1. 5.1 Increasing\decreasing, graphs and critical numbers

2. A function is increasing when its graph rises as it goes from left to right. A function is decreasing when its graph falls as it goes from left to right. inc dec

3. The inc\dec concept can be associated with the slope of the tangent line. The slope of the tan line is positive when the function is increasing. The slope of the tan line is and negative when decreasing

4. On an interval on which f is defined (a) if f ‘(x) > 0 for all x in an interval, then f is increasing on the interval. (b) if f ‘(x) < 0 for all x in an interval, then f is decreasing on the interval. (c) if f ‘(x) = 0 for all x in an interval then f is constant on that interval.

5. Find the intervals where f is increasing and decreasing

6. Slide 2.1- 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION: A critical value (or critical number) of a function f is any number c in the domain of f for which the tangent line at (c, f (c)) is horizontal or for which the derivative does not exist. That is, c is a critical value if f (c) exists and f (c) = 0 or f (c) does not exist. These are the x values where the function could change from increasing to decreasing or vice-versa.

7. Find the critical numbers for

8. Steps For Finding Increasing and Decreasing Intervals of a Function 1)Find the derivative 2)Find numbers that make the derivative equal to 0, and find numbers that make it undefined. These are the critical numbers. 3)Put the critical numbers and any x values where f is undefined on a number line, dividing the number line into sections. 4)Choose a number in each interval to test in the first derivative. Make a note of the sign you get. 5)Intervals that have first derivatives that are positive, are increasing, and intervals that have first derivatives that are negative, are decreasing.

9. Slide 2.1- 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 : Find the increasing and decreasing intervals for the funtion given by

10. Slide 2.1- 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 (continued): Find Derivative And set it = 0 These two critical values partition the number line into 3 intervals: A (– ∞, –1), B (–1, 2), and C (2, ∞).

11. Slide 2.1- 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1 (continued): 3 rd analyze the sign of f (x) in each interval. Test Valuex = –2x = 0x = 4 Sign of f (x) +–+ Result f is increasing on (–∞, –1] f is decreasing on [–1, 2] f is increasing on [2, ∞)

12. Find the intervals where f is increasing and decreasing. Since f ’(x) = 2x+5 it follows that f is increasing when 2x+5>0 or when x>-2.5 which is the interval

13. It is decreasing on It is increasing on

14. Find the intervals where the function is increasing and decreasing

15. A product has a profit function of for the production and sale of x units. Is the profit increasing or decreasing when 100 units have been sold?

16. Since C(x) is the cost for producing x units, the average cost for producing x units is C(x) divided by x. The marginal average cost would be found by taking the derivative.

17. Suppose a product has a cost function given by Find the average cost function. Over what interval is the average cost decreasing?

18. Assuming the graph is continuous (no break) at the point where the function changes from increasing to decreasing, that point is called a relative maximum point.

19. In the same manner, a relative minimum point occurs when the graph changes from decreasing to increasing.

20. Determine the critical numbers for each function and give the intervals where the function is increasing or decreasing.

21. Page 315 problems 46, 47, & 48