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

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)

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

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

Sums of infinities The antiderivative or indefinite integral Integration has an unlimited number of solutions. These are described by the integration constant

Assume Escherichia coli divides every 20 min. What is the change per hour? How does a population of bacteria change in time? First order recursive function Difference equation Differential equations contain the function and some of it’s derivatives Any process where the change in time is proportional to the actual value can be described by an exponential function. Examples: Radioactive decay,unbounded population growth, First order chemical reactions, mutations of genes, speciation processes, many biological chance processes

Allometric growth In many biological systems is growth proportional to actual values. A population of Escherichia coli of size growths twofold in 20 min. A population of size 1000 growths equally fast. Proportional growth results in allometric (power function) relationships. Relative growth rate

The unbounded bacterial growth process How much energy is necessary to produce a given number of bacteria? Energy use is proportional to the total amount of bacteria produced during the growth process What is if the time intervals get smaller and smaller? Gottfried Wilhelm Leibniz ( ) Archimedes (c. 287 BC – 212 BC) Sir Isaac Newton ( ) The Fields medal

tt f(t) The area under the function f(x)

tt f(x) Definite integral

tt f(x) What is the area under the sine curve from 0 to 2  ?

a b What is the length of the curve from a to b? What is the length of the function y = sin(x) from x = 0 to x = 2  ? cc xx yy

No simple analytical solution We use Taylor expansions for numerical calculations of definite integrals. Taylor approximations are generally better for smaller values of x.

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