Calculus

A detailed definition of calculus and its two main branches, differentiation and integration, along with specific examples of careers using advanced mathematics.

High school students fear it. College students dread it. It builds houses and stadiums, designs roller coasters, performs medical research, animates graphics, and determines mortgage payments. It's calculus.
What is Calculus?Calculus is Latin for stone, and the ancient Romans used stones for counting and arithmetic. In its most basic sense, calculus is just that - a form of counting. After advanced algebra and geometry, it is the next step in higher mathematics, and is used for solving complex problems that regular mathematics cannot complete.
Calculus was developed by two different men in the seventeenth century.Gottfried Wilhelm Leibniz (1646-1716), a self-taught German mathematician, and Isaac Newton (1642-1727), an English scientist, both developed calculus in the 1680s. While Leibniz invented it ten years later than Newton, he published his findings twenty years earlier, and that overlap led to decades of controversy about which man reached his conclusions first. Today it is generally agreed that both mendeveloped calculus independently.
Calculus is the mathematics of change, of calculating problems that are continually evolving. This is possible by breaking such problems into infinitesimal steps, solving each of those steps, and adding all the results. Rather than doing each step individually, calculus allows these computations to be done simultaneously.
Two Branches of Calculus
There are twoprimary branches of calculus: differential calculus and integral calculus. Differential calculus, or differentiation, is used primarily to determine the slope or steepness of a curve, also called a curve's derivative. Slope is a rate of change in a curve - a very steep curve is changing very fast - and calculus is used when a curve is very complicated, such as calculating the slope of a mountain orthe speed of a roller coaster.
Differential calculus involves any problem that may be graphed when the desired result is a single point on that graph. For example, if a rancher wants to construct a corral with a limited amount of fence, he can vary the lengths of the sides of the corral. Using calculus, he could determine which lengths would enclose the greatest area and make the largest corral. Agraph could be drawn using every possible combination of lengths, and the highest point on that graph - the maximum - would signify the greatest area.
Integral calculus, or integration, deals with areas and volumes of complex figures, such as determining the greatest amount of space or volume beneath a dome in a stadium design in order to incorporate as many seats as possible. To find the areabeneath a curve, integration breaks the area beneath the curve into minute pieces, determines the area of each piece, and adds them all together, or integrates them, into a final answer.
Another example of integration would be determining the exact volume of water necessary to fill a wading pool in a zoo exhibit. The pool may have varying depths and irregular edges, and it would be extremelydifficult to determine how much water is needed without being able to use the infinite sums that integration calculates.
What Is Calculus Used For?
Both differential and integral calculus are useful to a wide variety of careers. The examples below are only a fraction of the ways professionals regularly use calculus. Credit card companies use calculus to set the minimum payments due on credit cardstatements at the exact time the statement is processed by considering multiple variables such as changing interest rates and a fluctuating available balance.
Biologists use differential calculus to determine the exact rate of growth in a bacterial culture when different variables such as temperature and food source are changed. This research can help increase the rate of growth of necessary...