1. Accumulation AP Calculus AB Day 10
Instructional Focus: Model real world problems with a definite integral and derivatives.

2. Worksheet ROC and DI 1

3. is measured in hours from the time ride begins operation.
There are 700 people in line for a popular amusement-park ride when the ride begins operation in the morning. Once it begins operation, the ride accepts passengers until the park closes 8 hours later. While there is a line, people moves onto the ride at a rate of 800 people per hour. The graph above shows the rate, 𝑟 𝑡 , at which people arrive at the ride throughout the day. Time t
is measured in hours from
the time ride begins
operation.
2010 AB/BC 3 Modified
Rate of Change and Definite Integrals

4. How many people arrive at the ride between t = 0 and
t = 3? Show the computations that lead to your answer.

5. How many people arrive at the ride between t = 0 and
t = 3? Show the computations that lead to your answer.
0 3 𝑟 𝑡 𝑑𝑡 = =3200 people arrived in line
𝑟 𝑡 𝑑𝑡= people hour ∙hour=people
The rate times elapsed time give the number of people. You must accumulate (find the area under the curve) since the rate is not constant. Use two trapezoids or rectangles and triangles or count squares.

6. What is the value of 𝑟 ′ 2.5 ? Using appropriate units, what is the meaning of 𝑟 ′ in the context of this problem?

7. 𝑟 ′ 2.5 =− 800 2 =−400 people/hour hour
What is the value of 𝑟 ′ 2.5 ? Using appropriate units, what is the meaning of 𝑟 ′ in the context of this problem?
𝑟 ′ 2.5 =− =−400 people/hour hour
𝑟 ′ means the rate at which people arrive in line is decreasing at a rate of 400 people per hour per hour 2.5 hours after ride begins operation.
Slope of the segment of r(t) from points (2, 1200) and (4, 400).

8. Let 𝑤 𝑡 be the number of people waiting in line t hours after the ride begins operation. Complete the table below. Explain how you arrived at your answers. t
w(t)
700 1
950 2
1300

9. Let 𝑤 𝑡 be the number of people waiting in line t hours after the ride begins operation. Complete the table below. Explain how you arrived at your answers. t
w(t) ∆𝒘
700
given 1
950
0 1 𝑟 𝑡 𝑑𝑡 −800= −800=250 2
1300
1 2 𝑟 𝑡 𝑑𝑡 −800= −800=350
The number of people waiting in line is given by the number of people arriving in line minus the number of people moving onto the ride.

10. Is the number of people waiting in line 𝑤 𝑡 to get on the ride increasing or decreasing between t = 2 and t = 3? Justify your answer.

11. 2<𝑡<3⇒800<𝑟 𝑡 <1200⇒0< 𝑤 ′ 𝑡 <400
Is the number of people waiting in line 𝑤 𝑡 to get on the ride increasing or decreasing between t = 2 and t = 3? Justify your answer.
𝑤 ′ 𝑡 =𝑟 𝑡 −800
2<𝑡<3⇒800<𝑟 𝑡 <1200⇒0< 𝑤 ′ 𝑡 <400
The number of people in line is increasing between t = 2 and
t = 3.
The rate that the number of people in line is changing is the rate at which the people arrive in line minus the rate at which people move onto the ride.

12. What is the value of 𝑤 ′ 2.5 ? Using appropriate units, what is the meaning of 𝑤 ′ in the context of this problem?

13. 𝑤 ′ 2.5 =𝑟 2.5 −800=1000−800=200 people per hour
What is the value of 𝑤 ′ 2.5 ? Using appropriate units, what is the meaning of 𝑤 ′ in the context of this problem?
𝑤 ′ 2.5 =𝑟 2.5 −800=1000−800=200 people per hour
2.5 hours after the ride begins operation, the number of people in line is increasing at a rate of 200 people per hour.
The rate that the number of people in line is changing is the rate at which the people arrive in line minus the rate at which people move onto the ride.

14. What is the value of 𝑤 ′ 3.5 ? Using appropriate units, what is the meaning of 𝑤 ′ in the context of this problem?

15. 𝑤 ′ 3.5 =𝑟 3.5 −800=600−800=−200people per hour
What is the value of 𝑤 ′ 3.5 ? Using appropriate units, what is the meaning of 𝑤 ′ in the context of this problem?
𝑤 ′ 3.5 =𝑟 3.5 −800=600−800=−200people per hour
3.5 hours after the ride begins operation, the number of people in line is decreasing at a rate of 200 people per hour.
The rate that the number of people in line is changing is the rate at which the people arrive in line minus the rate at which people move onto the ride.

16. Is there a time when 𝑤 ′ 𝑡 =0? Justify your answer.

17. Is there a time when 𝑤 ′ 𝑡 =0? Justify your answer.
𝑤 ′ 𝑡 =0⇒𝑟 𝑡 =800⇒𝑡=3
The solution was determined from the graph of r(t).

18. When is the number of people waiting in line the largest
When is the number of people waiting in line the largest? Justify your answer.

19. When is the number of people waiting in line the largest
When is the number of people waiting in line the largest? Justify your answer.
The number of people waiting in line is the largest 3 hours after the ride opens. The number of people waiting in line is increasing anytime 𝑟 𝑡 >800 and decreasing anytime 𝑟 𝑡 <800. Graphically 𝑟 𝑡 >800 the first three hours after the ride opens and 𝑟 𝑡 <800 from 3 hours to 8 hours after the ride opens.

20. What is the earliest time when there is no longer a line
What is the earliest time when there is no longer a line? Justify your answer.

21. What is the earliest time when there is no longer a line
What is the earliest time when there is no longer a line? Justify your answer.
Let h be the earliest time when there is no longer a line.
𝑤 ℎ =0⇒ 0 ℎ 𝑟 𝑡 𝑑𝑡 −800ℎ=0⇒ 0 ℎ 𝑟 𝑡 𝑑𝑡 =800ℎ
It has already been determined that w(t) is a maximum at t = 3.
𝑤 3 = 𝑟 𝑡 𝑑𝑡 −800 3 = −2400=1500
𝑤 ℎ = ℎ 𝑟 𝑡 𝑑𝑡 −800 ℎ−3 =0
By guess and check, h = 7
𝑤 7 = 𝑟 𝑡 𝑑𝑡 −800 4 = −3200=0
There is no line 7 hours after the ride opens.

22. Worksheet ROC and DI 2

23. The graph of the velocity 𝑣 𝑡 , in ft/sec, of a car traveling on a straight road, for 0≤𝑡≤50, is shown. A table of values for 𝑣 𝑡 , at 5 second intervals of time t, is shown to the right of the graph.

24. During what intervals of time is the acceleration of the car positive
During what intervals of time is the acceleration of the car positive? Give a reason for your answer.

25. During what intervals of time is the acceleration of the car positive
During what intervals of time is the acceleration of the car positive? Give a reason for your answer.
Acceleration is positive on (0, 35) and (45, 50) because the velocity 𝑣 𝑡 is increasing on [0, 35] and [45, 50].
3 points:
1: (0, 35)
1: (45, 50)
1: reason
Note: ignore inclusion of endpoints

26. Find the average acceleration of the car, in ft/sec2, over the interval 0≤𝑡≤50.

27. Average Acceleration= 𝑣 50 −𝑣 0 50−0 = 72−0 50 = 72 50 or 1.44 ft/sec2
Find the average acceleration of the car, in ft/sec2, over the interval 0≤𝑡≤50.
Average Acceleration= 𝑣 50 −𝑣 0 50−0 = 72−0 50 = 72 50
or 1.44 ft/sec2
1: answer

28. Find one approximation for the acceleration of the car, in ft/sec2, at t = 40. Show the computations you used to arrive at your answer.

29. Difference quotient; e.g. 𝑣 45 −𝑣 40 45−40 = 60−75 5 =−3 ft/ sec 2 or
Find one approximation for the acceleration of the car, in ft/sec2, at t = 40. Show the computations you used to arrive at your answer.
Difference quotient; e.g.
𝑣 45 −𝑣 −40 = 60−75 5 =−3 ft/ sec 2 or
𝑣 40 −𝑣 −35 = 75−81 5 =− 6 5 ft/ sec 2 or
𝑣 45 −𝑣 −35 = 60−81 10 =− ft/ sec 2
or Slope of tangent line; e.g.
through 35, 90 and 40, 75 : 90−75 35−40 =−3 ft/ sec 2
2 points:
1: method
1: answer
Note: 0/2 if first point not earned

30. A Riemann sum is a sum of areas of rectangles drawn with base on the x-axis and height determined by the y-values on a curve. The first rectangle has a base from x = 0 to x = 10 or the interval [0, 10]. A midpoint Riemann sum will use the height determined by the x-coordinate at the midpoint of the interval [0, 10]. What are the units?

31. Approximate 𝑣 𝑡 𝑑𝑡 with a Riemann sum, using the midpoints of five subintervals of equal length. Using correct units, explain the meaning of this integral.

32. Approximate 𝑣 𝑡 𝑑𝑡 with a Riemann sum, using the midpoints of five subintervals of equal length. Using correct units, explain the meaning of this integral.
0 50 𝑣 𝑡 𝑑𝑡 ≈10 𝑣 5 +𝑣 15 +𝑣 25 +𝑣 35 +𝑣 45 = =2530 feet
This integral is the total distance traveled in feet over the time 0 to 50 seconds.
3 points:
1: midpoint Riemann sum
1: answer
1: meaning of integral

33. The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function R of time t. The graph of R and a table of selected values of 𝑅 𝑡 , for the time interval 0≤𝑡≤90 minutes, are shown.

34. Use data from the table to find an approximation for 𝑅 ′ 45
Use data from the table to find an approximation for 𝑅 ′ Show the computations that lead to your answer. Indicate units of measure.

35. Use data from the table to find an approximation for 𝑅 ′ 45
Use data from the table to find an approximation for 𝑅 ′ Show the computations that lead to your answer. Indicate units of measure.
𝑅 ′ 45 ≈ 𝑅 50 −𝑅 −40 = 55−40 10 =1.5 gal/ min 2
2 points
1: a difference quotient using numbers from table and interval that contains 45
1: 1.5 gal/min2

36. The rate of fuel consumption is increasing fastest at time t = 45 minutes. What is the value of 𝑅 ′′ 45 ? Explain your reasoning.

37. 𝑅 ′′ 45 =0 since 𝑅 ′ 𝑡 has a maximum at t = 45.
The rate of fuel consumption is increasing fastest at time t = 45 minutes. What is the value of 𝑅 ′′ 45 ? Explain your reasoning.
𝑅 ′′ 45 =0 since 𝑅 ′ 𝑡 has a maximum at t = 45.
2 points
1: 𝑅 ′′ 45 =0
1: reason

38. A Riemann sum is a sum of areas of rectangles drawn with base on the x-axis and height determined by the y-values on a curve. The first rectangle has a base from x = 0 to x = 30 or the interval [0, 30]. A left Riemann sum will use the height determined by the x-coordinate at the left edge of the interval [0, 30]. What are the units?

39. Approximate the value of 𝑅 𝑡 𝑑𝑡 using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation les than the value of 𝑅 𝑡 𝑑𝑡 ? Explain your reasoning.

40. Approximate the value of 𝑅 𝑡 𝑑𝑡 using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation les than the value of 𝑅 𝑡 𝑑𝑡 ? Explain your reasoning.
0 90 𝑅 𝑡 𝑑𝑡 ≈ =3700
Yes, this approximation is less because the graph of R is increasing on the interval.
2 points
1: value of left Riemann sum
1: “less” with reason

41. For 0<𝑏≤90 minutes, explain the meaning of 0 𝑏 𝑅 𝑡 𝑑𝑡 in terms of fuel consumption for the plane. Explain the meaning of 1 𝑏 0 𝑏 𝑅 𝑡 𝑑𝑡 in terms of fuel consumption for the plane. Indicate units of measure in both answers.

42. For 0<𝑏≤90 minutes, explain the meaning of 0 𝑏 𝑅 𝑡 𝑑𝑡 in terms of fuel consumption for the plane. Explain the meaning of 1 𝑏 0 𝑏 𝑅 𝑡 𝑑𝑡 in terms of fuel consumption for the plane. Indicate units of measure in both answers.
0 𝑏 𝑅 𝑡 𝑑𝑡 is the total amount of fuel in gallons consumed for the first b minutes.
1 𝑏 0 𝑏 𝑅 𝑡 𝑑𝑡 is the average value of the rate of fuel consumption in gallons/min during the first b minutes.
3 points
2: meanings
1: meaning of 0 𝑏 𝑅 𝑡 𝑑𝑡
1: meaning of 1 𝑏 0 𝑏 𝑅 𝑡 𝑑𝑡
<-1> if no reference to time b
1: units in both answers

43. The velocity of a particle moving along the x-axis is modeled by a differentiable function v, where the position x is measured in meters, and time t is measured in seconds. Selected values of 𝑣 𝑡 are given in the table. The particle is at position x = 7 meters when t = 0 seconds.

44. Estimate the acceleration of the particle at t = 36 seconds
Estimate the acceleration of the particle at t = 36 seconds. Show the computations that lead to your answer. Indicate units of measure.

45. 𝑎 36 = 𝑣 ′ 36 ≈ 𝑣 40 −𝑣 32 40−32 = 11 8 meters/ sec 2
Estimate the acceleration of the particle at t = 36 seconds. Show the computations that lead to your answer. Indicate units of measure.
𝑎 36 = 𝑣 ′ 36 ≈ 𝑣 40 −𝑣 −32 = meters/ sec 2
1: units in (a) and (b)
1: answer

46. A Riemann sum is a sum of areas of rectangles drawn with base on the x-axis and height determined by the y-values on a curve. The first rectangle has a base from t = 20 to t = 25 or the interval [20, 25]. A trapezoidal sum will use the average height determined by the t-coordinates at the left edge and right edge of the interval [20, 25]. What are the units? What does a negative velocity mean?

47. Using correct units, explain the meaning of 𝑣 𝑡 𝑑𝑡 in the context of this problem. Use a trapezoidal sum with the three subintervals indicated by the data in the table to approximate 𝑣 𝑡 𝑑𝑡 .

48. Using correct units, explain the meaning of 𝑣 𝑡 𝑑𝑡 in the context of this problem. Use a trapezoidal sum with the three subintervals indicated by the data in the table to approximate 𝑣 𝑡 𝑑𝑡 .
20 40 𝑣 𝑡 𝑑𝑡 is the particle’s change in position in meters from time t = 20 seconds to time t = 40 seconds
20 40 𝑣 𝑡 𝑑𝑡 ≈ 𝑣 20 +𝑣 ∙5+ 𝑣 25 +𝑣 ∙7+ 𝑣 32 +𝑣 ∙8=−75 meters
3 points
1: meaning of 𝑣 𝑡 𝑑𝑡
2: trapezoidal approximation

49. For 0≤𝑡≤40, must the particle change direction in any of the subintervals indicated by the data in the table? If so, identify the subintervals and explain your reasoning. If not, explain why not.

50. Therefore, the particle changes direction in the intervals
For 0≤𝑡≤40, must the particle change direction in any of the subintervals indicated by the data in the table? If so, identify the subintervals and explain your reasoning. If not, explain why not.
v(8) > 0 and v(20) < 0
v(32) < 0 and v(40) > 0
Therefore, the particle changes direction in the intervals
8 < t < 20 and 32 < t < 40
2 points
1: answer
1: explanation

51. Suppose that the acceleration of the particle is positive for 0<𝑡<8 seconds. Explain why the position of the particle at t = 8 seconds must be greater than x = 30 meters.

52. Since 𝑣 ′ 𝑡 =𝑎 𝑡 >0 for 0<𝑡<8, 𝑣 𝑡 ≥3 on this interval.
Suppose that the acceleration of the particle is positive for 0<𝑡<8 seconds. Explain why the position of the particle at t = 8 seconds must be greater than x = 30 meters.
Since 𝑣 ′ 𝑡 =𝑎 𝑡 >0 for 0<𝑡<8, 𝑣 𝑡 ≥3 on this interval.
Therefore, 𝑥 8 =𝑥 𝑣 𝑡 𝑑𝑡 ≥7+8∙3>30
2 points
1: 𝑣 ′ 𝑡 =𝑎 𝑡
1: explanation of 𝑥 8 >30

53. The volume of a spherical hot air balloon expands as the air insides the balloon is heated. The radius of the balloon, in feet, is modeled by a twice-differentiable function r of time t, where t is measured in minutes. For 0<𝑡<12, the graph of r is concave down. The table above gives selected values of the rate of change, 𝑟 ′ 𝑡 , of the radius of the balloon over the time interval 0≤𝑡≤12. The radius of the balloon is 30 feet when t = 5. (Note: The volume of a sphere of radius r is given by 𝑉= 4 3 𝜋 𝑟 3 .)

54. Estimate the radius of the balloon when t = 5
Estimate the radius of the balloon when t = 5.4 using the tangent line approximation at t = 5. Is your estimate greater than or less than the true value? Give a reason for your answer.

55. Estimate the radius of the balloon when t = 5
Estimate the radius of the balloon when t = 5.4 using the tangent line approximation at t = 5. Is your estimate greater than or less than the true value? Give a reason for your answer.
𝑟 5.4 ≈𝑟 5 + 𝑟 ′ 5 ∆𝑡= =30.8 ft
Since the graph of r is concave down on the interval 5<𝑡<5.4, this estimate is greater than 𝑟 5.4
2 points
1: estimate
1: conclusion with reason

56. Find the rate of change of the volume of the balloon with respect to time when t = 5. Indicate units of measure.

57. Find the rate of change of the volume of the balloon with respect to time when t = 5. Indicate units of measure.
𝑑𝑉 𝑑𝑡 = 𝜋 𝑟 2 𝑑𝑟 𝑑𝑡
𝑑𝑉 𝑑𝑡 𝑡=5 =4𝜋 =7200𝜋 ft 3 /min
3 points
2: 𝑑𝑉 𝑑𝑡
1: answer

58. A Riemann sum is a sum of areas of rectangles drawn with base on the x-axis and height determined by the y-values on a curve. The first rectangle has a base from t = 0 to t = 2 on the interval [0, 2]. A right Riemann sum will use the height determined by the t-coordinate at the right edge of the interval [0, 2]. What are the units?

59. Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate 𝑟 ′ 𝑡 𝑑𝑡 . Using correct units, explain the meaning of 𝑟 ′ 𝑡 𝑑𝑡 in terms of the radius of the balloon.

60. Use a right Riemann sum with the five subintervals indicated by the data in the table to approximate 𝑟 ′ 𝑡 𝑑𝑡 . Using correct units, explain the meaning of 𝑟 ′ 𝑡 𝑑𝑡 in terms of the radius of the balloon.
0 12 𝑟 ′ 𝑡 𝑑𝑡≈ =19.3 ft
0 12 𝑟 ′ 𝑡 𝑑𝑡 is the change in the radius, in feet, from t = 0 to t = 12 minutes.
2 points
1: approximation
1: explanation

61. Is your approximation in part (c) greater than or less than 𝑟 ′ 𝑡 𝑑𝑡 ? Give a reason for your answer.

62. Is your approximation in part (c) greater than or less than 𝑟 ′ 𝑡 𝑑𝑡 ? Give a reason for your answer.
Since r is concave down, 𝑟 ′ is decreasing on 0<𝑡<12. Therefore, this approximation, 19.3 ft, is less than 𝑟 ′ 𝑡 𝑑𝑡 .
1: conclusion with reason
1: units in (b) and (c)