1. Chapter 2 – Limits and Continuity

2. 2.1 Rates of Change and Limits
Example: A rock breaks loose from the top of a tall cliff. What is its average speed during the first two seconds of fall?

3. Example: Find the speed of the rock in the last example at the instant t = 2.
Def of a Limit

6. Properties of Limits

9. Polynomial and Rational Functions

11. Example: Calculate the following limits.

12. Ex: Use a graph to show that the following function does not exist.

13. One-Sided and Two-Sided Limits

14. Let’s examine the following graph to further explore right and left hand limits.

15. Find the following limits from the given graph.4 o 1 2 3

16. Sandwich Theorem

17. Ex: Show that

18. Homework:
p.66 (1- 63) odd

19. Explore problem number 75 on page 68

20. 2.2 Limits Involving Infinity

21. [-6,6] by [-5,5]
Ex: Find

22. Same properties hold for infinite limits as before.
Ex: Find

23. Example: Let and Show that while f and g are quite different for numerically small values of x, they are virtually identical for x large.

25. Let . Show that g(x) = x is a right end behavior model for f while
is a left end behavior model for f.

26. Example “Seeing” Limits as x→±∞

27. Homework:
p. 76 (1 – 55) odd

28. 2.3 Continuity o

29. Continuity at a Point
If a function f is not continuous at a point c , we say that f is discontinuous at c and c is a point of discontinuity of f.
Note that c need not be in the domain of f.

40. Homework:
p. 84 (1-43) odd

41. 2.4 Rates of Change and Tangent Lines
Example: Finding Average Rate of Change (Review)
Find the average rate of change of f(x) = x3 - x over the interval [1,3].
Page 88

44. Let f(x) = 1/x
Find the slope of the curve at x = a.
Where does the slope equal -1/4?
What happens to the tangent to the curve at the point (a,1/a) for different values of a?

45. Def: The normal line to a curve at a point is the line perpendicular to the tangent at that point.
Ex: Write an equation for the normal to the curve f(x) = 4 – x2 at x = 1.

47. Homework:
p. 92 (1 – 31)odd