Practical applications of functions Page 1 History has

Practical applications of functions Page 1

History has never seen two adversaries to fighting with greater determination and tenacity than Sir Isaac Newton and Gottfried Wilhelm von Leibniz Page 2

Newton Researched the graphics of moving objects and Introduced the terms Derivative of a function, angular factor, mechanical sense of the derivatives. Newton invented differential calculus, which he called the method of FLUX (infinitely Small increments) in 1665 - 1666 Page 3

Newton's kept this discovery for himself, leaving only written evidence at the Time to prove that he knew about It, without explaining publically how it worked. Looking for a universal language for all people he introduced many symbols in mathematics. Until his last breath Newton searched in the bible for a mathematical proof of the existence of God. Page 4

Leibniz introduces the indications for derivatives, he called Differential and describes methods for finding their mark, he also introduced a symbol for an integral and describes how to find the Area of a figure surrounded by graph of a function, Leibniz differential calculus developed fully ten years later before he had a chance to see the works of Newton in the Year 1676 … Page 5

The Englishman and German Argued over the creation of higher mathematics - their revolutionary method for determining the area under the curve. This major mathematical breakthrough is the backbone of the modern era. The world owes its knowledge to Leibniz, whose publication in 1684 describes the method of differential calculus. Furthermore, the system of Leibniz, in a sense clearer compared to Newton. Page 6

Derivative Let the function y = f (x) be defined in the definitional point x 0 of their field. The increase means the argument is described like this Δx = x-x 0, and the growth of the function like this Δy = f (x)-f (x 0). If there is a limit it is called a derivative of the function f (x) at the point x 0. http: //en. wikipedia. org/wiki/Derivative Page 7

The graph of the function (in black) and the tangential (in red). Difference of private tangent is equal to the derivative at a point. Page 8

Leibniz Indication Newton Indication Lagrange Indication Page 9

Area of a figure This formula is referred to as Newton-Leibniz formula. The geometrical meaning of the definite integral is the area of the figure between the two vertical line: x = a and x = b, is the abscissa and the graph of y = f (x). If the graph is above the abscissa axis, that are must be reported with a positive sign and if it’s under the abscissa axis – with a negative sign. Page 10

Example 1 The equation shows that the curved face of the triangle is a third, surrounded by the function y = x 2, Abscissa and the vertical line x = 1 Page 11

Example 2 The figure is a section of the graph of y = 1 / x, abscissa axis vertical making x = 1 and x = 2. The area of the colored figure is the natural logarithm of 2. Page 12

Leibniz quotes The present always hides in its debts the future. People despise the weakness and misery more than the bad habit. Libraries - these are the treasuries and all the riches of the human spirit. Charity - it is universally good implementation of which the wise, made in accordance with the needs of sound sense, derive the greatest amount of benefit. We create what we imagine. Proven example can Never be considered completely proven. He who is not one with himself is never truly free. Humans – they are all demigods! Page 13

Newton Quotes If I saw farther away than the others this is because I was sitting on the shoulders of giants – I look upon myself like I was child playing on the sea shore that has found rounded stones and more colorful shells while in front of me stood the infinite ocean of the truth. A genius is nothing more than perseverance in thinking and concentration all focused on a single goal. Page 14

Newton Quotes http: //rosi-velikimisli. blogspot. com/2010/06/blog-post_6269. html Leibniz Quotes http: //www. crossroadbg. com/gotfrid_vilhelm_fon_laibnic. html Areas of figures http: //stancho. roncho. net/High. Math/Dfnt. Integral/Intro. html Derivatives of functions http: //bg. wikipedia. org/wiki/%D 0%9 F%D 1%80%D 0%BE%D 0%B 8%D 0 %B 7%D 0%B 2%D 0%BE%D 0%B 4%D 0%BD%D 0%B 0. Newton and Leibniz http: //www. argumenti. net/? p=228 Page 15

Created by: Polina Staiikova Nadejda Gencheva Ivelina Georgieva SOU Jeleznik Martin Stoikov Page 16