethatisdescribed by thisfunction? Thereis no maximum. Populationincreaseisfaster and faster.

  14. Nicholson – Baileyapproach Differentialequation Differenceequation Wherearethemaxima of thisfunction? The global maximum of thefunction K=1. 5 a=0.01 b=0.5

  15. Seriesexpansions Geometricseries We try to expand a functioninto an arithmeticseries. We needthecoefficientsai. McLaurinseries

  16. Taylor series Binomialexpansion Pascal (binomial) coefficients

  17. Seriesexpansionsareused to numericallycomputeotherwiseintractablefunctions. Fast convergence In the natural sciences and mathsanglesarealwaysgiveninradians! Taylor seriesexpansion of logarithms Very slow convergence

  18. Home work and literature • Refresh: • Arithmetic, geometricseries • Limits of functions • Sums of series • Asymptotes • Derivative • Taylor series • Maxima and Minima • Stationarypoints • Prepare to thenextlecture: • Logistic growth • Lotka Volterra model • Sums of series • Asymptotes Literature: Mathe-online Logistic growth: http://en.wikipedia.org/wiki/Logistic_function http://www.otherwise.com/population/logistic.html

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