2. 2.1 Rates of Change and Limits

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

7. Secant Line y = f(x)

8. Solution:

9. Example: The function gives the height (in feet) of a ball thrown straight up as a function of time, t (in seconds). (a) Find the average rate of change of the height of the ball between 1 and t seconds. Solution: (b) Use the result found in part (a) to find the average rate of change of the height of the ball between 1 and 2 seconds. (a)

11. If t = 2, the average rate of change between 1 second and 2 seconds is: - 4(4(2) - 21) = 52 ft/second. The average rate of change between 1 second and t seconds is: -4(4t - 21) (b)

12. Example: How does the the function f(x)= x 2 -x+2 behave near x=2. x 2 f(x) 1.0 2.000000 3.0 8.000000 1.5 2.750000 2.5 5.750000 1.8 3.440000 2.2 4.640000 1.9 3.710000 2.1 4.310000 1.95 3.852500 2.05 4.152500 1.99 3.970100 2.01 4.030100 1.995 3.985025 2.001 4.003001 Before we give a definition of limit, let us look at the following examples. Solution:

13. We see that when x is close to 2 (x>2 or x<2), f(x) is close to 4. Then we can say that the limit of the function f(x) = x 2 -x+2 as x approaches 2 is equal to 4 And we write this as the notation :

14. Value of x f(x)=(x 2 –1)/(x – 1) 0.991.99 1.012.01 0.9991.999 1.0012.001 0.9999991.999999 1.0000012.000001 y 12x 1 2 3 0 Solution: The function f(x) is defined for all x except x=1

15. For any x=1 we simplify the formula of the function by So the graph of the function f is the line y=x+1 with the removable point (1,2) From the table we say that f(x) approaches the limit 2 as x approaches 1 And we write this as the notation

16. Definition: Informal Definition of limit Let f(x) be defined on an open interval about a, except possibly at a itself. If we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a but not equal to a. And we say “the limit of f(x) equals L” as x approaches a. And we write

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

18. 2 4 6 8 24 6 (3, 7) y x Solution: From graph we get

19. Try This Find: f(0)is undefined; and 2 Solution:

20. Try This 2 1 Solution: f(0)is undefined; and Find:

21. Try This Find the limit of f(x) as x approaches 3 where f is defined by: Solution:

22. (2, 7) (2, -1) does not exist, there is a jump at x =2. Solution:

23. Example: Consider Solution: The limit is 3

24. Example: Discuss the behavior of the following functions as x  0 The unit step function, it values jump at x=0. For negative values of x close to 0, U(x)=0. For positive values of x close to 0, U(x)=1. jump infinite oscillating

25. The values of g grow too large in the absolute values As x  0 and do not stay close to any real number The function’s values oscillate between +1 and -1 in every interval containing 0.

26. oscillating The function’s values oscillate between +1 and -1 in every interval containing 0. Example: Discuss the behavior of the following functions as x  0 Solution: