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4.2 The Mean Value Theorem. Rolle’s Theorem. Let f be a function that satisfies the following three conditions: f is continuous on the closed interval [a,b] . f is differentiable on the open interval (a,b) . f(a) = f(b) .

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Rolle s theorem l.jpg
Rolle’s Theorem

Let f be a function that satisfies the following three conditions:

  • f is continuous on the closed interval [a,b] .

  • f is differentiable on the open interval (a,b) .

  • f(a) = f(b) .

    Then there is a number c in (a,b) such that f ′(c)=0.

    Examples on the board.


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If f (x) is continuous over [a,b] and differentiable over (a,b), then at some point c between a and b:

The Mean Value Theorem says that at some point in the interval, the actual slope equals the average slope.

Mean Value Theorem for Derivatives

Note: The Mean Value Theorem only applies

over a closed interval.


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An illustration of the Mean Value Theorem.

Tangent parallel to chord.

Slope of tangent:

Slope of chord:


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Corollaries of the Mean Value Theorem

  • Corollary 1: If f ′(x)=0 for all x in an interval (a,b), then f is constant on (a,b) .

  • Corollary 2: If f ′ (x) = g′ (x) for all x in an interval (a,b), then f- g is constant on (a,b) , that is f(x)=g(x) + c where c is a constant.

    (see the next slide for an illustration of Corollary 2)


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Functions with the same derivative differ by a constant.

These two functions have the same slope at any value of x.


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