1. Tangent Lines

2. EX 1.2 Estimating the Slope of a Curve Estimate the slope of y = sin x at x = 0.

3. The Length Of a Curve

4. EX 1.3 Estimating the Length of a Curve Estimate the length of the curve y = sin x for 0 ≤ x ≤ π

5. The Concept Of Limit Consider the functions Both functions are undefined at x = 2. These two functions look quite different in the vicinity of x = 2.

6. The Concept Of Limit (Cont’d) Consider the function The limit of f (x) as x approaches 2 from the left is 4. The limit of f (x) as x approaches 2 from the right is 4. The limit of f (x) as x approaches 2 is 4.

7. The Concept Of Limit (Cont’d) Consider the function

8. The Concept Of Limit (Cont’d) A limit exists if and only if both one-sided limits exist and are equal.

9. EX 2.1 Determining Limits Graphically Use the graph in the following f igure to determine

10. EX 2.2 A Limit Where Two Factors Cancel

11. EX 2.3 A Limit That Does Not Exist

12. EX 2.4 Approximating the Value of a Limit

13. EX 2.5 A Case Where One-Sided Limits Are Needed Note that

14. Computation Of Limits

15. Computation Of Limits (Cont’d) Application Example

16. EX 3.1 Finding the Limit of a Polynomial

17. EX 3.2 Finding the Limit of a Rational Function

18. Note that Right answer EX 3.3 Finding a Limit by Factoring

19. Computation Of Limits (Cont’d)

21. EX 3.5 Finding a Limit By Rationalizing

22. EX 3.6 Evaluating a Limit of an Inverse Trigonometric Function

23. EX 3.7 A Limit of a Product That Is Not the product of the Limits

24. EX 3.7 (Cont’d)

25. Computation Of Limits (Cont’d)

26. EX 3.8 Using the Squeeze Theorem to Verify the Value of a Limit

27. EX 3.8 (Cont’d) By graphing By squeeze theorem

28. EX 3.9 A Limit for a Piecewise-Defined Function

29. Concept of velocity For an object moving in a straight line, whose position at time t is given by the function f (t), The instantaneous velocity of that object at time t = a

30. Concept of velocity (Cont’d) EX 3.10 Evaluating a Limit Describing Velocity we can’t simply substitute h = 0

31. - discontinuous cases Continuity And Its Consequence (a) (d) (b) (c)

32. Continuity And Its Consequence (Cont’d)

33. EX 4.1 Finding Where a Rational Function Is Continuous

34. EX 4.2 Removing a Discontinuity Make the function from example 4.1 continuous everywhere by redefining it at a single point.

35. EX 4.3 Nonremovable Discontinuities There is no way to redefine either function at x = 0 to make it continuous there.

36. Continuity And Its Consequence (Cont’d)

37. EX 4.4 Continuity for a Rational Function f will be continuous at all x where the denominator is not zero. Think about why you didn’t see anything peculiar about the graph at x =-1.

38. Continuity And Its Consequence (Cont’d)

39. EX 4.5 Continuity for a Composite Function Determine where is continuous.

40. Continuity And Its Consequence (Cont’d)

41. EX 4.6 Intervals Where a Function Is Continuous

42. EX 4.7 Interval of Continuity for a Logarithm

43. EX 4.8 Federal Tax Table

44. EX 4.8 Federal Tax Table (Cont’d)

45. Continuity And Its Consequence (Cont’d)

46. EX 4.9 Finding Zeros by the Method of Bisections

47. Limits Involving Infinity EX 5.1 A Simple Limit Revisited Remark

48. EX 5.2 A Function Whose One-Sided Limits Are Both Infinite

49. EX 5.5 A Limit Involving a Trigonometric Function

50. Limits at Infinity

51. EX 5.6 Finding Horizontal Asymptotes the line y = 2 is a horizontal asymptote. Since So

53. EX 5.7 A Limit of a Quotient That Is Not the Quotient of the Limits

54. EX 5.8 Finding Slant Asymptotes y→as

55. EX 5.9 Two Limits of an Exponential Function Evaluate (a) (b) and

56. EX 5.10 Two Limits of an Inverse TrigonometricF unction Note the graph of In similar way,

57. EX 5.11 Finding the Size of an Animal’s Pupils f (x) :the diameter of pupils x : the intensity of light find the diameter of the pupils with (a) minimum and (b) maximum light.

58. EX 5.12 Finding the Limiting Velocity of a Falling Object Note So

59. Formal Definition Of Limits

60. EX 6.2 Verifying a Limit Show Find δ such that

61. Formal Definition Of Limits (Cont’d)

65. Limits And Loss-Of-Significance Errors EX 7.1 A Limit with Unusual Graphical and Numerical Behavior Incorrect calculated value ?!

66. Computer Representation of Real Numbers All computing devices have finite memory and consequently have limitations on the size mantissa and exponent that they can store. This is called finite precision. Most calculator carry a 14-digit mantissa and a 3-digit exponent.

67. EX 7.2 Computer Representation of a Rational Number Determine how is stored internally on a 10-digit computer and how is stored internally on a 14-digit computer.

68. EX 7.3 A Computer Subtraction of Two “Close” Numbers A computer with 14-digit mantissa = 0. Exact value

69. EX 7.4 Another Subtraction of Two “Close” Numbers Exact Value 1 1 = 10,000,000 Calculate the following result by a

70. EX 7.5 Avoid A Loss-of-Significance Error Compute f (5 × 10 4 ) with 14-digit mantissa for

71. EX 7.5 (Cont’d) Rewrite the function as follows:

72. EX 7.6 Loss-of-Significance Involving a Trigonometric Function

73. EX 7.7 A Loss-of-Significance Error Involving a Sum Directly Calculate:

74. EX 7.7(Cont’d)