Calculus Basics: Limits, Functions, Derivatives, Integrals, and Series

Calculus is the study of change which focuses on limits, functions, derivaties, integrals, and infinite series. There are two main branches of calculus: differential calculus and integral calculus, which are connected by the fundamental theorem of calculus. It was discovered by two different men in the seventeenth century. Gottfried Wilhelm Leibniz – a self taught German mathematician – and Isaac Newton – an English scientist – both developed calculus in the 1680s.

Calculus is used in a wide variety of careers, from credit card companies to a physicist use calculus in their work.

In general, it is a form of mathematics which was developed from algebra and geometry. Integration and differentiation are an important concept in mathematics, and are the two main operations in calculus. Differential calculus is a subfield of calculus which concentrates over the study of how functions change when their inputs are changed.

The main focus in a differential calculus is the derivative which can be thought of as how much one quantity is changing in response to changes in some other quantity.

The process to find the derivative is called differentiation, the fundamental theorem of calculus states that the differentiation is the reverse process to integration. Derivatives are mainly applied in physics as it concerns with the way quantities change and evolve over time. The principles of integral were developed through the fundamental theorem if calculus individually by Newton and Leibniz.

What Professions Use Calculus

According to Bernhard Riemann, integral is based on a limiting procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs.

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During the 19th century more notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performs has been generalized. There are many modern concepts of integration but the most common one is based on the abstract mathematical theory known as Lebesgue integration, developed by Henri Lebesgue.

Fundamental theorem of calculus is the statement that differentiation and integration are inverse operations. The history of calculus goes back to ancient time, where the ideas of integral calculus are introduces but not the concept. The basic function of integral calculus was used by the Egyptian Moscow papyrus (1820 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum. The Greeks used the method of exhaustion, which prefigures the concept of the limit to calculate areas and volumes.

During the medieval time period, the Islamic mathematician was the first to derive the formula for the sum of the fourth powers of an arithmetic progression. In the modern time period, there was a debate on who developed calculus first between Newton and Leibniz. It was said that Newton was first to discover calculus but he did not open the idea to the world. He used the method of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth and many more problems.

He developed series expansions for functions, including fractional and irrational powers. He did not publish all these discoveries because at that time infinitesimal methods were still considered disreputable. When Gottfried Leibniz expresses the same concept of calculus he was originally accused of plagiarism by Newton. But now he is regarded as an independent inventor of and contributor to calculus. Newton was the first to apply calculus to general physics whereas Leibniz developed much of the notation used in calculus today. The two branches of calculus are used in many professions.

Credit card companies use calculus to set the minimum payments due on credit card statements at the exact time the statement is processed by considering multiple variables such as changing interest rates and a fluctuating available balance. Biologists use differential calculus to determine the exact rate of growth in a bacterial culture when different variables such as temperature and food source are changed. An electrical engineer uses integration to determine the exact length of power cable needed to connect two substations that are miles apart.

Because the cable is hung from poles, it is constantly curving. An architect will use integration to determine the amount of materials necessary to construct a curved dome over a new sports arena, as well as calculate the weight of that dome and determine the type of support structure required. Space flight engineers frequently use calculus when planning lengthy missions. To launch an exploratory probe, they must consider the different orbiting velocities of the Earth and the planet the probe is targeted for, as well as other gravitational influences like the sun and the moon.

A physicist uses calculus to find the centre of mass of a sports utility vehicle to design appropriate safety features that must adhere to federal specifications on different road surfaces and at different speeds. An operations research analyst will use calculus when observing different processes at a manufacturing corporation. By considering the value of different variables, they can help a company improve operative efficiency, increase production, and raise profits.

A graphics artist uses calculus to determine how different three-dimensional models will behave when subjected to rapidly changing conditions, which creates a realistic environment for movies or video games. Hence the use of calculus in the society is very common and important to many careers. Calculus, the rate of change, is divided into two main concepts – integration and differentiation calculus. Both of these concepts are related to the fundamental theorem of calculus, making the two inverses of each other. Today, both Newton and Leibniz are given credit for developing calculus independently and giving a range for what it is about.

There many uses of calculus in day to day life of a human’s profession, it is found deeply integrated in every branch of the physical sciences, such as physics and biology. Calculus has been one of the greatest inventions of modern society where it has so much to offer.