1. 3 - 1 © 2012 Pearson Education, Inc.. All rights reserved. Chapter 3 The Derivative

2. 3 - 2 © 2012 Pearson Education, Inc.. All rights reserved. Section 3.1 Limits

3. 3 - 3 © 2012 Pearson Education, Inc.. All rights reserved.

4. 3 - 4 © 2012 Pearson Education, Inc.. All rights reserved. Figure 2

5. 3 - 5 © 2012 Pearson Education, Inc.. All rights reserved. Notation *from Spivak’s Calculus

6. 3 - 6 © 2012 Pearson Education, Inc.. All rights reserved. Your Turn Solution: since The numerator also approaches 0 as x approaches −3, and 0/0 is meaningless. For x ≠ − 3 we can, however, simplify the function by rewriting the fraction as Now

7. 3 - 7 © 2012 Pearson Education, Inc.. All rights reserved. Left and Right

8. 3 - 8 © 2012 Pearson Education, Inc.. All rights reserved. Left and Right  What can you say about lim f(x) as x  10 if  lim f(x) as x  10 - (from the left) is 5  lim f(x) as x  10 + (from the right) is 5 ?

9. 3 - 9 © 2012 Pearson Education, Inc.. All rights reserved. infinity  lim 1/x as x  infinity ?

10. 3 - 10 © 2012 Pearson Education, Inc.. All rights reserved. Two Tools with Limits

11. 3 - 11 © 2012 Pearson Education, Inc.. All rights reserved. Your Turn 5 Suppose and find Solution:

12. 3 - 12 © 2012 Pearson Education, Inc.. All rights reserved.

13. 3 - 13 © 2012 Pearson Education, Inc.. All rights reserved. Your Turn 8 Solution: Here, the highest power of x is x 2, which is used to divide each term in the numerator and denominator.

14. 3 - 14 © 2012 Pearson Education, Inc.. All rights reserved. Your Turn

15. 3 - 15 © 2012 Pearson Education, Inc.. All rights reserved. Section 3.2 Continuity

16. 3 - 16 © 2012 Pearson Education, Inc.. All rights reserved. Figure 14

17. 3 - 17 © 2012 Pearson Education, Inc.. All rights reserved.

18. 3 - 18 © 2012 Pearson Education, Inc.. All rights reserved. Figure 15 - 16

19. 3 - 19 © 2012 Pearson Education, Inc.. All rights reserved. Figure 17

20. 3 - 20 © 2012 Pearson Education, Inc.. All rights reserved. Figure 18

21. 3 - 21 © 2012 Pearson Education, Inc.. All rights reserved. Figure 19

22. 3 - 22 © 2012 Pearson Education, Inc.. All rights reserved.

23. 3 - 23 © 2012 Pearson Education, Inc.. All rights reserved.

24. 3 - 24 © 2012 Pearson Education, Inc.. All rights reserved.

25. 3 - 25 © 2012 Pearson Education, Inc.. All rights reserved. Your Turn 1 Find all values x = a where the function is discontinuous. Solution: This root function is discontinuous wherever the radicand is negative. There is a discontinuity when 5x + 3 < 0

26. 3 - 26 © 2012 Pearson Education, Inc.. All rights reserved. Your Turn 2 Find all values of x where the piecewise function is discontinuous. Solution: Since each piece of this function is a polynomial, the only x-values where f might be discontinuous here are 0 and 3. We investigate at x = 0 first. From the left, where x-values are less than 0, From the right, where x-values are greater than 0 Continued

27. 3 - 27 © 2012 Pearson Education, Inc.. All rights reserved. Your Turn 2 Continued Because the limit does not exist, so f is discontinuous at x = 0 regardless of the value of f(0). Now let us investigate at x = 3. Thus, f is continuous at x = 3.

28. 3 - 28 © 2012 Pearson Education, Inc.. All rights reserved. Figure 20

29. 3 - 29 © 2012 Pearson Education, Inc.. All rights reserved. Figure 21

30. 3 - 30 © 2012 Pearson Education, Inc.. All rights reserved. Figure 22

31. 3 - 31 © 2012 Pearson Education, Inc.. All rights reserved. Section 3.3 Rates of Change

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33. 3 - 33 © 2012 Pearson Education, Inc.. All rights reserved. Figure 23

34. 3 - 34 © 2012 Pearson Education, Inc.. All rights reserved. Figure 24

35. 3 - 35 © 2012 Pearson Education, Inc.. All rights reserved.

36. 3 - 36 © 2012 Pearson Education, Inc.. All rights reserved. Figure 25

37. 3 - 37 © 2012 Pearson Education, Inc.. All rights reserved.

38. 3 - 38 © 2012 Pearson Education, Inc.. All rights reserved. Figure 26

39. 3 - 39 © 2012 Pearson Education, Inc.. All rights reserved. Section 3.4 Definition of the Derivative

40. 3 - 40 © 2012 Pearson Education, Inc.. All rights reserved. Figure 27

41. 3 - 41 © 2012 Pearson Education, Inc.. All rights reserved. Figure 28

42. 3 - 42 © 2012 Pearson Education, Inc.. All rights reserved. Figure 29

43. 3 - 43 © 2012 Pearson Education, Inc.. All rights reserved. Figure 30

44. 3 - 44 © 2012 Pearson Education, Inc.. All rights reserved.

45. 3 - 45 © 2012 Pearson Education, Inc.. All rights reserved. Figure 31

46. 3 - 46 © 2012 Pearson Education, Inc.. All rights reserved. Figure 32

47. 3 - 47 © 2012 Pearson Education, Inc.. All rights reserved. Figure 33 - 34

48. 3 - 48 © 2012 Pearson Education, Inc.. All rights reserved. Figure 35

49. 3 - 49 © 2012 Pearson Education, Inc.. All rights reserved. Figure 36 - 37

50. 3 - 50 © 2012 Pearson Education, Inc.. All rights reserved.

51. 3 - 51 © 2012 Pearson Education, Inc.. All rights reserved.

52. 3 - 52 © 2012 Pearson Education, Inc.. All rights reserved.

53. 3 - 53 © 2012 Pearson Education, Inc.. All rights reserved. Figure 38

54. 3 - 54 © 2012 Pearson Education, Inc.. All rights reserved.

55. 3 - 55 © 2012 Pearson Education, Inc.. All rights reserved.

56. 3 - 56 © 2012 Pearson Education, Inc.. All rights reserved. Figure 39

57. 3 - 57 © 2012 Pearson Education, Inc.. All rights reserved.

58. 3 - 58 © 2012 Pearson Education, Inc.. All rights reserved. Figure 40

59. 3 - 59 © 2012 Pearson Education, Inc.. All rights reserved. Figure 41 - 42

60. 3 - 60 © 2012 Pearson Education, Inc.. All rights reserved. Figure 43

61. 3 - 61 © 2012 Pearson Education, Inc.. All rights reserved.

62. 3 - 62 © 2012 Pearson Education, Inc.. All rights reserved. Figure 44

63. 3 - 63 © 2012 Pearson Education, Inc.. All rights reserved. Section 3.5 Graphical Differentiation

64. 3 - 64 © 2012 Pearson Education, Inc.. All rights reserved. Figure 45

65. 3 - 65 © 2012 Pearson Education, Inc.. All rights reserved. Figure 46

66. 3 - 66 © 2012 Pearson Education, Inc.. All rights reserved. Figure 47 - 48

67. 3 - 67 © 2012 Pearson Education, Inc.. All rights reserved. Figure 49

68. 3 - 68 © 2012 Pearson Education, Inc.. All rights reserved. Figure 50

69. 3 - 69 © 2012 Pearson Education, Inc.. All rights reserved. Figure 51

70. 3 - 70 © 2012 Pearson Education, Inc.. All rights reserved. Figure 52

71. 3 - 71 © 2012 Pearson Education, Inc.. All rights reserved. Figure 53

72. 3 - 72 © 2012 Pearson Education, Inc.. All rights reserved. Figure 54

73. 3 - 73 © 2012 Pearson Education, Inc.. All rights reserved. Chapter 3 Extended Application: A Model for Drugs Administered Intravenously

74. 3 - 74 © 2012 Pearson Education, Inc.. All rights reserved. Figure 55

75. 3 - 75 © 2012 Pearson Education, Inc.. All rights reserved. Figure 56

76. 3 - 76 © 2012 Pearson Education, Inc.. All rights reserved. Figure 57