Mathematics is the base of human civilization. From cutting vegetables to arranging books on the shelves, from tailoring clothes to motion of Planets - Mathematics applies everywhere.
If real scales are chosen on two mutually perpendicular axes X′OX and Y′OY (called the x axis and y axis respectively), you can locate any point in the plane determined by these lines, by the ordered pair of real numbers (a, b) called rectangular co-ordinates of the point.
AX = B
If the coefficients of matrix A is non-singular, then A−1 exists.
The Binomial and the Poisson distribution are the most useful theoretical distribution for discrete variables. They relate to the occurrence of distinct events. In order to have mathematical distribution suitable for dealing with quantities whose magnitude is continuously varying, a continuous distribution is needed. The normal distribution is also called the normal probability distribution, is the most useful theoretical distribution for continuous variables.
The values of random variables may be distributed according to some definite probability law which can be expressed mathematically and the corresponding probability distribution is called theoretical distribution. Theoretical distributions are based on expectations on the basis of previous experience.
Let f (x, y, c1) = 0 be an equation containing x, y and one arbitrary constant c1. If c1 is eliminated by differentiating f (x, y, c1) = 0 with respect to the independent variable once, you get a relation involving x, y and dy/dx, which is a differential equation of the first order.
The order of a differential equation is the order of the highest order derivative occurring in it. The degree of the differential equation is the degree of the highest order derivative which occurs in it, after the differential equation has been made free from radicals and fractions as far as the derivatives are concerned.
The study of calculus and its applications is best understood when it is studied through the geometrical representation of the functions involved. In order to investigate the nature of a function (graph) it is not possible to locate each and every point of the graph. But you can sketch the graph of the function and know its nature by certain specific properties and some special points.
Let y = f(x) be a differentiable function. Then the quantities dx and dy are called differentials. The differential dx is an independent variable that is dx can be given any real number as the value. The differential dy is then defined in terms of dx by the relation
Some of the most important applications of differential calculus are optimization problems, in which you are required to find the optimal (best) way of doing something. These problems can be reduced to finding the maximum or minimum values of a function. Many practical problems require to minimize a cost or maximize an area.
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