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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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## The Mean Value Theorem

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**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**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**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!

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