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Understanding the Mean Value Theorem in Calculus

Learn about the Mean Value Theorem, average speed, and instantaneous speed, along with examples to check hypotheses and find values. Solve for c and x values using the Theorem.

zachery-valentine
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Understanding the Mean Value Theorem in Calculus

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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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