Calculus

As he studied the skies in the late 17th century, Isaac Newton realized he would need more powerful mathematical tools than were then available in order to fully describe the universe around him. By the end of his pondering, he had invented the mathematical field of calculus to serve these ends. What a guy! So, what exactly is calculus? Calculus is comprised of three central mathematical concepts: limits, derivatives, and integrals. Though distinct in what they describe, each builds upon the previous. Calculus also usually includes a study of sequences and series of numbers, as these topics not only prepare students for more rigorous fields of mathematics such as real analysis, but also prove useful for many practical applications.

Limits: Limits describe the behavior of a function as its independent variable gets arbitrarily close to a particular value. While this at first may not seem interesting at first, it is the building block from which derivatives and integrals are formed. What happens as a function gets close to a point? What happens as it tends towards infinity? As it approaches an asymptote? The answers lie in limits.

Derivatives: The derivative of a function describes the instantaneous rate of change of a function. The derivative is itself a function, and when evaluated at a value, gives the instantaneous rate of change of the original function at the same value. More precisely, the derivative is the slope of the line that lies tangent to the function at a given point.

Integrals: The integral of a function can be understood in a few different ways. The first way is that it is an antiderivative - it "undoes" a derivative. The integral of a function is another function whose derivative is the original function. A more common definition is that the integral of a function describes the area under the curve, which is to say, the area between the function and the $x$-axis. While it may seem strange that rates of change and area are tightly coupled, you will find out that this is indeed the case!

Sequences and Series: Sequences are, well, long sequences of numbers described by particular patterns. They're not functions, but they can be described by functions. Series are the sums of sequences. Whether sequences and series converge to particular values, tend towards infinity, or do something else entirely has important implications for many theoretical and real-world applications, including those directly related to the above three concepts.