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1. 2 situations • It takes you 2 hours to drive the 140 miles to San Antonio. What can you say about your speed? • On the trip, you spot a cop in the median near Mathis. You nervously glance at your speedometer, and it says 78 mph. What does this mean compared to the previous computation?

2. 40 (Mathis) 0.75 A Graph 140 DistanceFrom Corpus (miles) 2 Time (hours)

3. What is the “Slope” of a Line?

4. y=f(x) (x1,f(x1)) What is slope?

5. y=f(x) (x1,f(x1)) What is slope?

6. y=f(x) (x1,f(x1)) What about the denominator?

7. Rate of Change Example Find the average rate of change of the function over the given interval.

8. Tangent Line Example Find the slope of the curve at the given point P and an equation of the tangent line at P.

9. Formal Definition Informally If there is no limit to how close f(x) gets to L as x gets close to a. (But you can’t demand we get L exactly.)

10. Consider this

11. New Notation

12. Discussion y x y=f(x)

13. Limit Theorems p. 65 No-brainers:

14. Limit Theorems p. 65 Nutshell version:

15. Limit Theorems p. 65 Nutshell version:

16. The Sandwich Theorem y=g(x) d c

17. Evaluating Limits: • Rule of thumb: • if you can just plug a into the numerator and denominator and those limits exist and the denominator is nonzero,you’re set • otherwise, you may have to try some algebraic tricks to find the limit There’s an exception to this rule.

18. When the “Plug” can shock you 2 -2 2 -2

19. Here, too! 2 -2 2 -2

20. Example: evaluate

21. Example: evaluate

22. Example: evaluate

23. Example: evaluate

24. Example: evaluate

25. Example: evaluate