Derivative and Integral - Essay Example

Derivative and Integral 1. LIMIT Let f be a function in the deleted neighborhood of of a and L be a real number. If to every positive number ε (however small) there exists a positive number δ such that |f(x)-L| < ε for all x such that 0 < |x-a| < δ ,we say that f(x) tends to Land x tends to a.We write it as Lim/x->a f(x) =L Lim/x->a f(x)=L means (i) the limit f(x) as x->a exists and (ii) the limit is equal to L 2. EXAMPLE 1. Consider f(x)=x-1.If we take a sequence of values 2.9,2.99,2.999,…..,3.01,3.002,3.0001…of x approaching 3 ,we observe that f(x) takes the respective sequence of values 1.9,1.992,1.9999,……,2.01,2.002,2.0001,….approaching 2. Now we see as x approaches 3,f(x) approaches 2.That is said as x tends to3,limit of f(x) is 2.We write it as Lim/x->3 f(x)=2. 2. Consider f(x)= x2-4/x-2 f(x) is not defined at x=2. If x is not equal to 2 then f(x)=x+2.If x takes sequence of values 1.9,1.995,1.9999,……2.01,2.002,2.0001,…approaching 2 then f(x)takes the respective sequence of values 3.9,3.995,3.9999,…..4.01,4.002,4.0001, approaching 4.As the difference between x and 2 is decreasing then the difference between f(x) and 4 is also decreasing correspondingly. Hence by making |x-2| sufficiently small |f(x)-4| can be made as small as we please. Suppose we have to make |f(x)-4|a f(x)=L Proof: Consider f(x)=mx+b for all x ε R As f(x) is defined for all values of x ε R,therefore,f(x) is defined in any deleted neighborhood of c Let ε>0 |f(x) - (mc+b)| < ε Substitute f(x) = mx+b |mx+b – (mc+b)| 0 |f(x) - (13)| < ε Substitute f(x) = 5x+3 |5x+3 – (13)| 0 [f(a+h)-f(a)]/h f’(x)=Lim/h->0 [m(x+h)+b-(mx+b)]/h f’(x)=Lim/h->0 [mx+mh+b-mx-b]/h f’(x)=Lim/h->0 m=m Therefore we have Dx(mx+b)=m 4. Real Number Example Prove that Dx(5x+3)=5 Let f(x)=5x+3 f(x+h)=5(x+h)+3 By definition of a derivative we have f’(x)=Lim/h->0 [f(a+h)-f(a)]/h f’(x)=Lim/h->0 [5(x+h)+3-(5x+3)]/h f’(x)=Lim/h->0 [5x+5h+3-5x-3]/h f’(x)=Lim/h->0 5=5 Therefore we have Dx(5x+3)=5 PART 3 INTEGRAL 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. a Definite Integral: ∫f(x) is called the definite Integral of f(x) from a to b and b is called the b and a is called the upper limit of the definite integral. Since c does not appear here, it is reasonably referred as definite integral. a ∫f(x)dx=F(a)-F(b) b Relationship between Derivative and Integral: Integral is an Anti Derivative of a function .This can be explained by a following example. Consider g(x)=4x3+15x2 Now the function defined by f1(x)=x4+5x3 is an anti derivative of g since f1’(x)=4x3+5.3x2=g(x) Also the function defined by f2(x)=x4+5x3+9 is an anti derivative of g since f2’(x)=4x3+5.3x2+0=g(x) Thus if f1 is an anti derivative (Integral) of g if f2(x)=f1(x)+c( a real number, then f2’(x)=f1’(x)=g(x) and f2 also is an anti derivative(Integral)of g BIBILIOGRAPHY Intermediate Mathematics –Volume I and II by V.Venkateshwar rao, N. Krishna murthy, B.V.S.S Sharma Publisher: Schand Company and Ltd ISBN for Vol I: 81-219-0662-8 ISBN or Vol II:81-219-1001-3 Read More