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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
slide5

Average Speed

= 78 mph

Instantaneous Speed =

78 mph

slide7

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

slide8

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

slide9

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

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].
example check that the hypothesis for the mvt is true for the function on the interval 0 5 21
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 22
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 23
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!

slide14

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

slide15

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

slide16

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

slide17

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

slide18

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!

slide19

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!

slide20

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!

slide21

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!

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