1. Unit 1 Limits

2. Slide 2 1.1 Limits Limit – Assume that a function f(x) is defined for all x near c (in some open interval containing c) but not necessarily at c. We say that, “the limit of f(x) as x approaches c is equal to L” * c and L are

3. Slide 3 1.1 Limits Existence of a Limit – The functions value must the same value from – If a function is not defined on one side of c, the limit

4. Slide 4 1.1 Examples Find f(-2), 1)2)

5. Slide 5 1.1 Examples Find f(-2), 3)4)

6. Slide 6 1.1 Examples The graph of f(x) is shown below. Use the graph to evaluate each limit.

7. Slide 7 1.1 Examples Piecewise Function – A function obtained by together multiple functions on. Sketch the graph and then evaluate each limit.

8. Slide 8 1.1 Limits Evaluating limits without graphing 1) Substitute the value you are into x. *If you get a definite answer, then the limit equals. *If you get, we have more work to do. 2) the expression. * *Multiply by 3)

9. Slide 9 1.1 Examples Determine the value of each limit without graphing.

10. Slide 10 1.1 Examples Determine the value of each limit without graphing.

11. Slide 11 1.1 Examples Determine the value of each limit without graphing.

12. Slide 12 1.1 Examples Determine the value of each limit without graphing.

13. Slide 13 1.2 Examples Determine the value of each limit without graphing.

14. Slide 14 1.2 Examples Determine the value of each limit without graphing.

15. Slide 15 1.3 More Evaluating Limits Evaluating limits If you substitute and get, then the limit equals either or. Evaluate each limit.

16. Slide 16 1.3 Examples Evaluate each limit.

17. Slide 17 1.3 Examples Investigate left and right hand limits, then evaluate.

18. Slide 18 1.3 Examples 6) Sketch a possible graph of f(x).

19. Slide 19 1.4 Limits to Infinity Limit of f as x approaches infinity – means the limit of f as x moves increasingly far to the. Limit of f as x approaches negative infinity – means the limit of f as x moves increasingly far to the.

20. Slide 20 1.4 Examples Evaluate each limit without graphing.

21. Slide 21 1.4 Examples Evaluate each limit without graphing.

22. Slide 22 1.5 Vertical and Horizontal Asymptotes Vertical Asymptote If, then f(x) has a vertical asymptote at. Horizontal Asymptote If, then f(x) has a horizontal asymptote at.

23. Slide 23 1.5 Examples Give the equations of all asymptotes.

24. Slide 24 1.5 Examples Give the equations of all asymptotes.

25. Slide 25 1.5 Examples 5) Sketch a graph of f(x).

26. Slide 26 1.6 Continuity A function f is continuous at x = c if and only if the following three conditions are met. 1) 2) 3)

27. Slide 27 1.6 Continuity Four types of discontinuities. 1) 2) 3) 4)

28. Slide 28 1.6 Examples Find each point of discontinuity and identify the type.

29. Slide 29 1.6 Examples Find each point of discontinuity and identify the type.

30. Slide 30 1.6 Examples 5) Explain why f is not continuous at x = 2.

31. Slide 31 1.6 Examples 6) Find the value of k that will make f continuous at x = –3.

32. Slide 32 1.6 Examples 7) Find the value of c that will make f continuous at x = –1.

33. Slide 33 1.6 Continuity Intermediate Value Theorem If f is continuous on [a, b] and w is between f(a) and f(b), then there exists at least one c on [a, b] such that.

34. Slide 34 1.7 Rate of Change Average Rate of Change Amount of change divided by the time it takes. * 1) Find the average rate of change of on [5, 21].

35. Slide 35 1.7 Rate of Change Sketch the graph of f(x) = 2x² on [0, 5]. Calculate the average rate of change for each interval. 2) [2, 4] 3) [2, 3] 4) [2, 2.5] 5) [2, 2.01]

36. Slide 36 1.7 Rate of Change What were we really finding? * To calculate: find the limit of the slope of the secant line as the points get closer and closer together. Write this limit using the points (a, f(a)) and (a + h, f(a + h))

37. Slide 37 1.7 Rate of Change 6) Find the instantaneous rate of change of f at x = 5.

38. Slide 38 1.7 Rate of Change 7) Find the slope of the tangent line at x = a.

39. Slide 39 1.7 Rate of Change Normal Line * 8) Write equations of the tangent and normal lines at x = -1.

40. Slide 40 1.8 Definition of the Derivative We use the derivative to help us find the slope of the tangent line. Notation: Prime: : 1)If f(6) = -5 and f’(6) = 2, write equations of the tangent and normal lines at x = 6.

41. Slide 41 1.8 Definition of the Derivative Definition of the Derivative: The derivative of a function f is another function f’ whose value at x is provided this limit exists. Alternate Form:

42. Slide 42 1.8 Examples f(x) = x² + 3x - 1 2)Use the definition of the derivative to find f’(-3). 3)Use the alternate form of the derivation to find f’(2).

43. Slide 43 1.8 Examples y = 3x² - 10x + 2 4) Find y’ at x = a. 5)Find y’ at x = 3.

44. Slide 44 1.8 Examples 6) Write equations of the tangent and normal lines at x = 3. 7) The graph of f(x) is shown below. Sketch the graph of f’(x).