# Write a system of inequalities for the graph of the function

Now we are going to take a look at function notation and how it is used in Algebra. The typical notation for a function is f x. Pre-calculus integration[ edit ] The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus ca.

## Functions & Graphing Calculator - Symbolab

This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was independently developed in China around the 3rd century AD by Liu Huiwho used it to find the area of the circle.

This method was later used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere Shea ; Katzpp.

The next significant advances in integral calculus did not begin to appear until the 17th century. Further steps were made in the early 17th century by Barrow and Torricelliwho provided the first hints of a connection between integration and differentiation.

Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers. Newton and Leibniz[ edit ] The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz.

The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals.

In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed.

Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern calculuswhose notation for integrals is drawn directly from the work of Leibniz.

Formalization[ edit ] While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them " ghosts of departed quantities ". Calculus acquired a firmer footing with the development of limits.

Integration was first rigorously formalized, using limits, by Riemann. Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis —to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integralfounded in measure theory a subfield of real analysis. Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system.

Historical notation[ edit ] Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with. The modern notation for the indefinite integral was introduced by Gottfried Wilhelm Leibniz in Burtonp.Improve your math knowledge with free questions in "Write inequalities from graphs" and thousands of other math skills.

Section Logarithm Functions.