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3.3 1 st Derivative Test

3.3 1 st Derivative Test. Increasing or Decreasing? : If f  ( x ) > 0 in an interval, then f is increasing in the interval. If f  ( x ) < 0 in an interval, then f is decreasing in the interval. 1st Derivative Test c is critical number of f :

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3.3 1 st Derivative Test

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  1. 3.3 1st Derivative Test

  2. Increasing or Decreasing?: • If f (x) > 0 in an interval, then f is increasing in the interval. • If f (x) < 0 in an interval, then f is decreasing in the interval.

  3. 1st Derivative Test • c is critical number of f: • If f changes from + to – at c, then f(c) is a local max. • If f changes from – to + at c, then f(c) is a local min.

  4. Ex 1: Find where f (x) is increasing or decreasing:

  5. Ex 2: Find the local min & local max values of the function:

  6. HW – 3.3 pg. 186#1 – 7 odds,#17 – 45 EOO, #55 – 63 odds

  7. 3.4 Concavity Test

  8. Concave Up or Down?: • Concave up: holds water • Inc @ an Increasing rate • Dec @ a Decreasing rate

  9. Concave Up or Down?: • Concave down: spills water • Inc @ a Decreasing rate • Dec @ an Increasing rate

  10. Concavity Test • f (x) > 0 in an interval, then f is concave up in the interval. • f (x) < 0 in an interval, then f is concave down in the interval.

  11. Point of Inflection • point where f changes concavity. • where f changes from increasing to decreasing or vice versa. • where f changes sign.

  12. Ex 1: Find where f (x) is concave up or concave down:

  13. Ex 2: Find the points of inflection of the function:

  14. HW – 3.4 pg. 195 # 1 – 5 odds,#11 – 39 EOO,#49 – 56 all

  15. 2nd Derivative Test • c is critical number of f: • If f (c) = 0 & f (c) > 0, then f(c) is a local min. • If f (c) = 0 & f (c) < 0, then f(c) is a local max.

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