1. CHAPTER 2 UEQ: WHAT IS THE RELATIONSHIP BETWEEN LIMITS, GRAPHS, FUNCTIONS AND CONTINUITY?

2. SECTION 0 DAY 1: FACTORING REVIEW EQ: How do you factor all types of equations?

3. RULES OF FACTORING

4. Find the Monomial Factor Examples: y 3 + 3y4x + 87xy 2 + 28x 3 y 2

5. Examples: 6x 2 – 5x – 253x 2 – 10x + 8 x – 3x 2 + 4

6. Examples: 3x 2 y – 27yx 4 - 81

7. Examples: 6x 2 – 5x – 253x 2 – 10x + 8

8. (x – 1) 2 – 4(x -1) – 5 = 0 Let u = ________

9. HOMEWORK Page(s): worksheet 2.1a

10. SECTION 1 DAY 1: RATES OF CHANGE AND LIMITS EQ: What is the difference between average and instantaneous rates of change?

11. LIMIT—A VALUE THAT A FUNCTION APPROACHES AT A GIVEN POINT/X-VALUE Sketch f(x) = -2x 2 + 8x – 4 Then evaluate the limit as x 3 Complete the Square y + 4 = -2(x 2 – 4 ) y + 4 - 8 = -2(x 2 – 4 + 4) y – 4 = -2(x – 2) 2 y = -2(x – 2) 2 + 4 Vertex (-2, 4) Opens down Using the Quad. Formula it crosses the x at 3.4 and.6 XYXY 24 2.53.5 2.93.1 Y 4 Y -4 Y 4 3.5 Y -4 -.5 Y 4 3.5 2.38 Y -4 -.5 1.58 Are these close enough to determine the value of the limit? 2.992.043.011.96 Why did we do this when we could have simply substituted the 3?

12. Simple definitions: Continuity—a graph that can be drawn without lifting your pencil Discontinuity—a graph for which you MUST lift your pencil to complete the drawing If f(x) is continuous you may substitute the exact value to determine the limit If f(x) is discontinuous you must use the chart.

13. RULES FOR LIMITS: #1 #1 The limit of a constant times f(x) = the constant time the limit of f(x)

14. HOMEWORK Page(s): worksheet 2.1b

15. SECTION 1 DAY 2: RATES OF CHANGE AND LIMITS EQ: What is the difference between average and instantaneous rates of change?

16. WHAT DO YOU DO IF YOU DON’T KNOW IF THE FUNCTION IS CONTINUOUS OR DISCONTINUOUS?

17. RULE: Why:

18. EXAMPLES: This is removeable discontinuity (causes a hole in the graph)

19. JUMP DISCONTINUITY

20. RULE

21. HOMEWORK Page(s): 62 – 63 18-36 Even + 46, & 48

22. SECTION 1 DAY 3: RATES OF CHANGE AND LIMITS EQ: What is the difference between average and instantaneous rates of change?

23. AVERAGE AND INSTANTANEOUS SPEED

24. Instantaneous Speed –the speed at one point (the tangent line at the given point)  If we allow h to be a small changes in x, then we can approximate the slope at a given point using the limit  Find the slope of the tangent line when x = 3

25. FORMULA FOR INSTANTANEOUS SPEED

26. EXAMPLES A projectile is shot upwards with a speed of 100 m/sec and moves according to the law y = 100t – 5t 2 a) What is the average speedb) What is the instantaneous speed during the 1 st 2 seconds? at 1 second?

27. HOMEWORK Page(s): 64 57 & 58

28. SECTION 1 DAY 5: RATES OF CHANGE AND LIMITS EQ: What is the difference between average and instantaneous rates of change?

29. ALTERNATE VIEWS OF LIMITS—GEOMETRIC In the following figure, the unit circle is centered at O. BQ is a vertical tangent line and the length of BP is the same as the length BQ. The line QP extended cuts the horizontal axis at E. What happens to the E as Q approaches B? Q P B E O

30. ALTERNATE VIEWS OF LIMITS—ALGEBRAIC

31. REVIEW SHEET, THEN QUIZ