Integration by Parts If u and v be two functions of x, then the integral of product of these two functions is given by . In applying the above rule care has to be taken in the selection of the first function (u) and the second function (v) depending on which function can be integrated easily. Read more about Integration by Parts[…]
Definite Integral Problems Suppose f : R→R is a continuous periodic function with periodic T. let a ϵ R. Then prove that for any positive integer n, Problems: 1. Evaluate Solution: let us I = … (1) = , … (2) Equations 1 + 2 2I . , 2I = . 2I = , I Read more about Definite Integral Problems[…]
Problems on Differential Equation 1. Find the order and degree of Solution: Given that That is The highest differentiation is order that power is degree, hence Order is 2, degree = 1 2. Find the order of the family of the differential equation obtained by eliminating the arbitrary constant b and c from xy Read more about Problems on Differential Equation[…]
Torque on an Electric Dipole in a Uniform Electric Field Consider a dipole with charges +q and –q forming a dipole since they are a distance d away from each other. Let it be placed in a uniform electric field of strength E such that the axis of the dipole forms an angle θ with Read more about Torque on an Electric Dipole in a Uniform Electric Field[…]
Differential Equations An equation involving one dependent variable, one or more independent variable and the differential coefficient (derivatives) of dependent variable with respect to independent variable is called differential equations. Examples: 1. , 2. . Types of differential equations 1. Ordinanry differential equation: A differential equations is said to be an ordinary differential equation. If Read more about Differential Equations[…]
Indefinite Integration – Direct Subtitution Definition: If f(x) and g(x) are two functions such that f'(x) = g(x) then f(x) is called antiderivative or primitive of g(x) with respect to x Note1: If f(x) is an antiderivative of g(x) then f(x)+c is also an antiderivative of g(x) for all c ϵ wR If F(x) is an Read more about Indefinite Integration – Direct Subtitution[…]
Newton-Leibnitz Formula, Integral Function and its Properties Let f(x) be a continuous function defined on [a, b], then a function ф(x) defined by for all x ϵ [a, b] Is called the integral function of the function f(x)x Property I: The integral function of an integrable function is always continuous. Property II: If ф(x) is differentiable Read more about Newton-Leibnitz Formula, Integral Function and its Properties[…]
MAXIMA AND MINIMA PROBLEMS 1. Without using the derivative, show that f(x) = (½)x is strictly decreasing on R. Solution: Given that f(x) = (½)x Let x₁, x₂ ϵ R Let x₁ < x₂ ⇒ (½)x₁ > (½)x₂ ⇒ f(x₁) > f(x₂). ∴ f is strictly decreasing on R. 2. Without using the derivative, show Read more about MAXIMA AND MINIMA PROBLEMS[…]
Global Maxima or Minima Global Maxima or Minima in [a, b]: Global maxima or minima of f(x) in [a, b] is basically the greatest or least value of f(x) in [a, b]. Mathematically, it is written as: The function f(x) has a global maximum at the point ‘a’ in the interval I if f (a) ≥ Read more about Global Maxima or Minima[…]
Maxima and Minima Maximum: Let f (x) be function with domain D ⊂ R then f (x) is said to attain the maximum value at a point a ϵ D if f (x) ≤ f (a) ∀ x ϵ D. The point is called the point maximum f (a) is known as Maximum value of Read more about Maxima and Minima[…]