# Calculus

Calculus has been contributed to by many individuals throughout centuries going all the way back to Archimedes, and in the 17th century, there was “remarkable activity in which researchers throughout Europe contributed novel solutions and competed with each other to arrive at important new methods” [13]. During this time, there were a few main problems that needed to be solved. These were calculating areas, volumes, and tangent lines to curves, and it was from the methods developed to solve these problems that much of calculus grew [13]. Two men in particular stand out during this time period, Isaac Newton from England and Gottfried Wilhelm Leibniz from Germany. As far as we know, these men came to their groundbreaking conclusions on calculus independently of one another. Although there is much debate as to which of them should be given credit for “creating” calculus, one thing is clear. Calculus would not be what it is today or have had such a historical impact if it were not for the hard work of these brilliant men. Newton introduced what is now known as infinite series, and he also confirmed the inverse relationship between area and tangents by his use of infinitesimals [13]. Additionally, “the operations of differentiation and integration emerged in his work as analytic processes that could be applied generally to investigate curves” [13]. Leibniz explored the sums and differences of sequences, both finite and infinite, and found the infinite series that equals [13]. He also devised some of the mathematical notation that we still use today, such as for integrals, and he introduced the differential used in finding derivatives and integrals [13]. Both Newton and Leibniz contributed much more than what has been mentioned above, and calculus would not be what it is without them. But it is also important to remember that many people over many years contributed to calculus. Newton and Leibniz did not work in a vacuum but were influenced by those before them and inspired many people after them to make calculus into what it is today.

Calculus helped solve some of the oldest mathematical problems in history. These were the problems of finding tangent lines, velocity, acceleration, maximums, minimums, and area [15]. But the impact of calculus does not stop there. Amazingly, the world would be a very different place if it were not for calculus. Calculus can be seen in many branches of math as well as in engineering, biology, physics, economics, medicine and so much more [12]. Calculus is very useful when it comes to explaining how things change related to time. In fact, Albert Einstein would not have been able to introduce his theory of relativity without calculus since his theory describes how time and space change relative to each other [12]. Calculus can also be seen in areas having to do with motion, such as when dealing with electricity or vehicle development in engineering [12]. It is also used in economics when making economic predictions and is used for calculating trends in data, such as in birth and death rates [12]. The real-world applications of calculus just keep going on and on.

In this paper, we will begin by discussing limits and continuity. We will learn how to evaluate limits analytically and how to determine whether a function is continuous. We will then examine derivatives which find their foundation in limits. We will discuss some rules of differentiation and take an in-depth look at the chain rule and implicit differentiation. We will then look at integrals which are to derivatives what multiplication is to division. We will examine integration rules and focus in on integration by parts and trigonometric substitution. Then we will introduce more applications of calculus through derivatives and integrals and work through a couple of real-world problems demonstrating more techniques of differentiation and integration. We will briefly discuss multivariate calculus and how to differentiate using partial differentiation. Then we will take a break from these topics to look at sequences and series and some theorems of how to determine whether they converge or diverge. Then we will wrap up this calculus section by taking an introductory look at differential equations. So, without further loss of time, let’s begin our journey into the beautiful world of calculus.