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The Mean Value Theorem - PowerPoint PPT Presentation

The Mean Value Theorem. Average Speed. = 78 mph. Average Speed. = 78 mph. Instantaneous Speed =. 78 mph . The Mean Value Theorem. Let f be continuous on [a, b] and differentiable on (a, b). Then there exists a point c in (a, b) such that:. a. c. b. The Mean Value Theorem.

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Presentation Transcript

The Mean Value Theorem

= 78 mph

= 78 mph

Instantaneous Speed =

78 mph

Let f be continuous on [a, b] and differentiable on (a, b). Then there exists a point c in

(a, b) such that:

a

c

b

Let f be continuous on [a, b] and differentiable on (a, b). Then there exists a point c in

(a, b) such that:

a

c

b

Let f be continuous on [a, b] and differentiable on (a, b). Then there exists a point c in

(a, b) such that:

a

c

b

Example Check that the hypothesis for the MVT is true for the function on the interval [0.5, 2].

Example Check that the hypothesis for the MVT is true for the function on the interval [0.5, 2].

f(x) is not continuous at x = 0!

Example Check that the hypothesis for the MVT is true for the function on the interval [0.5, 2].

f(x) is not continuous at x = 0!

That’s okay because 0 is not in [0.5, 2]!

Example Check that the hypothesis for the MVT is true for the function on the interval [0.5, 2].

f(x) is not continuous at x = 0!

That’s okay because 0 is not in [0.5, 2]!

f(x) is not differentiable at x = 0, again, that’s okay!

Example If c is the value defined by the Mean Value Theorem, then for on [0, 1], what is the value of c?

Example If c is the value defined by the Mean Value Theorem, then for on [0, 1], what is the value of c?

Example If c is the value defined by the Mean Value Theorem, then for on [0, 1], what is the value of c?

Example If c is the value defined by the Mean Value Theorem, then for on [0, 1], what is the value of c?

Example If c is the value defined by the Mean Value Theorem, then for on [0, 1], what is the value of c?

Now solve for x!

Example If c is the value defined by the Mean Value Theorem, then for on [0, 1], what is the value of c?

Now solve for x!

Example If c is the value defined by the Mean Value Theorem, then for on [0, 1], what is the value of c?

Now solve for x!

Now find the roots using your calculator!

Example If c is the value defined by the Mean Value Theorem, then for on [0, 1], what is the value of c?

Now solve for x!

Now find the roots using your calculator!

This the c value of x that satisfies the MVT!