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Dr. Malay Kumar Ghorui Associate Professor Dept. Of Mathematics B.B. College Asansol

Dr. Malay Kumar Ghorui Associate Professor Dept. Of Mathematics B.B. College Asansol. Expansion of functions. Rolle's Theorem. Geometrical significance of Rolle’s theorem.

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Dr. Malay Kumar Ghorui Associate Professor Dept. Of Mathematics B.B. College Asansol

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  1. Dr. Malay Kumar Ghorui Associate Professor Dept. Of Mathematics B.B. College Asansol

  2. Expansion of functions Rolle's Theorem

  3. Geometrical significance of Rolle’s theorem If the graph of y=f(x) has equal ordinate at the two points A,B and if the graph be continuous throughout the interval from A to B and if the curve has a tangent at every point on it from A to B except possibly at the two points A and B, then there must exist at least one point on the curve between A and B, where the tangent is parallel to x-axis.

  4. Lagrange's Mean Value Theorem (MVT)or First MVT of Differential calculus:

  5. Geometrical interpretation of Lagrange's MVT Let A and B be two points on the graph of y=f(x) corresponding to x = a and x = b respectively. Then the slope of the chord AB =f(b)-f(a)/(b –a) and we know that geometrically f , (η) represent the gradient of the tangent at the point x = η on the curve. Thus MVT geometrically indicated the fact that there exists at least one point η in (a, b), the tangent at which on the curve is parallel to the chord joining the points x = a and x = b Increasing and Decreasing functions: A function f defined on an interval I is said to be Defn-1: increasing in I if f(x1) <or =f(x2) whenever x1 and x2 in I and x1 < x2 Defn-2: strictly increasing in I if f(x1) <or =f(x2) whenever x1 and x2 in I and x1 < x2 Defn-3: decreasing in I if f(x1) >or =f(x2) whenever x1 and x2 in I and x1 < x2 Defn-4: strictly decreasing in I if f(x1) >f(x2) whenever x1 and x2 in I and x1 < x2 Defn-5: monotone in an interval I if it is either increasing or decreasing in I. Defn-6: strictly monotone in an interval I if it is either strictly increasing or strictly decreasing in I.

  6. Cauchy's Mean Value Theorem (MVT)or Second MVT of Differential calculus:

  7. Taylor's Theorem with Lagrange's form of Remainder :

  8. Taylor's Theorem with Cauchy's form of Remainder :

  9. Taylor's Theorem with Generalized form of Remainder:

  10. Maclaurin's Theorem with Lagrange's form of Remainder :

  11. Maclaurin's Theorem with Cauchy's form of Remainder:

  12. Maclaurin's Theorem with Generalized form of Remainder :

  13. Important Problems on this Chapter

  14. End of the chapter

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