Differential and Integral Calculus Essay Example | Topics and Well Written Essays - 2000 words

Differential and Integral Calculus - Essay Example Calculus is widely used in physical, biological and social sciences. Examples of its applications in physical sciences are like studying the speed of falling body, rates of change in a chemical reaction, or rate of decay in a radioactive reaction. In the biological sciences its applications include solving the problem of rate of growth of bacteria as a function of time. In social sciences Calculus has its applications in the study of probability and statistics. The two main branches of Calculus are Differential Calculus and Integral Calculus. Differential Calculus deals with rates of change while studying or solving a problem and Integral Calculus involves summations of special type. One helps to find the slope of tangent to a curve at a certain point while the other is used to find the area covered by a curve and two points on it. As the entire natural world is in a constant motion and thus a change, mathematical analysis provides us the means to investigate the process of change, motion and dependence of quantities upon each other. Consider the motion of a body moving in a straight line whose position is given by a number expressing the distance and direction from a fixed point, the origin. Now if we specify the position of this body at each instant of time, it is equivalent to defining a function of some real numbers representing time to some corresponding real numbers representing position. Now consider the following three scenarios: 1. What will be function to give the velocity at each instant 2. If only velocity is known at each instant, find the distance travelled during a particular interval of time. 3. If only the function giving the velocity at each instant is known, what would be the function giving the position at each instant These are the basic problems which are generally addressed by Calculus. DIFFERENTIAL CALCULUS The two main concepts in Calculus are limits of a function and continuity. Limit of a Sequence If n is a set of integers greater than 0 then consecutive points of a sequence, in our example 2-1/n, when plotted on a number line the sequence will come out to be as 1,1.5,1.66,1.75, 1.8, , 2-1/n, . or 1, 3/2, 5/3, 7/4, 9/5, , 2-1/n, .. Sequence I It is worth noting that as our sequence progresses it seems that we get closer and closer to 2 or our sequence appears to be approaching 2 as it progresses further and further but at no point does it appear to be exactly equal to 2. If x is a variable with above sequence as its range then it is said that xapproaches 2 as limit, or, x tends to 2 as limit and it is written as x 2. Limit of a Function Continuing with our example of Sequence I above, if function of x f(x) = x2 then all our results would be approaching a value of 4 as in (1)2, (3/2)2, (5/3)2, (7/4)2, (9/5)2, , (2-1/n)2, .. or 1, 9/4/ 25/9, 49/16, 81/25, whereas x 2. Like in another example of a sequence emerging from 2+1/10n the terms of the sequence are 2.1, 2.01, 2.001, 2.0001, .., 2+1/10n Sequence II Here again x2. It can be easily demonstrated that x24 as squaring