2. Average Speed Example: Suppose you drive 200 miles in 4 hours. What is your average speed? Since d = rt, = 50 mph

3.  The moment you look at your speedometer, you see your instantaneous speed. Example A rock breaks loose from the top of a tall cliff. What is the speed of the rock at 2 seconds? We can calculate the average speed of the rock from 2 seconds to a time slightly later than 2 seconds (t = 2 + Δt, where Δt is a slight change in time.)

4. Example A rock breaks loose from the top of a tall cliff. What is the speed of the rock at 2 seconds? Free fall equation: y = 16t 2 We cannot use this formula to calculate the speed at the exact instant t = 2 because that would require letting Δt = 0, and that would give 0/0. However, we can look at what is happening when Δt is close to 0.

5. Length of Δt (seconds)Average Speed (ft/sec) 180 0.165.6 0.0164.16 0.00164.016 0.000164.0016 0.0000164.00016 What is happening? As Δt gets smaller, the rock’s average speed gets closer to 64 ft/sec.

6. Algebraically:

7. Now, when Δt is 0, our average speed is 64 ft/sec

8.  Let f be a function defined on a open interval containing a, except possibly at a itself. Then, there exists a such that WHAT THE CRAP??????

9.  The function f has a limit L as x approaches c if any positive number (ε), there is a positive number σ such that Still, WHAT THE CRAP?????? We read, “The limit as x approaches c of a function is L.”

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

11. Properties of Limits If L, M, c, and k are real numbers and and then Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. 1.) Sum Rule: 2.) Difference Rule: 3.) Product Rule:

12. Properties of Limits 4.) Constant Multiple Rule: 5.) Quotient Rule: 6.) Power Rule:

13.  Useand and the properties of limits to find the following limits: a.) b.)

14.  If f is a continuous function on an open interval containing the number a, then (In other words, you can many times substitute the number x is approaching into the function to find the limit.) Techniques for Evaluating Limits: 1.) Substituting Directly Ex: Find the limit:

15. 2.) Using Properties of Limits Ex: Find the limit: (product rule)

16. 3.) Factoring & Simplifying Ex: Find the limit: What happens if we just substitute in the limit? When something like this happens, we need to see if we can factor & simplify! HOLY COWCULUS!!!

17. 4.) Using the conjugate Ex: Find the limit: What happens if we just substitute in the limit? We must simplify again.

18. 5.) Use a table or graph Ex: Find the limit: What happens if we just substitute in the limit? As x approaches 0, you can see that the graph of f(x) approaches 3. Therefore the limit is 3. (You can also see this in your table.)

19.  If f, g, and h are functions defined on some open interval containing a such that g(x) ≤ f(x) ≤ h(x) for all x in the interval except at possibly at a itself, and then, h(x) g(x) f(x)

20. Ex: Find the limit: sin oscillates between -1 and 1, so Now, let’s get the problem to look like the one given.

21. Ex: Find the limit: Therefore, by the Sandwich Theorem,

23.  In order for a limit to exist, the limit from the left must approach the same value as the limit from the right. If then and are called one-sided limits

24. 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!

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

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

27.  Section 2.1 (#7, 11, 15, 19, 21, 23, 27, 31- 36, 37, 43, 49, 63)