The Basics of Calculus

Calculus is the study of change.
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Calculus has been around since ancient times and, in its simplest form, is used for counting. Its importance in the world of mathematics is in filling the void of solving complex problems when more simple math cannot provide the answer. What many people do not realize is that calculus is taught because it is used in everyday life outside of high school and college classrooms. From designing a building to calculating loan payments, calculus surrounds us.


Two 17th century men, Gottfried Wilhelm Liebniz and Sir Isaac Newton are often credited with working to develop calculus principles. However, due to discrepancies on which man developed conclusions first, it has been deemed that the two worked independently of each other on the subject. Other claims regarding the origins of this type of mathematics include the Greeks working on the main ideas that form the basis for calculus as far back as 450 BC.


Calculus consists of two main branches called differential and integral calculus. Differential calculus deals with derivatives and their applications. Integral calculus implies a form of mathematics that identifies volumes, areas and solutions to equations. Differential calculus is a study of functions and the rate of change within functions when variables are altered. Integral calculus concentrates on determining mathematical answers such as total size or value.


A major feature of differential calculus is the use of graphs. Any problem in which the answer is defined as one point on a graph is where differential calculus is involved. It usually identifies the steepness of a curve, commonly known as the slope. In real world applications, the steepness of a curve could be represented by things such as a hill or bridge. Integral calculus takes the next step by working to solve questions such as “how much water would it take to fill a pool?” Numbers and variables are “integrated” into a more complex equation or formula in order to arrive at the final answer.


Calculus has numerous real-world applications. When there is a more complex problem to solve or it involves unusual shapes or sizes, calculus becomes the tool to arrive at the solution. For example, if there is an unusual roof to be built such as the roofs that extend over sports stadiums, the designers will utilize calculus tools to plan for the size and strength of the structure. For any professional trying to determine work, area, volume, gradient, or surface area, calculus will provide the answer.


In differential calculus, measuring the rate of change at any given point on a curve is called the derivative. Often, it is described as measuring the slope of a line in equations. Let’s say the line is straight on a graph, with the graph having an X and Y coordinate. The slope (m) is defined as the difference in Y divided by the difference in X. Here is the differential calculus equation: (Y2-Y1) Slope = m = (X2-X1) Integral calculus involves calculating areas. When calculating an area, this process of “integration” results in a formula known as the integral. Some will refer to the integral as the anti-derivative found in differential calculus. Below is a simple form of integral calculus: For a function of the form k * xn , the integral equals k * x(n+1) (n+1) These formulas, while simple and basic, provide rudimentary examples for introducing the wide and expansive mathematical world known as calculus.

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