Derivatives and Definite Integral

Derivatives and Definite Integrals

Word Count (excluding title and works cited page): 628

Calculus pioneers of the seventeenth century such as Leibniz, Newton, Barrow, Fermat, Pascal, Cavelieri, and Wallis sought to find solutions to puzzling mathematical problems. Specifically, they expressed the functions for derivatives and definite integrals. Their areas of interest involved discussions on tangents, velocity and acceleration, maximums and minimums, and area. This introductory paper shall briefly introduce four specific questions related to these problems and the solutions that were sought.

In calculus, how a function changes in response to input is measured using a derivative. The derivative of a function is the result of mathematical differentiation. It measures the instantaneous rate of change of one certain quantity in relationship to another and is expressed as df (x)/dx. It can be interpreted geometrically as the slope of the curve of a mathematical function f (x) plotted as a function of x. The integral of a function can be geometrically interpreted as the area under the curve of the mathematical function f (x) plotted as a function of x (Nave).

How does the tangent line evolve from the secant line and how does the derivative relate to the tangent line? A primary problem in calculus is to discover the slope of a curve. To do this, mathematicians use tangent and secant lines. Segments drawn in tangent to or through a curved line create angles which help define and measure the slope of a curve. The slope of a curve is found at a point. The method is to draw a secant line through that point and to first find the slope of the secant line. The formula is ?y/?x. The next step is to slide the second point along the curve toward the first. The secant approaches the limit and, as the two points meet, it becomes a tangent line. The slope of the tangent becomes the slope of the curve and is called the derivative. It is written dy/dx .

How are derivatives used to solve velocity and acceleration problems? Another common use of derivatives is the ability to define concepts relating to velocity and acceleration. This is done with a "time derivative" which defines the rate of change over time. The instantaneous velocity of an object is calculated by the coordinate derivative relative to time. To know how quickly the velocity of a given object will change in the course of time, another value called acceleration is defined. Thus, acceleration is the time-derivative of an object's velocity.

How are derivatives used to solve maximum and minimum problems? The maxima and the minima, also known as the extremum, are the values of the largest and smallest limits that a function may take in a given point. They are expressed as a set of greatest and least values. Derivatives are used to determine these points. The derivative of a function is interpreted geometrically as the slope of the curve of the mathematical function y (t), whereas the function of t is plotted. The derivative is noted as positive when a function increases toward a maximum; the maximum being horizontal at zero. It is considered negative just beyond the maximum. The second derivative notates the rate of change. It is called negative since the process of the slope, as described, is always getting smaller. The second derivative is always negative and corresponds to a maximum.