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Math 140

Math 140. 3.1 – Increasing and Decreasing Functions; Relative Extrema. Let be defined on the interval . Suppose and are in that interval . is ___________________ on that interval if when is ___________________ on that interval if when .

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Math 140

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  1. Math 140 3.1 – Increasing and Decreasing Functions; Relative Extrema

  2. Let be defined on the interval . Suppose and are in that interval. is ___________________ on that interval if when is ___________________ on that interval if when

  3. Let be defined on the interval . Suppose and are in that interval. is ___________________ on that interval if when is ___________________ on that interval if when increasing

  4. Let be defined on the interval . Suppose and are in that interval. is ___________________ on that interval if when is ___________________ on that interval if when increasing decreasing

  5. Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE)

  6. Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE)

  7. Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE) In other words, where might change signs?

  8. Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE) In other words, where might change signs?

  9. Recall: is increasing where , and decreasing where . Where might change from increasing to decreasing, or decreasing to increasing? 1. When 2. When does not exist (DNE) In other words, where might change signs?

  10. 1. When

  11. 1. When

  12. 1. When

  13. 1. When

  14. 1. When

  15. 2. When does not exist (DNE)

  16. 2. When does not exist (DNE)

  17. 2. When does not exist (DNE)

  18. Ex 1. Find the intervals of increase and decrease for

  19. Ex 1. Find the intervals of increase and decrease for

  20. Relative Extrema

  21. Relative Extrema

  22. Relative Extrema

  23. Relative Extrema

  24. Relative Extrema

  25. Relative Extrema

  26. If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur.

  27. If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number

  28. If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number

  29. If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number critical point

  30. If is defined, and either or DNE, then is called a ________________. Also, is called a __________________. Critical numbers give us -values where relative extremamight occur. critical number critical point

  31. First Derivative Test for Relative Extrema Let be a critical number. Relative max at

  32. First Derivative Test for Relative Extrema Let be a critical number. Relative max at Relative min at

  33. First Derivative Test for Relative Extrema Let be a critical number. Not a relative extremum

  34. First Derivative Test for Relative Extrema Let be a critical number. Not a relative extremum Not a relative extremum

  35. Ex 2. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.

  36. Ex 2. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.

  37. Ex 3. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.

  38. Ex 3. Find all critical numbers for , and classify each critical point as a relative maximum, relative minimum, or neither.

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