1. The Derivative Objective: We will explore tangent lines, velocity, and general rates of change and explore their relationships.

2. Tangent Lines We are going to look at a secant line (PQ) to a curve and talk about its slope. This is defined as:

3. Definition 2.1.1 We will now look at what happens as point Q approaches point P. We will look at this as a limit; the limit as x approaches x 0. This will be defined as:

4. Definition 2.1.1 We will now look at what happens as point Q approaches point P. We will look at this as a limit, the limit as x approaches x 0. This will be defined as: and the equation of the tangent line to the curve at the point is:

5. Example 1 Use definition 2.1.1 to find an equation for the tangent line to the parabola at the point P(1, 1).

6. Example 1 Use definition 2.1.1 to find an equation for the tangent line to the parabola at the point P(1, 1). P(1, 1) & Q(2, 4) P(1, 1) & Q(1.5, 2.25) P(1, 1) & Q(1.1, 1.21) P(1, 1) & Q(1.01, 1.0201)

7. Example 1 Use definition 2.1.1 to find an equation for the tangent line to the parabola at the point P(1, 1). You can also use a point to the right and a point to the left of the given point to estimate the slope of the tangent line. P(2, 4) & Q(0, 0)

8. Example 1 We have the point, we need the slope. We will use our definition of the slope of a tangent line to find this by first substituting a 1 for.

9. Example 1 Now we use the fact that and find f(1) and substitute where necessary.

10. Example 1 Now, we find the limit.

11. Example 1 We now have the point and the slope, so the equation of the tangent line is:

12. The Difference Quotient There is another formula that is commonly used to find the slope of a tangent line. This is called the Difference Quotient and we define h as the difference between and. The equation now becomes:

13. Example 2 We will use this formula to find the slope of the tangent line in example 1. Again, we start by replacing with a 1.

14. Example 2 We again use to evaluate

15. Example 2 Do the algebra to find the limit.

16. Example 3 Find the equation of the tangent line to the curve y=2/x at the point (2, 1) on this curve.

17. Example 3 Find the equation of the tangent line to the curve y=2/x at the point (2, 1) on this curve.

18. Example 3 Find the equation of the tangent line to the curve y=2/x at the point (2, 1) on this curve.

19. Example 3 Find the equation of the tangent line to the curve y=2/x at the point (2, 1) on this curve.

20. Example 3 Find the equation of the tangent line to the curve y=2/x at the point (2, 1) on this curve.

21. Example 3 Find the equation of the tangent line to the curve y=2/x at the point (2, 1) on this curve. We will use a different equation this time.

22. Example 4 We have been finding the slope of the tangent line at a specific point. We will now find the slope of the tangent line at a general point.

23. Example 4 Find the slopes of the tangent lines to the curve at

24. Example 4 Find the slopes of the tangent lines to the curve at

25. Example 4 Find the slopes of the tangent lines to the curve at

26. Example 4 Find the slopes of the tangent lines to the curve at

27. Example 4 Find the slopes of the tangent lines to the curve at

28. Example 4 Find the slopes of the tangent lines to the curve at

29. Example 4 The slopes of the tangent lines are: at x = 1 at x = 4 at x = 9

30. Velocity When we talk about the motion of an object, we want its speed and direction. Together, we call this velocity. Movement to the right or up is considered positive velocity and movement to the left or down is considered negative velocity. We will explore these meanings with a position vs. time curve, with the horizontal axis being time (t) and the vertical axis position (s). The movement of the particle will be called Rectilinear Motion.

31. Position vs. Time Curve We will look at two typical position vs. time curves.

32. Position vs. Time Curve We will look at two typical position vs. time curves. The first is for a car that starts at the origin and moves only in the positive direction. Movement to the right is considered positive and movement to the left is considered negative. In this case s increases as t increases.

33. Position vs. Time Curve We will look at two typical position vs. time curves. The second is for a ball that is thrown straight up in the positive direction and falls straight down in the negative direction.

34. Displacement and Average Velocity The key to describing the velocity of a particle in rectilinear motion is the notion of displacement, or change in position. This differs from distance traveled. Since movement to the right is positive and movement to the left is negative, you could travel 10 units to the right and then 8 units to the left and your displacement would be +2 and your distance traveled would be 18.

35. Average Velocity vs. Average Speed We will define average velocity as: And we will define average speed as :

36. Example 5 Suppose that is the position function of a particle, where s is in meters and t in seconds. Find the displacements and average velocities of the particle over the time intervals: [0,1] and [1,3].

37. Example 5 Suppose that is the position function of a particle, where s is in meters and t in seconds. Find the displacements and average velocities of the particle over the time intervals: [0,1] and [1,3]. f(0) = 1, f(1) = 2, displacement is 1. f(1) = 2, f(3) = -8 displacement is -10.

38. Instantaneous Velocity Instead of looking at velocity over a time interval, we will now look at velocity at one point and we will call this instantaneous velocity, which describes the behavior of the particle at a specific instant in time.

39. Example 6 Consider the particle in Ex. 5, whose position function is. Find the particle’s instantaneous velocity at time t = 2s.

40. Example 6 Consider the particle in Ex. 5, whose position function is. Find the particle’s instantaneous velocity at time t = 2s. As a first approximation to the particle’s instantaneous velocity at time t = 2, let us recall the average velocity from t = 2 to t = 3 is -5 m/s. To improve this approximation we will compute the average velocity over a succession of smaller and smaller time intervals.

41. Example 6 Consider the particle in Ex. 5, whose position function is. Find the particle’s instantaneous velocity at time t = 2s. The average velocities in this table appear to be approaching a limit of -3 m/s. Let’s confirm this.

42. Example 6 Consider the particle in Ex. 5, whose position function is. Find the particle’s instantaneous velocity at time t = 2s. The average velocities in this table appear to be approaching a limit of -3 m/s. Let’s confirm this.

43. Example 6 Consider the particle in Ex. 5, whose position function is. Find the particle’s instantaneous velocity at time t = 2s. The average velocities in this table appear to be approaching a limit of -3 m/s. Let’s confirm this.

44. Instantaneous Velocity We define instantaneous velocity as: And instantaneous speed as:

45. Slopes and Rates of Change Velocity can be viewed as rate of change- the rate of change of position with respect to time. Rates of change occur in other applications as well.

46. Slopes and Rates of Change Velocity can be viewed as rate of change- the rate of change of position with respect to time. Rates of change occur in other applications as well. A microbiologist might be interested in the rate of change at which the number of bacteria in a colony changes with time.

47. Slopes and Rates of Change Velocity can be viewed as rate of change- the rate of change of position with respect to time. Rates of change occur in other applications as well. A microbiologist might be interested in the rate of change at which the number of bacteria in a colony changes with time. An economist might be interested in the rate of change at which production cost changes with the quantity of a product that is manufactured.

48. Example 7 Find the rate of change of y with respect to x.

49. Example 7 Find the rate of change of y with respect to x. The rate of change for the equation on the left is 2, and the rate of change for the equation on the right is -5.

50. Average rate of change The average rate of change is the same thing as the slope of the secant line, so we define it as: or

51. Instantaneous rate of change The instantaneous rate of change is the same thing as the slope of the tangent line, so we define it as: or

52. Example 9 Let 1)Find the average rate of change of y with respect to x over the interval [3,5] 2)Find the instantaneous rate of change of y with respect to x when x = - 4.

53. Example 9 Let 1)Find the average rate of change of y with respect to x over the interval [3,5]

54. Example 9 Let 1)Find the instantaneous rate of change of y with respect to x when x = -4

55. Example 10 The limiting factor in athletic endurance is cardiac output, that is, the volume of blood that the heart can pump per unit of time during an athletic competition. The figure shows a stress-test graph of cardiac output in liters of blood vs workload for 1 minute of lifting.

56. Example 10 Use the secant line shown on the graph below to estimate the average rate of change of cardiac output with respect to workload as the workload increases from 300 to 1200.

57. Example 10 Use the secant line shown on the graph below to estimate the average rate of change of cardiac output with respect to workload as the workload increases from 300 to 1200.

58. Example 10 Use the tangent line on the graph below to estimate the instantaneous rate of change of cardiac output with respect to workload at the point where the workload is 300 kg*m.

59. Example 10 Use the tangent line on the graph below to estimate the instantaneous rate of change of cardiac output with respect to workload at the point where the workload is 300 kg*m.

60. Homework Section 2.1 Pages 140-141 1-10 all 11-27 odd

61. #11a Find the average rate of change of y with respect to x over the interval [x 0, x 1 ].

62. #11a Find the average rate of change of y with respect to x over the interval [x 0, x 1 ]. Slope of the secant line. (0, 0) and (1, 2)

63. #11b Find the instantaneous rate of change of y with respect to x at a specified value of x 0.

64. #11b Find the instantaneous rate of change of y with respect to x at a specified value of x 0. Tangent line.

65. #11c Find the instantaneous rate of change of y with respect to x at an arbitrary value of x 0.

66. #11c Find the instantaneous rate of change of y with respect to x at an arbitrary value of x 0.

67. #11d Graph the function and the two lines found in parts a and b.

68. #12a Find the average rate of change of y with respect to x over the interval [x 0, x 1 ].

69. #12a Find the average rate of change of y with respect to x over the interval [x 0, x 1 ]. Slope of the secant line. (1, 1) and (2, 8)

70. #12b Find the instantaneous rate of change of y with respect to x at the specified value of x 0.

71. #12b Find the instantaneous rate of change of y with respect to x at the specified value of x 0. Tangent line.

72. #12c Find the instantaneous rate of change of y with respect to x at an arbitrary value of x 0.