Derivative and Integral Essay Example | Topics and Well Written Essays - 1500 words

Derivative and Integral - Essay Example Let a is a number in the domain of f and Lim/h->0 [f(a+h)-f(a)]/h exists, then f is said to be differentiable at a. This limit is called the derivative of f at a and is denoted by f’(a).For all x at which f(x) is differentiable ,f’(x) is a function called the derived function of f(x) .The domain of f’(x) is the subset of f(x).f’(x) is sometimes called as the derivative or the differential coefficient of f(x) at x.The process of obtaining the derivative of f is called Differentiation.f’(x) is sometimes denoted by dy/dx or Dy   or Df(x) or d/dx f(x).2. EXAMPLEIf f(x) is a continuous function of x and if x varies, f(x) also varies correspondingly. But the variation in the function may not be uniform sometimes slowly and sometimes rapidly. Geometrically, this problem is equivalent of that of finding a tangent line to the graph of the function.The function F(x) is called the anti derivative of the function f(x) on the interval (a,b) if at all points of this interval F’(x)=f(x)Definition:Indefinite Integral: If the function F(x) is an anti derivative of f(x), then F(x) +c is called the indefinite integral of the function f(x).It is denoted by ∠«f(x)dx. Since c is an arbitrary constant the integral is reasonably referred to as indefinite integral.Thus by definition ,∠«f(x)dx= F(x)+ C if F’(x)=f(x).f(x) is called the integrand and c is called the constant of integration. x is the variable of integration. The process of obtaining the integral is called as Integration.Definite Integral:... Let >0 |f(x) - (13)| < Substitute f(x) = 5x+3 |5x+3 - (13)| < |5x-10|2 (5x+3)=13 PART 2 1. DERIVATIVE Let a is a number in the domain of f and Lim/h->0 [f(a+h)-f(a)]/h exists, then f is said to be differentiable at a. This limit is called the derivative of f at a and is denoted by f'(a).For all x at which f(x) is differentiable ,f'(x) is a function called the derived function of f(x) .The domain of f'(x) is the subset of f(x). f'(x) is sometimes called as the derivative or the differential coefficient of f(x) at x.The process of obtaining the derivative of f is called Differentiation. f'(x) is sometimes denoted by dy/dx or Dy or Df(x) or d/dx f(x). 2. EXAMPLE If f(x) is a continuous function of x and if x varies, f(x) also varies correspondingly. But the variation in the function may not be uniform sometimes slowly and sometimes rapidly. Geometrically, this problem is equivalent of that of finding a tangent line to the graph of the function. For Example, velocity is derived from the position function and acceleration is derived from the velocity function. Each of velocity at a point, acceleration at a point etc., is an instantaneous rate of change ,but not the average rate of change, which relates to a finite interval of space or time .This is obtained by applying the limit concept to the problem of determining the instantaneous rate of change of a function. This is done by finding the