1. 2.1 Rates of Change and Limits Average and Instantaneous Speed –A moving body’s average speed during an interval of time is found by dividing the distance covered by the elapsed time. The unit of measure is length per unit time – ex. Miles per hour, etc.

2. Finding an Average Speed A rock breaks loose from the top of a tall cliff. What is its average speed during the first 2 seconds of fall? Experiments show that objects dropped from rest to free fall will fall y = 16t² feet in the first t seconds. For the first 2 seconds of, we change t = 0 to t = 2.

3. Finding an Instantaneous Speed Find the speed of the rock in example 1 at the instant t = 2. Since we cannot use h = 0 because it will give us an undefined answer, evaluate the formula at values close to 0. See the table 2.1 on p. 60 in your textbook. Notice, the average speed approaches the limiting value of 64 ft/sec.

4. Average Speeds over Short Time Intervals Starting at t = 2

5. Finding an Instantaneous Speed Confirm algebraically: So, we can see why the average speed has the limiting value of 64 + 16(0) = 64 ft/sec as h approaches 0.

6. Limits Most limits of interest in the real world can be viewed as numerical limits of values of functions. A calculator can suggest the limits, and calculus can give the mathematics for confirming the limits analytically.

7. Properties of Limits

9. Using Properties of Limits Use the observations and and the properties of limits to find the following limits. a.b.

10. Polynomial and Rational Functions

11. Using Theorem 2 a. b.

12. Using the Product Rule Determine

13. Exploring a Nonexistent Limit Use a graph to show that does not exist. Notice that the denominator is 0 when x is replaced by 2, so we cannot use substitution. The graph suggests that as x approaches 2 from either side, the absolute values get very large. This suggests that the limit does not exist.

14. One-Sided and Two-Sided Limits Limits can approach a function from opposite sides. Right-hand limit – limit approaches from the right side. Left-hand limit – limit approaches from the left side.

15. One-sided and Two-sided Limits

16. Exploring Right- and Left-Hand Limits

18. Sandwich Theorem

19. Using the Sandwich Theorem Show that

20. Homework!!!!! Textbook p. 66 – 67 #1, 2, 5, 6, 7 – 14, 20 – 28 even, 37, 40 – 44 even.