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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. 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)

  6. 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.

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

  8. Taylor series Binomialexpansion Pascal (binomial) coefficients

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

  10. Sums of infinities Theantiderivativeorindefiniteintegral Integrationhas an unlimitednumber of solutions. Thesearedescribed by theintegrationconstant

  11. Howdoes a population of bacteriachangein time? AssumeEscherichia coli dividesevery 20 min. Whatisthechange per hour? First order recursivefunction Differentialequationscontainthefunction and some of it’sderivatives Differenceequation Any processwherethechangein time isproportional to theactualvaluecan be described by an exponentialfunction. Examples: Radioactivedecay, unboundedpopulation growth, First order chemical reactions , mutations of genes, speciationprocesses, many biologicalchanceprocesses

  12. Allometric growth In many biological systems is growth proportional to actualvalues. A population of Escherichia coli of size 1 000 000growthstwofoldin 20 min. A population of size 1000 growthsequallyfast. Relative growth rate Proportional growth resultsinallometric(powerfunction) relationships.

  13. Theunboundedbacterial growth process How much energy isnecessary to produce a givennumber of bacteria? Energy useisproportional to thetotalamount of bacteriaproducedduringthe growth process Whatisifthe time intervalsgetsmaller and smaller? Gottfried Wilhelm Leibniz (1646-1716) Sir Isaac Newton (1643-1727) Archimedes (c. 287 BC – 212 BC) The Fields medal

  14. f(t) Dt Thearea under thefunction f(x)

  15. f(x) Dt Definiteintegral

  16. f(x) Dt Whatisthearea under the sine curvefrom 0 to 2p?

  17. b Dc Dy a Dx Whatisthelength of thecurvefrom a to b? What is the length of the function y = sin(x) from x = 0to x = 2p?

  18. No simpleanalyticalsolution We use Taylor expansions for numericalcalculations of definiteintegrals. Taylor approximationsaregenerallybetter for smallervalues of x.

  19. y y Whatisthevolume of a rotation body? x x What is the volume of the body generated by the rotation of y = x2 from x = 1 to x = 2 y What is the volume of sphere? x

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