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Welcome to my Wiki on Calculus. My name is Ryan James and I have been teaching AP Calculus for 2 years. The purpose of this wiki is to share more information about Calculus that I don't have time in class to elaborate on.

__What is this //Calculus// you speak of?__

The word 'Calculus' comes from Latin for a small stone used for counting. Although we don't use stones for counting anymore, we can still use a stone to explain some big concepts in Calculus.

Suppose a stone is thrown in the air. If you watched the stone fly through the air, it would go up and then go down (due to gravity) and it would look very parabolic in nature. Side note: A parabola is a type of graph that is typically used in describing bodies in motion and optics. An example of a parabola is shown below:



If you want to know how fast the rock is traveling at any time, you just need to draw a tangent line through that point on the graph. This sounds very easy but is actually a difficult task since each point has a unique tangent line. So, here comes Calculus. With Differential Calculus, you can find the slopes of these tangent lines very easy and this will give you how fast the rock is traveling at that time. Differential Calculus is called this because the process to find these slopes is called 'taking the derivative' or 'differentiation'.

Now that we have a nice looking graph, suppose that we want to find the area of the graph that is above the x-axis. Now, there are a lot of geometry formulas for area that can get us a close estimate to the area, but there are not any that will give us the exact area for this shape. With Integral Calculus, you can find the area under any graph by using integration, or anti-differentiation. Integration is the inverse of differentiation. The big piece of the puzzle that connects these two branches of calculus is the Fundamental Theorem of Calculus:



__History of Calculus:__

So now, for a little history about where Calculus came from.

Most of the credit for discovering Calculus comes from Isaac Newton, but it was Isaac Newton and Gottfried Leibniz who discovered Calculus during the same time in two different countries. Newton once said, "If I have seen anything, it is because I stood on the shoulders of giants." This quote gives insight to the story of Calculus.

The foundational ideas of Calculus start in Greece with Eudoxus (c. 408-355 BC) who developed the method of exhaustion which gives us the idea of very tiny changes. This idea is what later mathematicians base their 'limit' off of. Then after Eudoxus, there was Archimedes who wrote //The Method//, which outlined how to find the area of geometrical shapes that had no specific area formulas for. This treatise was lost until 1906. What happened to the palimpsest that contained //The Method// was that during the 10th century is was copied from Archimedes original work which is now lost, and then in the 12th century a monk tried to clean the palimpsest and was reused as a Christian text. In 1906, Johan Heiburg discovered that this was palimpsest was actually the mathematical work of Archimedes. The important of this palimpsest was that Bonaventura Cavalieri used many of the same methods that Archimedes had shown in //The Method// in his own work. If //The Method// had not been lost, the methods that Archimedes had outlined would not have had to been rediscovered by Cavalieri in 1635. Following Cavalieri, the work of Pierre de Fermat, John Wallis, Isaac Barrow, and James Gregory all proving certain theorems in Calculus such as the second part of the Fundamental Theorem of Calculus in 1675.

Right about this time in history, the Calculus wars are about to start. In 1687 Newton published __Principia Mathematica__, which is considered as one of the first Physics textbooks. Newton claimed that all of the Physics was explained by his method of fluxions (differential calculus) that he had written in 1671 in his __Method of Fluxions__, which was not published until 1736. In 1684 was when Leibniz published __Nova Methodus Pro Maximis et Minimis__ (translates to 'New Methods for Maximums and Minimums')//.// There was no official record of who discovered Calculus first only the publication dates. Since Newton was the head of the Royal Society in England, he used that position to accuse Leibniz of plagiarism. Leibniz never was able to clear his name of this while he was alive. It is accepted now that Newton and Leibniz both independently discovered Calculus.

Calculus was still not finished though. Mathematicians were not convinced that Calculus was bulletproof and sought to make it so. Due to the work of Bolzano, Cauchy, Weiestrass, and Riemann in the early 1800s, Calculus was made rigorous and a new branch of mathematics were discovered, Real Analysis. What we see today in Calculus is the work of all these mathematicians working over 2000 years perfecting the subject.

media type="youtube" key="X9t-u87df3o" height="349" width="560" Professor Gilbert Strang from MIT talking about Calculus

__Limits__

Limits are truly the foundation of Calculus. Without limits there would not be derivatives, integrals, or infinite series. All of these are based on the notion of a limit. The basic idea of a limit is: what is your //y//-value approaching as your //x// approaches a certain number. The notation for limits looks like:



The mathematical statement above reads: the limit of //f(x)// as //x// approaches //a// is equal to //L//. So the //y//-value that the limit is approaching is //L.// The //x//-value that the limit is approaching is //a//. In a limit problem, you are given the left side of the equation and asked to find //L//. So far this has been very general. This is how Calculus was first explained. Then, when Bolzano and Weierstrass brought some rigor into Calculus, they could algebraically prove that these limits work. Here is a proof that I have written to show how rigorous Calculus can get:



media type="youtube" key="W0VWO4asgmk" height="349" width="425" An Introduction to Limits by Khan Academy which can be found on iTunes U.

media type="youtube" key="ryLdyDrBfvI" height="349" width="425"

An Introduction to Limits by Professor David Jerison at MIT.