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The Mean Value Theorem

This article explores the Mean Value Theorem (MVT), which states that for a continuous function on a closed interval [a, b] and differentiable on (a, b), there exists at least one point c in (a, b) where the instantaneous speed equals the average speed. In this context, we analyze an example with an average speed of 78 mph over specific intervals. We discuss the conditions of continuity and differentiability, and verify the hypotheses of the MVT by exploring practical applications and calculations to find the value of c.

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The Mean Value Theorem

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  1. The Mean Value Theorem

  2. Average Speed = 78 mph

  3. Average Speed = 78 mph Instantaneous Speed = 78 mph

  4. 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

  5. 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

  6. 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

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

  8. 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!

  9. 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]!

  10. 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!

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

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

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

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

  15. 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!

  16. 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!

  17. 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!

  18. 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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