Warm-Up: December 13, 2012

1. Warm-Up: December 13, 2012 • Find all possible antiderivatives of

2. NASA!!!! • If you are a female junior, you could work with NASA scientists for a week this summer. • https://wish.aerospacescholars.org/

3. Homework Questions?

4. Connecting f’ and f’’ with the Graph of f Section 4.3

5. First Derivative • At a critical point, c, of a continuous function, f: • If f’ changes from positive to negative at c, then f has a local maximum at c. • If f’ changes from negative to positive at c, then f has a local minimum at c. • If f’ does not change sign at c, then f has no local extreme value at c.

6. First Derivative • At a left endpoint, a, of a continuous function, f: • If f’<0 for x>a, then f has a local maximum value at a. • If f’>0 for x>a, then f has a local minimum value at a.

7. First Derivative • At a right endpoint, b, of a continuous function, f: • If f’<0 for x<b, then f has a local minimum value at b. • If f’>0 for x<b, then f has a local maximum value at b.

8. Concavity • The graph of a differentiable function y=f(x) is: • concave up if y’ is increasing (if y’’>0) • concave down if y’ is decreasing (if y’’<0)

9. Points of Inflection • A point where the graph of a function has a tangent line and where the concavity changes is a point of inflection. • Also called an inflection point. • Points of inflection can occur (but do not always occur) where y’’=0 or where y’’ fails to exist. • The sign of y’’ must change around the point.

10. Example 1 • Use analytic methods to find the intervals on which the function is (a) increasing, (b) decreasing, (c) concave up, (d) concave down. Then find any (e) local extreme values, and (f) inflection points.

11. Assignment • Read Section 4.3 (pages 194-203) • Page 204 Exercises #7-27 odd • Page 203 Exercises #1-5 odd, 29-41 odd • Read Section 4.4 (pages 206-213)

12. Warm-Up: December 14, 2012 • Find all inflection points of

13. Homework Questions?

14. Point of Inflection • Points of inflection can occur (but do not always occur) where y’’=0 or where y’’ fails to exist. • The sign of y’’ must change around the point.

15. Second Derivative Test • If f’(c)=0 and f’’(c)<0, then f has a local maximum at x=c. • If f’(c)=0 and f’’(c)>0, then f has a local minimum at x=c.

16. Page 203 Exercises #2, 4, 6

17. Assignment • Read Section 4.3 (pages 194-203) • Page 204 Exercises #7-27 odd • Page 203 Exercises #1-5 odd, 29-41 odd • Read Section 4.4 (pages 206-213) • Quiz on 4.1-4.3 on Tuesday