1. 2.1 Rates of Change and Limits Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2007 Grand Teton National Park, Wyoming

2. Chapter 2 Section 1 Rates of Change and Limits Powerpoint Reflections

3. Suppose you drive 200 miles, and it takes you 4 hours. Then your average speed is: If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.

4. A rock falls from a high cliff. The position of the rock is given by: After 2 seconds: average speed: What is the instantaneous speed at 2 seconds?

5. for some very small change in t where h = some very small change in t We can use the TI-89 to evaluate this expression for smaller and smaller values of h.

6. 1 80 0.165.6.0164.16.00164.016.0001 64.0016.0000164.0002 We can see that the velocity approaches 64 ft/sec as h becomes very small. We say that the velocity has a limiting value of 64 as h approaches zero. (Note that h never actually becomes zero.)

7. What is a limiting value? Or limit? When does a limit exist? How do you evaluate limits?

8. The limit as h approaches zero: 0 Since the 16 is unchanged as h approaches zero, we can factor 16 out.

9. Consider: What happens as x approaches zero? Graphically: WINDOW Y= GRAPH

10. Looks like y=1

11. Numerically: TblSet You can scroll down to see more values. TABLE

12. You can scroll down to see more values. TABLE It appears that the limit of as x approaches zero is 1

13. Limit notation: “The limit of f of x as x approaches c is L.” So:

14. The limit of a function refers to the value that the function approaches, not the actual value (if any). not 1

15. Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. (See your book for details.) For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only.

16. 1234 1 2 At x=1:left hand limit right hand limit value of the function does not exist because the left and right hand limits do not match!

17. At x=2:left hand limit right hand limit value of the function because the left and right hand limits match. 1234 1 2

18. At x=3:left hand limit right hand limit value of the function because the left and right hand limits match. 1234 1 2

19. The Sandwich Theorem: Show that: The maximum value of sine is 1, soThe minimum value of sine is -1, soSo:

20. By the sandwich theorem: Y= WINDOW

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