You Gotta Know These Ideas from Calculus
Calculus is a subfield of mathematics that concerns continuous changes. Although there are several precursors, including Archimedes’ use of the method of exhaustion to derive the area under a parabola, modern calculus is generally considered to have begun with the work of Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Although they carried out their investigations independently, Newton’s claims that Leibniz had copied his unpublished notes helped stoke a long-running controversy over the origins of calculus that has become one of the most famous examples of a scientific priority dispute.
- A limit is a value that a function approaches as the input approaches another value, even if the function is not equal to that output value (or even defined there). For example, the limit of 1/x as x gets infinity large (“approaches infinity”) is 0, even though the function never actually reaches 0. The definition of a limit is formalized in a way traditionally written with the Greek letters epsilon and delta, a technique developed by Augustin-Louis Cauchy.
- Continuity is informally defined as the ability of a function’s graph to be drawn without lifting the pen. Properly, a function is continuous at a given point if the function has a limit at that point and is equal to its limit there. It is also common to speak of “continuous functions,” which are functions that are continuous at every point. Polynomials are continuous, as are exponential and logarithmic functions, sine, and cosine. However, the other four trigonometric functions (cosecant, secant, tangent, and cotangent) are not continuous at some points (the ones at which their graphs have asymptotes), and rational functions generally have some points of discontinuity.
- The derivative is an operation that takes a function and results in another function that gives the original function’s rate of change. Taking a derivative is called differentiation and is carried out with respect to the input variable that is changing. The derivative is calculated as the limit of a tangent line between two points on the function’s graph as the two points get arbitrarily close together. At any given point, the derivative’s value is equal to the slope of the tangent line to the function’s graph at that point. The derivative of a function f(x) is notated as f′(x) (Lagrange’s notation, pronounced “f prime of x”) or df/dx (Leibniz notation, pronounced “d f d x). Differentiation can be repeated, e.g,. differentiating a function twice gives its second derivative (f′′(x) or d²f/dx²).
- A function is called differentiable if its derivative can be evaluated. Like for continuity, this can be stated about a specific point or the whole function. The absolute value function is an example of a function that is differentiable in some places but not others: it is differentiable everywhere except at an input of 0, where its graph has a sharp corner. Differentiability is a stronger property than continuity: a function can only be differentiable if it is continuous, but it could be only continuous and not differentiable (the absolute value function being an example of that). If a function’s derivative can be taken infinitely many times, it is called smooth.
- Definite integration is an operation that can be interpreted as giving the area under a curve, or more precisely, the signed area between the curve and the x-axis (signed area meaning that area below the axis counts as negative). An integral is denoted by a “long S” symbol, ∫. Definite integration is performed between two endpoints (e.g., the integral between 0 and 5), and those endpoints are written above and below, or next to, the integral symbol.
- Riemann sums are a way to formalize the definition of an integral. A Riemann sum approximates the area under a curve by partitioning the area into narrow rectangles that extend between the x-axis and the function’s graph, finding the areas of the rectangles, and adding those up. For well-behaved functions, the limit can be taken as the rectangles get infinitely thin, giving the definite integral.
- The fundamental theorem of calculus roughly states that integration and differentiation are opposite operations. This means that most basic rules for integration can be found by reversing the basic rules for differentiation given above. The formal statement of the theorem is usually given in two parts. The first part states that integrating a function gives an antiderivative of the function (i.e., a function whose derivative is the original function). The second part states that the definite integral of f(x) from a to b can be calculated as F(b) − F(a), where F is an antiderivative of f.
- The chain rule is used to find the derivative of the composition of two functions, such as sin(x²). It states that the derivative of f(g(x)) is f′(g(x)) · g′(x). In Leibniz notation, it is written as df/dx = df/dg dg/dx and thus resembles canceling two fractions. Integrating both sides of this result gives a corresponding rule for integration by substitution.
- The product rule is used to find the derivative of the product of two functions, such as x sin x. It states that the derivative of f(x) g(x) is f′(x) g(x) + f(x) g′(x). Integrating both sides of this result gives a corresponding rule for integrating the product of two functions, called integration by parts. It can be combined with the chain rule to find the quotient rule (the way to differentiate a function written as one function divided by another).
- Taylor series is a way of approximating a differentiable function using an infinite sum of monomials (which can be truncated to get a polynomial). Coefficients of Taylor series are found using nth derivatives and the factorial function. Taylor series can be based around any point of a function, and when the chosen point is x = 0, it is called a Maclaurin series. Taylor’s theorem gives a bound on the error resulting from truncating the Taylor series to a finite number of terms; the error can be expressed as a remainder term in the Lagrange form, Cauchy form, or integral form.
- Differential equations are equations that relate a function to its derivatives, or even multiple derivatives (e.g., a function to its first and second derivatives). They are widely used throughout the sciences to model behavior like radioactive decay, population growth, predator-prey relations, waves, fluid flow, and more.
This article was contributed by NAQT editor Joseph Krol.