You use geometry for manipulating shapes. You use algebra for doing math with unknown numbers. You use trigonometry for studying triangles, and by extension, waves. So what's calculus for?

Calculus is the study of change.

When you differentiate something, you get the derivative, which represents the rate at which something increases or decreases.

Integration is combining derivatives, or minute changes, into a whole result.

Using these, you can handle changing values in anything: acceleration, electromagnetism, money... calculus can handle them all.

Useful, eh?

A set of ideas and operations used to manipulate functions. The two most basic manipulations are called derivatives and integrations. A ten-year-old child can imitate calculus using successive difference methods (though the ten-year-old probably wouldn't call it that). See also difference of powers. Successive differences are easiest to use for polynomials with integer-only coefficients (e.g. x1, x2, 3x5, etc.). For f(x)=x2 (x squared), the first n+1 values for f(m) where m=0,1,2,...,n-1,n are as follows:


To use successive differences, subtract any value from the value after it:


This operation can be repeated as often as desired. Each successive difference set is a pseudo-derivative of the set prior to it. In this example, the next successive difference would be a series of 2's. Note that increasingly complex functions are increasingly difficult to get nice results from. Also note that whenever a successive difference set has values which are all the same (other than zero), they will very likely be related to the factorial of the order of the original function. This leads to some interesting factorial theory . . .

Seeing as this is a node about calculus, some mention should be made of the fundamental theorem of calculus (two parts):

Let f be a continuously differentiable function on a finite interval [a,b]. Then,

  / b
  \  f'(x)dx = f(b)-f(a).
 / a

Here, f'(x) is the derivative of f with respect to x. The second half is similar:

Let f(x) be a continuous function on a finite interval [a,b] and define F(x) in the following manner:

        / x
F(x)=   \  f(t)dt.
       / a

Then F is continuously differentiable on [a,b], and F'(x)=f(x).

On the surface, these may seem to be obvious, but they must be meticulously proven at some point or other before mathematicians trust them.

In addition to being a useful form of math, it is also a calcified deposit that forms on teeth. Supragingival calculus forms above the gingivae (above the gumline), Subgingival calculus forms beneath the crest of the gingivae (below the gumline). You should brush and floss your teeth to prevent this.

This is a review from my collection of math book reviews.

Calculus by Michael Spivak

This is a book everyone should read. If you don't know calculus and have the time, read it and do all the exercises. Parts 1 and 2 are where I finally learned what a limit was, after three years of "explanations" from bad calculus books. The whole thing is the most coherently envisioned and explained treatment of one-variable calculus I've seen; Spivak's plan and the nature of the insight he's trying to impart are evident throughout.

The book has flaws, of course. The exercises get a little monotonous because Spivak has a few tricks he likes to use repeatedly to construct them. There is perhaps too little material on applications, but this can be found in other books (try Apostol's Calculus, or Differential and integral calculus by Courant if you're brave). Also, Spivak sometimes avoids sophistication at the expense of clarity, as in the proofs of Three Hard Theorems in chapter 8 (where a lot of epsilon-pushing takes the place of the words "compact" and "connected"). Nevertheless, this is the best calculus book overall, and I've seen it do a wonderful job of brain rectification on many people.

Addendum from Pete Clark, one of my co-reviewers: Yes, it's good, although perhaps more of the affection comes from more advanced students who flip back through it? Most of my exposure to this book comes from tutoring and grading for 161 [the University of Chicago honors first-year calculus class], but I seriously believe that working as many problems as possible (it must be acknowledged that many of them are difficult for first year students, and a few of them are really hard!) is invaluable for developing the mathematical maturity and epsilonic technique that no math major should be without.

Calculus in the present sense was developed in the 17th century concurrently by Leibniz and Newton. However, there were many foreshadowings of Calculus long before, such as Zeno's Paradoxes. Especially of note are Archimedes discovery of how to find the tangent to a spiral and the discovery of power series in present day Kerala, India by the astronomer Mahadeva. Many discoveries that lead up to Calculus had been made earlier in the millenium and the science of Calculus was later refined by others. One of the most notable of these later mathematicians was Weierstrass, who formulated the Epsilon-Delta definition of limits that is taught in Calculus classes today. The science of Calculus made possible several things that were impossible before. For example, Kepler's laws of motion, which took him years to derive, became less than a day's work with the advent of Calculus. Modern-day students of Calculus are assigned the project of deriving Kepler's laws. Calculus has advanced much farther in it's four centuries of existence than it's creators could have imagined. Today it holds an important place in the study of Mathematics and Physics.

During the seventeenth century, European mathematicians were at work on four major problems. These four problems gave birth to the subject of Calculus. The problems were the tangent line problem, the velocity and acceleration problem, the minimum and maximum problem, and the area problem. Each of these four problems involves the idea of limits.

The tangent line problem

There is a given function (f), and a point (P} on its graph. The idea of this problem is to find the equation of the tangent line to the graph at that point. This problem is equivalent to finding the slope of the tangent line at that point. This may be approximated by using a line through the point of tangency and a second point on the curve (Q)—this gives us a secant line.

As point Q approaches point P, the secant line will become a better and better approximation of the tangent line. This uses the concept of limits—the limit as Q approaches P will give you the slope of the tangent line. In other words, choosing points closer and closer to the point of tangency would give you more accurate approximations. The derivative of a function gives us the slope of the tangent line to the function.

Although partial solutions to this problem were given by Pierre de Fermat (1601-1665), Rene Descartes (1596-1650), Christian Huygens (1629-1695), and Isaac Barrow (1630-1677), credit for the first general solution is usually given to Sir Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716).

The velocity and acceleration problem

The velocity and acceleration of a particle can be found by using Calculus. This was one of the problems faced by mathematicians in the seventeenth century. The derivative of a function can not only be used to determine slopes, but also to determine the rate of change between two variables. This may be used to describe the motion of an object moving in a straight line. This is the position function, which, if differentiated (or the derivative of it is found) gives us the velocity function. In other words, the velocity function is the derivative of the position function. You may also find the acceleration function by finding the derivative of the velocity function. So the velocity and acceleration problem helped in the development of Calculus.

The minimum and maximum problem

What if we want to examine a function by finding where it is increasing? Where it is decreasing? What is the behavior of its concavity? When does it have a maximum point? Where does it have a minimum point? All of these questions were answered with the development of Calculus. The minimum or maximum of the function must occur at a critical point, or a critical number. If we find the derivative of a function, its zeros are called critical numbers.

Now, we must analyze the behavior of the function. The values over which the derivative is positive equates into the actual function increasing. When the derivative is negative, the function is decreasing. If the function is increasing, and then changes to decreasing, that point is a relative maximum of the function. Similarly, if the function is decreasing, and then changes to increasing, that point is a relative minimum. An easier way to analyze the minimum and maximum problem is to graph the derivative. If the point to the left of the critical number is a negative, and the point to the right of it is a positive, then the critical number is a minimum of the function. Similarly, if the point to the left of the critical point is a positive, and the point to the right is a negative, the point is a maximum of the function.

We may also analyze concavity. If the second derivative of the function is positive over a given interval, then the function is concave up over that given interval. If the second derivative is negative, then the function is concave down.

The area problem

This classic Calculus problem is used to find the area of a plane region that is bounded by the graphs of functions. Like the tangent line problem, the limit concept is applied here. To approximate the area of the plane region underneath the graph, one may break the region up into several rectangles, and sum up the values of the rectangles. This method is a form of the Riemann Sums. This would give an approximation of the area of the graph. Now, if the amount of rectangles is increased, the approximation will become more and more precise. The area will therefore be the sum of the areas of the rectangles as the number of rectangles increases without bound. In other words, the limit as the number of rectangles approaches infinity, will give you the area of the region. This eventually leads into the idea of integration.

The method of limits which includes derivatives and integrals is also known as analysis. Judging from Webster 1913's definition of calculus which is much broader, it seems that analysis is a more accurate term for this method.

In modern literature, 'calculus' is often used as a synonym for analysis, but apparently only for the more mainstream branches of real and vector variables. For instance complex analysis and tensor analysis retain the original term.

Webster 1913's broader definition is substantiated by the etymology of the word calculus. Originally its only meaning was stone or pebble, which when used in kinds of abacus gained a connection with mathematics.

Cal"cu*lus (?), n.; pl. Calculi (#) [L, calculus. See Calculate, and Calcule.]

1. Med.

Any solid concretion, formed in any part of the body, but most frequent in the organs that act as reservoirs, and in the passages connected with them; as, biliary calculi; urinary calculi, etc.

2. Math.

A method of computation; any process of reasoning by the use of symbols; any branch of mathematics that may involve calculation.

Barycentric calculus, a method of treating geometry by defining a point as the center of gravity of certain other points to which coefficients or weights are ascribed. -- Calculus of functions, that branch of mathematics which treats of the forms of functions that shall satisfy given conditions. -- Calculus of operations, that branch of mathematical logic that treats of all operations that satisfy given conditions. -- Calculus of probabilities, the science that treats of the computation of the probabilities of events, or the application of numbers to chance. -- Calculus of variations, a branch of mathematics in which the laws of dependence which bind the variable quantities together are themselves subject to change. -- Differential calculus, a method of investigating mathematical questions by using the ratio of certain indefinitely small quantities called differentials. The problems are primarily of this form: to find how the change in some variable quantity alters at each instant the value of a quantity dependent upon it. -- Exponential calculus, that part of algebra which treats of exponents. -- Imaginary calculus, a method of investigating the relations of real or imaginary quantities by the use of the imaginary symbols and quantities of algebra. -- Integral calculus, a method which in the reverse of the differential, the primary object of which is to learn from the known ratio of the indefinitely small changes of two or more magnitudes, the relation of the magnitudes themselves, or, in other words, from having the differential of an algebraic expression to find the expression itself.

© Webster 1913.

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