Notes

Differential Equations Basic Concepts of Differential Equations * Differential equation: an equation involving a derivative of the dependent variable with respect to the independent variable is called a differential. * The order of the highest order derivative occurring in a differential equation is called the order of the differential equation. * The degree of a ... Read more

Application of Integrals Areas Under Simple Curves * Area of the region bounded by the curve, y = ƒ(x), the coordinates x = a and x = b, and the X- axis = \int_{a}^{b} ƒ(x) dx. * Area of the region bounded by the curve, x = ɡ(y), the coordinates y = c and y ... Read more

Integrals Integration * Integration is the inverse process of differentiation. If \frac{d}{dx} (F(x)) = ɡ(x), then ∫ g (x) dx  = F(x) + C * Where C is an arbitrary constant known as the constant of integration. * Geometrical interpretation of indefinite integrals: * An indefinite integral is a collection of a family of curves, ... Read more

Rate of Change of Quantities Rate of Change: The change in one quantity with respect to time is known as rate of change of quantity. * If a quantity y varies with another quantity x, satisfying y = ƒ(x), then \frac{dy}{dx} represents the rate of change quantity with respect to x. * If x = ... Read more

Continuity of a Function * A function ƒ(x) is continuous at x = c, if ƒ(x) is defined at x = c and lim_{x \rightarrow } c ƒ ( x) =n ƒ(c). * If ƒ is not continuous at x = c, then we say ƒ is discontinuous at c, and c is called a ... Read more

Determinants * Let A be the set of square matrices and B is a set of real or complex numbers. If a function, ƒ: A → B is defined by ƒ(x) = b, such that X ∈ A and b ∈ B, then ƒ(x) is called the determinant of X. * The determinant of A ... Read more

Fundamental of Matrices * A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. Column matrix: A matrix is said to be column matrix if it has only one column. Row matrix: A matrix is said to be a row ... Read more

Basic Concept * The principal value branch of sin^{-1} is [ - \frac{π}{2} , \frac{π}{2}  ]. * The principal value branch of cos^{-1} is [ 0 , π ]. * The principal value branch of cosec^{-1}is [  -\frac{π}{2} , \frac{π}{2}  ]  - { 0 }. * The principal value branch of tan^{-1} is ( - ... Read more