Differential and Integral Calculus - Essay Example

Calculus Calculus is mathematical means used to study variations in physical quantities. Four main scientific and mathematical problems, mentioned below, of seventeenth century stressed the need of a new mathematical tool to study and analyze those problems. 1. To find the tangent line to a curve at a point. 2. Find the length of a curve, the area of a region and the volume of a solid. 3. Find the maximum and minimum values of a quantity, like maximum and minimum distance of a planet from the sun. Also, to find the maximum range attainable for a projectile by varying its angle of fire. 4. Find instantaneous velocity and acceleration of a body when given a formula for distance travelled in any specified amount of time, and conversely to find the distance travelled in any specified time period when given a formula that gives instantaneous velocity and acceleration. These were the problems addressed by the geniuses of seventeenth century, mainly Gottfried Wilhelm Leibniz and Sir Isaac Newton. 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 the terms comes out to be in the form of 4.41, 4.0401, 4.004001, , (2+1/10n)2, . . It can be said that it is appropriate to expect x2 to approach 4 as limit as x approaches 2 as limit. Or, the limit of x2 is 4 as x approaches2. It is written as. Continuity To put it informally, a function is said to be continuous if it can be plotted without lifting the pen from the paper in a manner that there are no gaps, jumps or breaks. Below are a few graphical examples of continuous and discontinuous functions. Rate of Change and Slope In linear functions the slope does not vary over the domain of the function and provides an exact value of the rate of change in the value of y or f(x) with respect to change in the value of x, the independent variable. This is not the case in nonlinear functions and the rate of change in y is not constant with respect to change in x. However, rate of change in nonlinear functions can be approximated by taking the average rate of change over small intervals. Considering the two point A and B on f in figure above, the straight line joining the two points is termed as the secant line. At point A the independent variable has a value x and the corresponding value of dependent variable can be determined by evaluating f(x). At point B the independent variable has changed to x+x and the dependent variable can be determined by evaluating f(x+x). In moving from point A to B the x value changes by (x+x) - x or x. Similarly, the corresponding change in y value is y = f(x+x) - f(x). Thus, the slope of this function or the ratio of changes will be, this is also termed as difference quotient. Consider a function y = f(x) = x2 passing through two points whose coordinates are given as (-2, 4) and (3, 9). Applying our knowledge thus far gained we can find a general equation for the difference quotient and the slope of the line passing through these two points. In our case, = = In our example x = -2 and x = 5, therefore the slope or the difference quotient comes out to be 1. Derivative of a Function As we shorten the extent of x and bring it closer and closer to x such that it approaches to be very close to x but not equal to it we say that the function is approaching its limit and the rate of change so deduced is not the average rate of change but the rate of change of a particular instant. The average rate of change is measured over an interval whereas instantaneous rate of change is the change at a particular instant. It is a snapshot of what is happening at a particular instant. Moreover, if the two points A and B are brought closer to each other such that x 0, the secant line becomes the tangent of the curve and the slope of this tangent is the slope of the curve at a particular instant. The derivative of a function is thus defined to be, = Rules of Differentiation 1. Constant Rule: If , where c is any constant, then 2. Power Rule: If , where n is a real number, then 3. Constant times a function: If , where c is constant and g is a differentiable function, then 4. Sum or difference of functions: If , where u and v are differentiable, then 5. Product Rule: If , where u and v are differentiable, then 6. Quotient Rule: If , where u and v are differentiable and , then INTEGRAL CALCULUS The second major area in the field of Calculus is Integral Calculus. Differential Calculus is mainly used to study rates of change and tangent slopes, whereas Integral Calculus mainly concerns itself with calculation of areas under curves or other defined boundaries. Mainly, it is about finding the original function if the derivative of the function is known. As there are rules of finding derivatives of functions, Integral Calculus has its own set of rules and exceptions. Antiderivatives We now know how to find the derivative () of a given function f. Often there will be situations where we might be knowing the derivative of a function and will be required to find the original function f from . One way to do that is trial and error but fortunately the rules of integration come in handy in these situations. The method of finding antiderivatives is called integration and the functions thus obtained are termed as indefinite integrals denoted in the form as shown below. Rules of Integration 1. Constant Rule: , where k is any constant 2. Power Rule: , where 3. Constant times function: , where k is any constant 4. If and exist, then 5. Power Rule Exception: 6. 7. , where There are many more rules available for differentiation and integration both. Bibliography Anton, Howard. Calculus with Analytic Geometry. Ed. Nancy Prinz. 5th Edition. New York: John Wiley & Sons, Inc., 1995. Ayers, Frank. Theory and Problems of Differential and Integral Calculus. 2nd Edition. New York: McGraw-Hill Book Company, 1964. Encyclopdia Britannica, Inc. "Calculus." Ultimate Reference Suite. Encyclopdia Britannica. Chicago, 2008. Microsoft Corporation. "Calculus." Microsoft Encarta 2008. Redmond: Microsoft Corporation, 2007. Read More