4.3 1st & 2nd Derivative Tests

1. 4.3 1st & 2nd Derivative Tests

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

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

6. 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.

8. 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.

9. 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.

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

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

12. 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.

14. 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.

15. HW – 4.3 pg. 3021, 5 – 10 all, 25 – 49 EOO