Thursday, 9 May 2013

CALCULUS !! Backbone of Higher Mathematics

In day to day life we are often interested in the extent to which a change in one quantity aects a change in another related quantity. This is called a rate of change. For example, if you own a motor car you might be interested in how much a change in the amount of fuel used aects how far you have traveled  This rate of change is called fuel consumption. If your car has high fuel consumption then a large change in the amount of fuel in your tank is accompanied by a small change in the distance you have traveled  Sprinters are interested in how a change in time is related to a change in their position. This rate of change is called velocity. Other rates of change may not have special names like fuel consumption or velocity, but are nonetheless important.
Thus in layman’s language Calculus is the mathematical study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.
Origins of Calculus-                                             
The discovery of calculus is often attributed to two men, Isaac Newton and Gottfried Leibniz, who independently developed its foundations. Although they both were instrumental in its creation, they thought of the fundamental concepts in very different ways. While Newton considered variables changing with time, Leibniz thought of the variables x and y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between successive values of these sequences. Leibniz knew that dy/dx gives the tangent but he did not use it as a defining property. On the other hand, Newton used quantities x' and y', which were finite quantities, to compute the tangent. Of course neither Leibniz nor Newton thought in terms of functions, but both always thought in terms of graphs. For Newton the calculus was geometrical while Leibniz took it towards analysis.
The development of Calculus can roughly be described along a timeline which goes through three periods: Anticipation, Development, and Rigorization. In the Anticipation stage techniques were being used by mathematicians that involved infinite processes to find areas under curves or maximize certain quantities. In the Development stage Newton and Leibniz created the foundations of Calculus and brought all of these techniques together under the umbrella of the derivative and integral. However, their methods were not always logically sound, and it took mathematicians a long time during the Rigorization stage to justify them and put Calculus on a sound mathematical foundation.
Applications of Calculus-
You can look at differential calculus as the mathematics of motion and change. Integral calculus covers the accumulation of quantities, such as areas under a curve. The two ideas work inversely together.
 Calculus is deeply integrated in every branch of the physical sciences, such as physics and biology. It is found in computer science, statistics, and engineering; in economics, business, and medicine. Modern developments such as architecture, aviation, and other technologies all make use of what calculus can offer.

Finding the Slope of a Curve
Calculus can give us a generalized method of finding the slope of a curve. The slope of a line is fairly elementary, using some basic algebra it can be found. Although when we are dealing with a curve it is a different story. Calculus allows us to find out how steeply a curve will tilt at any given time. This can be very useful in any area of study.

Calculating the Area of Any Shape
Although we do have standard methods to calculate the area of some shapes, calculus allows us to do much more. Trying to find the area on a shape like this would be very difficult if it wasn't for calculus.

Calculating Complicated X-intercepts
Without an idea like the Intermediate Value Theorem it would be exceptionally hard to find or even know that a root existed in some functions. Using Newton’s Method you can also calculate an irrational root to any degree of accuracy, something your calculator would not be able to tell you if it wasn't for calculus.

Visualizing Graphs
Using calculus you can practically graph any function or equation you would like. In fact you can find out the maximum and minimum values, where it increases and decreases and much more without even graphing a point, all using calculus.

Finding the Average of a Function
A function can represent many things. One example is the path of an airplane. Using calculus you can calculate its average cruising altitude, velocity and acceleration. Same goes for a car, bus, or anything else that moves along a path. Now what would you do without a speedometer on your car?

Calculating Optimal Values
By using the optimization of functions in just a few steps you can answer very practical and useful questions such as: “You have square piece of cardboard, with sides 1 meter in length. Using that piece of card board, you can make a box, what are the dimensions of a box containing the maximal volume?” These types of problems are a wonderful result of what calculus can do for us.

- Ishan Arora ( Childhood friend ) 

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