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Section 4.1

Section 4.1. Extrema on an Interval. Relative Extrema. f(x). Relative Maximum . Relative Minimum . Relative Extrema. f(x). Relative Extrema. f(x). Relative Extrema. f(x). Three Examples. Relative Extrema. Can typically think of these as the “peaks” and “valleys” of a graph.

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Section 4.1

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  1. Section 4.1 Extrema on an Interval

  2. Relative Extrema f(x) Relative Maximum Relative Minimum

  3. Relative Extrema f(x)

  4. Relative Extrema f(x)

  5. Relative Extrema f(x)

  6. Three Examples

  7. Relative Extrema • Can typically think of these as the “peaks” and “valleys” of a graph.

  8. Some Examples for Discussion

  9. Example 1 Find the value of the derivative (if it exists) at each given extremum.

  10. Example 1 (cont.) Find the value of the derivative (if it exists) at each given extremum.

  11. Critical Numbers • -values where the derivative is 0 or undefined. • Provide possible locations of relative extrema.

  12. Critical Numbers (cont.) • is a critical number in both pictures.

  13. Example 2 Approximate the critical numbers. What takes place at each one?

  14. Example 3 Find any critical numbers of the function.

  15. Example 3 (cont.) Find any critical numbers of the function. • ,

  16. Example 3 (cont.) Find any critical numbers of the function.

  17. Locating Extrema

  18. Example 4 Locate the absolute extrema of the function on the closed interval.

  19. Example 4 (cont.) Locate the absolute extrema of the function on the closed interval.

  20. Example 4 (cont.) Locate the absolute extrema of the function on the closed interval.

  21. Example 4 (cont.) Locate the absolute extrema of the function on the closed interval.

  22. Minimum on the open interval?

  23. Section 4.2 Rolle’s Theorem and The Mean Value Theorem

  24. Rolle’s Theorem • Michel Rolle (1652-1719)

  25. When can Rolle’sThm. be applied?

  26. Example 1 Explain why Rolle’s Theorem does not apply.

  27. Example 1 (cont.) Explain why Rolle’s Theorem does not apply.

  28. Example 2 Determine if Rolle’s can be applied and, if so, find all for which . If not, state why.

  29. Example 2 (cont.) Determine if Rolle’s can be applied and, if so, find all for which . If not, state why.

  30. The Mean Value Theorem

  31. Example 3 Determine if MVT can be applied and, if so, find all for which . If not, state why.

  32. Example 3 (cont.) Determine if MVT can be applied and, if so, find all for which . If not, state why.

  33. Example 3 (cont.) Determine if MVT can be applied and, if so, find all for which . If not, state why.

  34. Questions? Be sure to be practicing the given problem sets!

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