Building Intuition for the Derivative How do you wish the derivative was explained to you? The derivative is the heart of calculus, buried inside this definition: But what does it mean?
This is a sub-article to Calculus and History of mathematics. Contents Development of calculus Integral calculus Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Moscow papyrus c.
Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea.
At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create. Antiphon and later Eudoxus are generally credited with implementing the method of exhaustionwhich made it possible to compute the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes.
Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. It should not be thought that infinitesimals were put on rigorous footing during this time, however.
Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. In the third century Liu Hui wrote his Nine Chapters and also Haidao suanjing Sea Island Mathematical Manualwhich dealt with using the Pythagorean theorem already stated in the Nine Chaptersknown in China as the Gougu theorem, to measure the size of things.
He discovered the usage of Cavalieri's principle to find an accurate formula for the volume of a cylinder, showing a grasp of elementary concepts associated with the differential and integral calculus. In the 11th century, the Chinese polymathShen Kuodeveloped 'packing' equations that dealt with integration.
Indian mathematicians produced a number of works with some ideas of calculus. The formula for the sum of the cubes was first written by Aryabhata circa AD, in order to find the volume of a cubewhich was an important step in the development of integral calculus.
While solving this problem, he was the first mathematician to derive the formula for the sum of the fourth powersusing a method that is readily generalizable for determining the general formula for the sum of any integral powers.
He performed an integration in order to find the volume of a paraboloidand was able to generalize his result for the integrals of polynomials up to the fourth degree.
He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree. At around the same time, there was also a great deal of work being done by Japanese mathematiciansparticularly Kowa Seki.
While these methods of finding areas were made largely obsolete by the development of the fundamental theorems by Newton and Leibniz, they still show that a sophisticated knowledge of mathematics existed in 17th century Japan.
Differential calculus The Greek mathematician Archimedes was the first to find the tangent to a curve, other than a circle, in a method akin to differential calculus. While studying the spiral, he separated a point's motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together thereby finding the tangent to the curve.
To determine this, he finds a maximum value for the function. However, his work was never followed up on in either Europe or the Islamic world. In particular, in Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed a method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation.
The mean value theorem in its modern form was stated by Augustin Louis Cauchy also after the founding of modern calculus. Mathematical analysis Main article: Mathematical analysis Greek mathematicians such as Eudoxus and Archimedes made informal use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids.
In the 15th century, a German cardinal named Nicholas of Cusa argued that rules made for finite quantities lose their validity when applied to infinite ones, thus putting to rest Zeno's paradoxes.AP Calculus AB Course Overview of a graphing calculator in AP Calculus AB is considered an integral part of the course.
The Big Ideas of AP Calculus derivative, estimate derivatives from tables and graphs, and apply various derivative rules and properties. Students should. In basic calculus, we learn rules and formulas for differentiation, which is the method by which we calculate the derivative of a function, and integration, which is the process by which we.
Calculus is the branch of mathematics studying the rate of change of quantities and the length, area and volume of objects. With the ability to answer questions from single and multivariable calculus, Wolfram|Alpha is a great tool for computing limits, derivatives and integrals and their applications, including tangent lines, extrema, arc length and much more.
Introduction and Summary The derivative is the first of the two main tools of calculus (the second being the integral).The derivative is the instantaneous rate of change of a function at a point in its domain.
Summary of foundations of integration foundations of integral calculus as the mechanics of nding antiderivatives and evaluating de nite integrals. The fundamental theorem of calculus (ftc). the derivative F0of the integral F is the original function f.
This theorem ftc 1 can be thought of as the inverse to ftc when we just take the. A BRIEF SUMMARY OF CALCULUS calculus: the derivative and the deﬂnite integral. These concepts are deﬂned as the solutions derivative and the deﬂnite integral in a very broad way.
After studying this chapter, you should be .