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Lesson 10: What does f' say about f?

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Lesson 10: What does f' say about f?

1. 1. Section 2.9 What does f say about f ? Math 1a October 17, 2007 Announcements Midterm I review session 10/21, 7:30pm in Hall D?
2. 2. Outline Increasing and Decreasing functions Concavity and the second derivative
3. 3. Deﬁnition Let f be a function deﬁned on and interval I . f is called increasing if f (x1 ) < f (x2 ) whenever x1 < x2 for all x1 and x2 in I .
4. 4. Deﬁnition Let f be a function deﬁned on and interval I . f is called increasing if f (x1 ) < f (x2 ) whenever x1 < x2 for all x1 and x2 in I . f is called decreasing if f (x1 ) > f (x2 ) whenever x1 < x2 for all x1 and x2 in I .
5. 5. Fact If f is increasing on (a, b), then f (x) ≥ 0 for all x in (a, b)
6. 6. Fact If f is increasing on (a, b), then f (x) ≥ 0 for all x in (a, b) If f is decreasing on (a, b), then f (x) ≤ 0 for all x in (a, b).
7. 7. Fact If f (x) > 0 for all x in (a, b), then f is increasing on (a, b). If f (x) < 0 for all x in (a, b), then f is decreasing on (a, b).
8. 8. Outline Increasing and Decreasing functions Concavity and the second derivative
9. 9. Deﬁnition A function is called concave up on an interval if f is increasing on that interval.
10. 10. Deﬁnition A function is called concave up on an interval if f is increasing on that interval. A function is called concave down on an interval if f is decreasing on that interval.
11. 11. Fact If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b)
12. 12. Fact If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b) If f is concave down on (a, b), then f (x) ≤ 0 for all x in (a, b).
13. 13. Fact If f (x) > 0 for all x in (a, b), then f is concave up on (a, b). If f (x) < 0 for all x in (a, b), then f is concave down on (a, b).