Warm-Up: December 11, 2012

1. Warm-Up: December 11, 2012 • Find the slope of the line that connects the endpoints of the graph of:

2. Homework Questions?

3. Mean Value Theorem Section 4.2

4. Mean Value Theorem • If y=f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b), then there is at least one point c in (a, b) at which

5. One Physical Interpretation – Speed • The instantaneous speed at some interior point must equal the average speed over the entire interval. • If a car’s average speed for a trip is 30 mph, then its speedometer must have read 30 mph at some point during the trip.

6. Rolle’s Theorem • Special case of the Mean Value Theorem

7. Increasing and Decreasing • Let f be a function defined on an interval I and let x1 and x2 be any two points in I. • A function is called monotonic if it is increasing everywhere inside an interval or if it is decreasing everywhere inside an interval

8. Increasing and Decreasing • Let f be continuous on [a, b] and differentiable on (a, b).

9. Example 1 • Find: • The local extrema • The intervals on which the function is increasing • The intervals on which the function is decreasing

10. Assignment • Read Section 4.2 (pages 186-191) • Page 192 #1-19 odd • Page 192 #25-41 odd • Read Section 4.3 (pages 194-203)

11. Warm-Up: December 12, 2012 • Find: • The local extrema • The intervals on which the function is increasing • The intervals on which the function is decreasing

12. Homework Questions?

13. Continuing 4.2 notes

14. Consequences of the MVT • If f’(x)=0 for all x, then there is a constant C such that f(x)=C for all x. • If f’(x)=g’(x) for all x, then there is a constant C such that f(x)=g(x)+C

15. Antiderivative • A function F(x) is an antiderivative of f(x) if F’(x)=f(x). • The process of finding an antiderivative is antidifferentiation. • There are an infinite number of antiderivatives of any given function, that only differ by a constant, C. Differentiation Antidifferentiation

16. Example 2 • Find all possible functions f with the given derivative.

17. Example 3 • Find the function with the given derivative whose graph passes through the point P.

18. You-Try #3 • Find the function with the given derivative whose graph passes through the point P.

19. Assignment • Read Section 4.2 (pages 186-191) • Page 192 #1-19 odd • Page 192 #25-41 odd • Read Section 4.3 (pages 194-203)