What is Calculus all about?

Answer 1

Calculus is the study of how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models.
Calculus is an important branch of mathematics. The word stems from the nascent development of mathematics: the early Greeks used pebbles arranged in patterns to learn arithmetic and geometry, and the Latin word for "pebble" is "calculus." Calculus is built on two major complementary ideas, both of which rely critically on the concept of limits. The first is differential calculus, which is concerned with the instantaneous rate of change of quantities with respect to other quantities, or more precisely, the local behavior of functions. This can be illustrated by the slope of a function's graph. The second is integral calculus, which studies the accumulation of quantities, such as areas under a curve, linear distance travelled, or volume displaced. These two processes act inversely to each other, as shown by the fundamental theorem of calculus.

Differential calculus typically provides a way to derive the acceleration and velocity of a free-falling body at a particular moment while integral calculus problems are used to compute areas and volumes, to find the amount of a liquid pumped by a pump with a set power input but varying conditions of pumping losses and pressure, or to find the amount of parking lot plowed by a snowplow of given power with varying rates of snowfall.

In Europe, fundamental advances in calculus during the 17th and 18th century had a deep impact on the ensuing development of physics. Today, calculus is used in every branch of the physical sciences, in computer science, in statistics, and in engineering; in economics, business, and medicine; and as a general method whenever the goal is an optimal solution to a problem that can be given in mathematical form.

The origins of integral calculus are generally regarded as going back no further than to the time of the ancient Greeks, circa 200 BC. The Hellenic mathematician Eudoxus is generally credited with the method of exhaustion, which made it possible to compute the areas of regions and the volumes of solids. Archimedes developed this idea further, inventing heuristics which resemble integral calculus. After him, the development of calculus did not advance appreciably for over 500 years.[1]

In India, the mathematician-astronomer Aryabhata in 499 used infinitesimals and expressed an astronomical problem in the form of a basic differential equation.[2] Manjula in the 10th century elaborated on this differential equation in a commentary. This equation eventually led Bhaskara in the 12th century to develop a number of ideas that are foundational to the development of modern calculus, perhaps including an early form of the theorem now known as "Rolle's theorem".[3] He was also the first to define the notion of the derivative as a limit. In the 14th century, Madhava, along with other mathematician-astronomers of the Kerala School, studied infinite series, power series, Taylor series, differentiation, integration, and the mean value theorem.[4] Yuktibhasa, which some consider to be the first text on calculus, summarizes these results.[5][6][7][8] These developments would not be duplicated in Europe until much later.

Calculus started making great strides in Europe towards the end of the early modern period and into the first years of the eighteenth century. This was a time of major innovation in Europe, making accessible answers to old questions. Calculus provided a new method in mathematical physics. Several mathematicians contributed to this breakthrough, notably John Wallis and Isaac Barrow. James Gregory proved a special case of the second fundamental theorem of calculus in 1668. In Japan at around this time, Seki Kowa expanded further upon Eudoxus's method of exhaustion.

Leibniz and Newton pulled these ideas together into a coherent whole and they are usually credited with the probably independent and nearly simultaneous "invention" of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today; he often spent days determining appropriate symbols for concepts. The fundamental insight that both Newton and Leibniz had was the fundamental theorem of calculus. Virtually all modern methods of symbolic integration follow from this theorem, and it proved indispensable in the development of modern mathematics and physics. For example, see Integration by parts and Integration by substitution.

When Newton and Leibniz first published their results, there was some controversy over whether Leibniz's work was independent of Newton's. While Newton derived his results years before Leibniz, it was only some time after Leibniz published in 1684 that Newton published. Later, Newton would claim that Leibniz got the idea from Newton's notes on the subject; however examination of the papers of Leibniz and Newton show they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. This controversy between Leibniz and Newton divided English-speaking mathematicians from those in Europe for many years, which slowed the development of mathematical analysis. Today, both Newton and Leibniz are given credit for independently developing calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the science of fluxions". Some others who contributed important ideas are Descartes, Barrow, Fermat, Huygens, and Wallis.

Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. In the 19th century, calculus was put on a much more rigorous footing by Cauchy, Riemann, Weierstrass, and others. It was also during this time period that the ideas of calculus were generalized to Euclidean space and the complex plane. Calculus continues to be further generalized, such as with the development of the Lebesgue integral in 1900.
The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula


for an object moving at constant speed. The speed of a car, as measured by the speedometer, is the derivative of the car's distance traveled as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation.
Visit http://en.wikipedia.org/wiki/Calculus for more!

Answer 2

Calculus is focused around two fundamental concepts. The derivative, and the integral. The derivative can be thought of as exactly how fast a graph is changing at a certain point. The integral is used to find the area under curves, among other things. Those are the basics anyway. I'm just now finishing up Calculus 3 and we are talking about vector fields in space, surface integrals, and other more complicated things.

Answer 3

What distinguishes calculus from all the forerunners is the introduction of infinitesimals, which Newton used to describe change. For example, if I want to describe the slope of a function at some point, I use the expression

dy/dx

where dy is the infinitesimal change in y, and dx is the infinitesimal change in x. Likewise, if I want to compute the area under a function, I use the expression

S y dx

where S is the integral sign, y is the function, and dx is the infinitesimal element of x. It's the summation of all the rectangles of areas y dx.

From this basic concepts, a enormously fecund branch of mathematics has been developed, and as others have already said here, it's really about the mathematics of change.

Answer 4

Oh Boy. Calculus is the study of things that are changing. Example, a rocketship weights 100,000 pounds, including 60,000 pounds of fuel. The engine generates a thrust of 200,000 pounds and burns 1000 pounds of fuel each second. After 30 seconds, how fast is the rocket going and how high up is it? Ordinary arithmatic and algebra cannot solve such problems. The invention of calculus is credited to Isaac Newton who developed it to help him study the orbits of planets.

Answer 5

Its about the rate of change mostly. For example, say you have a balloon and you are filling it with water. In calculus, you get to use formulas that tell you how fast the balloon is getting bigger, and how much water is in the balloon, the surface area, the change in diameter, how much time it takes to fill it up to some size, etc all kinds of things related to that, but not only for balloons all kinds of things that involve a change in something you can measure.

Answer 6

Technically it is about forces rates of change. A great visual concept is to think of a roller coaster in space. The track is the equation, the roller coaster is the 'tangent' force at any point. If the track breaks at point P, then the car would fly off the track at the exact rate of change that it was enduring at the point the track broke. (fly off at a straight line..forever), on earth and in physics, gravity plays a role and brings the car crashing to the ground.. physics suck.

Answer 7

Hehehe. They really don't teach anything in undergraduate math anymore, do they?

Calculus (both differential and integral) is all about the study of limits (either at 0 or infinity) and the convergence of sequences of functions at those limits to finite values (or the divergence of those sequences to undefined values ☺)

As such, it is the first 'introduction' to the branch of mathematics known as 'Analysis'.


Doug

Answer 8

Calculus is the study of continously changing quantities.

Answer 9

from my experience, calculus is the mathematics of change. using calculus you and find the rate at which variables are being manipulated...it is the equations of equations.

Answer 10

I think it's about calculating.