The derivative Lecture 5 Handling a changing world x 2 -x 1 y 2 -y 1 The derivative x 2 -x 1 y 2 -y 1 x1x1 x2x2 y1y1 y2y2 The derivative describes the change in the slope of functions Aryabhata ( ) Bhaskara II ( ) The first Indian satellite

u Four basic rules to calculate derivatives b Local minimum Stationary point, point of equilibrium Mean value theorem

 y=30-10  x=15-5 The derivative of a linear function y=ax equals its slope a  y=0 The derivative of a constant y=b is always zero. A constant doesn’t change.

dy dx The importance of e

The approximation of a small increase How much larger is a ball of 100 cm radius if we extend its radius to 105 cm? The true value is  V = 0.66m 3.

Rule of l’Hospital The value of a function at a point x can be approximated by its tangent at x.

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

Populations of bacteria can sometimes be modelled by a general trigonometric function: a: amplitude b: wavelength; 1/b: frequency c: shift on x-axis d: shift on y-axis a b d c The time series of population growth of a bacterium is modelled by At what times t does this population have maximum sizes?

Maximum and minimum change Point of maximum change Point of inflection f’=0 Positive sense Negative sense At the point of inflection the first derivative has a maximum or minimum. To find the point of inflection the second derivative has to be zero. 4/3

The most important growth process is the logistic growth (Pearl Verhulst model) tWeight The growth of Saccharomyces cerevisiae (Carlson 1913) Logistic growth K The function is symmetric around the point of fastest growth.

The most important growth process is the logistic growth (Pearl Verhulst model) The process converges to an upper limit defined by the carrying capacity K The population growths fastest at K/2 Saccharomyces cerevisiae Maximum population size is at Differential equation t0t0

The change of populations in time The Nicholson – Bailey approach to fluctuations of animal populations in time First order recursive function K=0.95 a=0.05 b=2.0 K=1. 5 a=0.01 b=0.5 K=2. 0 a=1.2 b=3.9 K=3. 0 a=3.0 b=6.0 A simple deterministic process (function) is able to generate a quasi random (pseudochaotic) pattern. Hence, seemingly complicated fluctuations of populations in time might be driven by very simple ecological processes

Recursive functions of n th order First order recursive function How fast does a population increase that is described by this function? There is no maximum. Population increase is faster and faster. Exponential model of population growth

Nicholson – Bailey approach Difference equation Differential equation Where are the maxima of this function? The global maximum of the function K=1. 5 a=0.01 b=0.5

Series expansions Geometric series We try to expand a function into an arithmetic series. We need the coefficients a i. McLaurin series

Taylor series Binomial expansion Pascal (binomial) coefficients

Series expansions are used to numerically compute otherwise intractable functions. Fast convergence Taylor series expansion of logarithms In the natural sciences and maths angles are always given in radians! Very slow convergence

Home work and literature Refresh: Arithmetic, geometric series Limits of functions Sums of series Asymptotes Derivative Taylor series Maxima and Minima Stationary points Prepare to the next lecture: Logistic growth Lotka Volterra model Sums of series Asymptotes Literature: Mathe-online Logistic growth: function n/logistic.html