2.1 Rates of Change & Limits

1. 2.1 Rates of Change & Limits

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

3. Instantaneous Speed • 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. Instantaneous Speed 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 = 16t2 We cannot use this formula to calculate the speed at the exact instant t = 2because 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. Instantaneous Speed What is happening? As Δt gets smaller, the rock’s average speed gets closer to 64 ft/sec.

6. Instantaneous Speed Algebraically:

7. Instantaneous Speed Algebraically: Now, when Δt is 0, our average speed is 64 ft/sec

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

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

13. Example • Use and and the properties of limits to find the following limits: a.) b.)

14. Evaluating Limits • If fis 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. Limiting Techniques: 2.) Using Properties of Limits (product rule) Ex: Find the limit:

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

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

18. Limiting Techniques 5.) Use a table or graph What happens if we just substitute in the limit? Ex: Find 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. 6. Sandwich (Squeeze) Theorem • 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 h(x) • then, f(x) g(x)

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

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

23. Existence of a Limit • 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. 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:

25. 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:

26. 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:

27. Suggested HW Probs: • Section 2.1 (#7, 11, 15, 19, 21, 23, 27, 31-36, 37, 43, 49, 63)