Exploring Rates of Change and Limits in Mathematics

1. Rates of Change and Limits 2.1

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.

4. Average vs. Instantaneous Velocity A rock breaks loose from the top of a tall cliff. What is the average velocity during the first 2 seconds of fall? Find the instantaneous velocity at t = 2 seconds.

5. 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? How can we find this?

6. for some very small change in t where h = some very small change in t We can use the TI-83/TI84 to evaluate this expression for smaller and smaller values of h.

7. 1 80 0.1 65.6 .01 64.16 .001 64.016 64.0016 .0001 .00001 64.0002 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.)

8. 0 The limit as h approaches zero:

9. Definition of Limit continued

10. Definition of Limit continued

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

12. If we graph , it appears that

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

16. Example Properties ofLimits

17. Polynomial and Rational Functions

18. Example Limits

19. Examples

20. Examples

21. Evaluating Limits As with polynomials, limits of many familiar functions can be found by substitution at points where they are defined. This includes trigonometric functions, exponential and logarithmic functions, and composites of these functions.

22. Example Limits

23. Example Limits [-6,6] by [-10,10]

24. One-Sided and Two-Sided Limits

25. One Sided Limits

26. does not exist because the left and right hand limits do not match! left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=1:

27. because the left and right hand limits match. left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=2:

28. because the left and right hand limits match. left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=3:

29. Example One-Sided and Two-Sided Limits Find the following limits from the given graph. 4 o 3 1 2

30. Sandwich Theorem

31. Sandwich Theorem

32. Example

33. Show that: The maximum value of sine is 1, so The minimum value of sine is -1, so So: The Sandwich Theorem:

38. The TI-83 contains the command , but it is important that you understand the function rather than just entering it in your calculator. “Step functions” are sometimes used to describe real-life situations. Our book refers to one such function: This is the Greatest Integer Function.

39. Greatest Integer Function:

40. Greatest Integer Function:

41. Greatest Integer Function:

42. Greatest Integer Function:

43. Some books use Greatest Integer Function: The greatest integer function is also called the floor function. The notation for the floor function is: