Mean-Value Theorem

  • Rolle’s Theorem: Let a function be f: [a, b]®R, such that
  1. f is continuous over the interval [a, b],
  2. f is differentiable over the interval (a, b) and
  3. f(a) = f(b).
  • Then there exists at least one real number, c in (a, b) such that
  • f’(c) = 0.
  • Geometrical Interpretation of Rolle’s Theorem: There is at least one point in [a, b] at which the tangent is parallel to x-axis. There may exist more than one point in the [a, b] at which the tangents are parallel to x-axis.
  • Lagrange’s Mean Value Theorem: Let a function f: [a, b]®R, such that f  is continuous over the interval [a, b] and f  is differentiable over the interval (a, b). Then there exists at least one real number c in (a, b) such that f’(c) = [f(b) – f(a)]/( b- a).
  • Lagrange’s mean value theorem asserts that there is at least one point lying between ‘x= a’ and ‘x= b’ at which the tangent is parallel to the chord AB where coordinates of A are (a, y¬1) and coordinates of B are (b, y1).The Mean Value Theorem is useful in finding the specific point of a continuous function that has the same slope a
  • s the secant line.
  • It is used to find the instantaneous rate and widely used to prove other theorems in calculus like Fundamental Theorem of Calculus.

   Keywords:  Rolle’s Theorem, Geometrical Interpretation of Rolle’s theorem, Lagrange’s Mean Value Theorem

 

To Access the full content, Please Purchase

  • Q1

    Marks:2
    Answer:

    View Answer
  • Q2

    Marks:2
    Answer:

    View Answer
  • Q3

    Marks:2
    Answer:

    View Answer
  • Q4

    State the Lagrange’s Mean Value Theorem.

    Marks:1
    Answer:

    Theorem: If a function f(x) is continuous in the closed interval [a, b], derivable in the open interval (a, b), then there exists at least one real number c in (a, b) such that

    View Answer
  • Q5

    State the Rolle’s Theorem.

    Marks:1
    Answer:

    Theorem: If a function f(x) is continuous in the closed interval [a, b], derivable in the open interval (a, b) and f(a) = f(b), then there exists one real number c in (a, b) such that f’(c) = 0.

    View Answer