1. 2.1 Rates of Change and Limits

2. 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.

3. 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?

4. 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.

5. 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.)

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

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

8. Looks like y=1

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

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

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

12. 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.

13. 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!

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

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

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

17. By the sandwich theorem: Y= WINDOW

18. 