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4.2 The Mean Value Theorem

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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
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.

slide3

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.

slide4

An illustration of the Mean Value Theorem.

Tangent parallel to chord.

Slope of tangent:

Slope of chord:

corollaries of the mean value theorem
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)

slide6

Functions with the same derivative differ by a constant.

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