Mathematics – Differential Calculus
Differential calculus, founded in part by Sir Isaac Newton and Gottfried Wilhelm Leibniz, describes the behavior of a function in infinitely small parts.
The information available about, for instance, a hike up a mountain to a certain height above sea level is usually of interest to a hiker, but perhaps how steep the mountain is and how difficult the climb will be is of the most importance. Differential calculus focuses on this aspect: the steep- ness-or slope-of graphs of functions.
Slope, secant, difference quotient
The term slope describes the steepness of a straight line; a curved line has a constantly changing slope. A straight line that cuts the graph of a function f(x) at two points (x, and x
2) is called the secant, and the difference quotient (..) is used to approximate the slope of a specific section of the curve.
Boundary value, tangent, derivative To precisely measure the slope of a curve at a particular section, the secant should cut the curve to produce an infinitesimal section. Such an infinitesimal straight line is called the tangent (from the Latin tangere “to touch”) and appears to touch the “back” of the graph of a function. The slope of the tangent line is
the “limit” (from the Latin limus for “border”) of the slope of the secant and is expressed mathematically as://m ^=^=f(x)
This differntial calculation gives the slope of the curve at this point, called the derivative of the graph at point x and written as f'(x) or df/dx.
Differential calculus finds application in real processes by using the derivative to specify the rate of a process at different points. In business economics, marginal costs are determined by taking the derivative of the total costs as a function of the quantity produced. Companies are then able to determine how far they should be able to lower the cost of a product through infinitely small steps before profits are affected.
DEVELOPMENT OF CALCULUS
Calculus was independently developed by Gottfried Wilhelm Leibniz and Isaac Newton in the late 17th century. Leibniz based his theory on the use of geometric processes to solve mathematical problems. He viewed a curve as being made up of infinitely small segments, whereby the slope of the tangent could be calculated for each segment. Similarly, a curved surface could be seen as the sum of an infinite number of tiny rectangles Leibniz thus recognized the relationship between differential and integral calculus Newton, on the other hand, was more interested in solving a physics problem: how to determine the instantaneous speed of an accelerating object.
He viewed a curve as a reflection of constant acceleration and imagined a point as an infinitely small segment of a line. The time interval between observations of an object’s motion can be reduced to the point that the change in speed disappears. Thus, acceleration or deceleration can be calculated as the sum of the Instantaneous speeds of the observed object.
Leibniz was later accused of stealing Newton’s ideas from correspondence exchanged by the two, and the Royal Society of London, influenced by Newton, erroneously pronounced him guilty. However, Leibniz’ system became the dominant form of calculus, thanks to its elegant notation and simplicity of calculation.