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Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH

Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH. IncreasingDecreasing. 1. f ’(x) > 0 for all x in I. f(x) increasing. 2. f(x) decreasing. f ’ (x) < 0 for all x in I. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH. F091.

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Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH

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  1. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH Increasing\Decreasing 1 f ’(x) > 0 for all x in I f(x) increasing 2 f(x) decreasing f ’ (x) < 0 for all x in I

  2. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH F091

  3. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH F101

  4. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH F102

  5. How do you know that: local maximum or local minimum First Derivative Test: 1 _ + 2 First Derivative Test: 1 _ + 2

  6. F081

  7. F083

  8. F102

  9. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH Concavity 1 f ’’(x) > 0 for all x in I f(x) concave Up 2 f(x) concave Down f ’’(x) < 0 for all x in I

  10. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH Example: Discuss the curve with respect to concavity, Concavity 1 f ’’(x) > 0 for all x in I f(x) concave Up 2 f(x) concave Down f ’’(x) < 0 for all x in I

  11. Concavity Test 1 f ’’(x) > 0 for all x in I f(x) concave Up

  12. Concavity Test 1 f ’’(x) < 0 for all x in I f(x) concave Down

  13. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH F092

  14. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH Inflection point: 1 _ + _ 2 +

  15. Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH Example: Find all inflection points of

  16. F091

  17. How do you know that: local maximum or local minimum second Derivative Test: 1 2 second Derivative Test: 1 2

  18. F081

  19. F091

  20. F091

  21. F083

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