GENERAL THEOREMS,INCREASE AND DECRESE OF A FUNCTION,INEQUALITIES AND APPROXIMATIONS.

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Presentation transcript:

GENERAL THEOREMS,INCREASE AND DECRESE OF A FUNCTION,INEQUALITIES AND APPROXIMATIONS

Rolle’s Theorem If a function f is (i) continous in [a,b] (ii) Derivable in (a,b) (iii)f(a)=f(b) then there exist atleast one real number such that

Lagrange’s Mean Value Theorem If a function f is (i) continous in the closed interval [a,b] (ii)Derivable in the open interval (a,b) then at least one value c of x in (a,b) such that

Application of Lagranges Mean Value Theorem If f is continous in [a,b] and in (a,b) then f is constant in [a,b]

Cauchy’s Mean Value Theorem If f and g be two function defined on [a,b] such that (i) f and g are continous in [a,b] (ii) f and g are derivable in (a,b) (iii) for any x in (a,b) then at least one such that

Geometrical Interpertation of C.M.V. If a curve is continous between the two points P and Q and has tangent at every point, the there exists at least one point R on the curve where the tangent at R is parallel to chord PQ.

Taylor’s Theorem with the Lagrange’s form of Remainder If a function f(x) is defined on [a,b] and (i) are continous in [a,b] (ii) exist in (a,b),then there exist atleast one real number c in (a,b) such that Lagrange’s Remainder

Taylor’s Theorem with Cauchy’s form of Remainder If a function f(x) is defined on [a,b] and (i) are continous in [a,b] (ii) exist in (a,b),then there exist atleast one real number t (0<t<1) such that Cauchy’s Remainder

Maclaurin’s Theorem If a function f(x) is defined on [0,x] and (i) are continous in [0,x] (ii) exist in (0,x),then there exist atleast one real number c in (0,x) such that

Increasing and Decreasing Function Let f be a real valued function defined on and be a non empty subset of S, then f is called increasing on iff Decreasing on iff

f is called strictly increasing on iff f is called strictly decreasing on iff

If f is real valued derivable in (a,b) and (i)If f is increasing in (a,b),then (ii)If f is decreasing in (a,b),then

If f is continuous in [a,b] and x in (a,b) then f is strctly increasing in [a,b] If f is continuous in [a,b] and x in (a,b) then f is strctly decreasing in [a,b]

Error in Linear Approximation If f(x) is twice differentiable in a nbd. N of then for we have