1. Accumulation Problems
Section 7.5
Calculus AP/Dual, Revised ©2018
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§7.5A: Accumulation

2. Application Problems
Understand the question. It is often not necessary to as much computation as it seems at first or as all the reading may seem to indicate.
Use FTC may help differentiating, 𝑭.
Explain the meaning of a derivative or definite integral or its value in terms of the context of the problem.
There are problems with one rate and with 2 rates (in-out problems) of change work together
Max/min and increasing/decreasing analysis.
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§7.5A: Accumulation

3. Based on the 2nd FTC
Accumulation is the net change using a definite integral
Rate of change of a quantity over an interval interpreted as the change of the quantity over the interval
Equation: 𝒇 𝒕 =𝒇 𝒂 + 𝒂 𝒕 𝒇 ′ 𝒙 𝒅𝒙
Known as: Final Value = Starting Value + Net Change
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§7.5A: Accumulation

4. 2nd FTC to Accumulation
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§7.5A: Accumulation

5. Equation
By applying the First and Second FTC, the equation follows: 𝒇 𝒕 =𝒇 𝒂 + 𝒂 𝒕 𝒇 ′ 𝒙 𝒅𝒙 Final Value = Starting Value + Accumulated Change 𝒂 𝒕 𝒇 ′ 𝒙 𝒅𝒙 =𝒇 𝒕 −𝒇 𝒂
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§7.5A: Accumulation

6. Labeling Answers T = Time U = Units N = Noun A = Answer
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§7.5A: Accumulation

7. Example 1 - Non Calc (2009 AB6)
The derivative of a function 𝒇 is defined by 𝒇 ′ 𝒙 = 𝒈 𝒙 𝒇𝒐𝒓 −𝟒≤𝒙≤𝟎 𝟓 𝒆 −𝒙/𝟑 −𝟑 𝒇𝒐𝒓 𝟎<𝒙≤𝟒 The graph of the continuous function , 𝒇 ′ shown in the figure above, has 𝒙-intercepts at 𝒙=−𝟐 and 𝒙=𝟑𝐥𝐧 𝟓 𝟑 . The graph of 𝒈 on −𝟒≤𝒙≤𝟎 is a semicircle, and 𝒇 𝟎 =𝟓. (a) Solve for 𝒇 −𝟒 and (b) 𝒇 𝟒
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§7.5A: Accumulation

8. Example 1a - Non Calc (2009 AB6)
The derivative of a function 𝒇 is defined by 𝒇 ′ 𝒙 = 𝒈 𝒙 𝒇𝒐𝒓 −𝟒≤𝒙≤𝟎 𝟓 𝒆 −𝒙/𝟑 −𝟑 𝒇𝒐𝒓 𝟎<𝒙≤𝟒 The graph of the continuous function , 𝒇 ′ shown in the figure above, has 𝒙-intercepts at 𝒙=−𝟐 and 𝒙=𝟑𝐥𝐧 𝟓 𝟑 . The graph of 𝒈 on −𝟒≤𝒙≤𝟎 is a semicircle, and 𝒇 𝟎 =𝟓. (a) Solve for 𝒇 −𝟒
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§7.5A: Accumulation

9. Example 1a - Non Calc (2009 AB6)
The derivative of a function 𝒇 is defined by 𝒇 ′ 𝒙 = 𝒈 𝒙 𝒇𝒐𝒓 −𝟒≤𝒙≤𝟎 𝟓 𝒆 −𝒙/𝟑 −𝟑 𝒇𝒐𝒓 𝟎<𝒙≤𝟒 The graph of the continuous function , 𝒇 ′ shown in the figure above, has 𝒙-intercepts at 𝒙=−𝟐 and 𝒙=𝟑𝐥𝐧 𝟓 𝟑 . The graph of 𝒈 on −𝟒≤𝒙≤𝟎 is a semicircle, and 𝒇 𝟎 =𝟓. (a) Solve for 𝒇 −𝟒
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§7.5A: Accumulation

10. Example 1b - Non Calc (2009 AB6)
The derivative of a function 𝒇 is defined by 𝒇 ′ 𝒙 = 𝒈 𝒙 𝒇𝒐𝒓 −𝟒≤𝒙≤𝟎 𝟓 𝒆 −𝒙/𝟑 −𝟑 𝒇𝒐𝒓 𝟎<𝒙≤𝟒 The graph of the continuous function , 𝒇 ′ shown in the figure above, has 𝒙-intercepts at 𝒙=−𝟐 and 𝒙=𝟑𝐥𝐧 𝟓 𝟑 . The graph of 𝒈 on −𝟒≤𝒙≤𝟎 is a semicircle, and 𝒇 𝟎 =𝟓. (b) Solve for 𝒇 𝟒
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§7.5A: Accumulation

11. Example 1b - Non Calc (2009 AB6)
The derivative of a function 𝒇 is defined by 𝒇 ′ 𝒙 = 𝒈 𝒙 𝒇𝒐𝒓 −𝟒≤𝒙≤𝟎 𝟓 𝒆 −𝒙/𝟑 −𝟑 𝒇𝒐𝒓 𝟎<𝒙≤𝟒 The graph of the continuous function , 𝒇 ′ shown in the figure above, has 𝒙-intercepts at 𝒙=−𝟐 and 𝒙=𝟑𝐥𝐧 𝟓 𝟑 . The graph of 𝒈 on −𝟒≤𝒙≤𝟎 is a semicircle, and 𝒇 𝟎 =𝟓. (b) Solve for 𝒇 𝟒
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§7.5A: Accumulation

12. Example 1b - Non Calc (2009 AB6)
The derivative of a function 𝒇 is defined by 𝒇 ′ 𝒙 = 𝒈 𝒙 𝒇𝒐𝒓 −𝟒≤𝒙≤𝟎 𝟓 𝒆 −𝒙/𝟑 −𝟑 𝒇𝒐𝒓 𝟎<𝒙≤𝟒 The graph of the continuous function , 𝒇 ′ shown in the figure above, has 𝒙-intercepts at 𝒙=−𝟐 and 𝒙=𝟑𝐥𝐧 𝟓 𝟑 . The graph of 𝒈 on −𝟒≤𝒙≤𝟎 is a semicircle, and 𝒇 𝟎 =𝟓. (b) Solve for 𝒇 𝟒
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§7.5A: Accumulation

13. Example 2 – Non Calc (2000 AB4)
Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at a rate of 𝒕+𝟏 gallons per minute for 𝟎≤𝒕≤𝟏𝟐𝟎 minutes. At time 𝒕=𝟎, the tank contains 30 gallons of water.
How many gallons of water leak out of the tank time from 𝒕=𝟎 to 𝒕=𝟑 minutes?
How many gallons of water are in the tank at time 𝒕=𝟑 minutes?
Write an expression for 𝑨 𝒕 , the total number of gallons of water in the tank at time 𝒕.
At what time 𝒕, for 𝟎<𝒕<𝟏𝟐𝟎, is the amount of water in the tank a maximum? Justify answer.
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§7.5A: Accumulation

14. Example 2a – Non Calc (2000 AB4)
Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at a rate of 𝒕+𝟏 gallons per minute for 𝟎≤𝒕≤𝟏𝟐𝟎 minutes. At time 𝒕=𝟎, the tank contains 30 gallons of water.
How many gallons of water leak out of the tank time from 𝒕=𝟎 to 𝒕=𝟑 minutes?
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§7.5A: Accumulation

15. Example 2b – Non Calc (2000 AB4)
Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at a rate of 𝒕+𝟏 gallons per minute for 𝟎≤𝒕≤𝟏𝟐𝟎 minutes. At time 𝒕=𝟎, the tank contains 30 gallons of water. b) How many gallons of water are in the tank at time 𝒕=𝟑 minutes?
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§7.5A: Accumulation

16. Example 2c – Non Calc (2000 AB4)
Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at a rate of 𝒕+𝟏 gallons per minute for 𝟎≤𝒕≤𝟏𝟐𝟎 minutes. At time 𝒕=𝟎, the tank contains 30 gallons of water. c) Write an expression for 𝑨 𝒕 , the total number of gallons of water in the tank at time 𝒕.
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§7.5A: Accumulation

17. Example 2d – Non Calc (2000 AB4)
Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at a rate of 𝒕+𝟏 gallons per minute for 𝟎≤𝒕≤𝟏𝟐𝟎 minutes. At time 𝒕=𝟎, the tank contains 30 gallons of water. d) At what time 𝒕, for 𝟎<𝒕<𝟏𝟐𝟎, is the amount of water in the tank a maximum? Justify answer.
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§7.5A: Accumulation

18. Example 2 – Score Sheet
Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at a rate of 𝒕+𝟏 gallons per minute for 𝟎≤𝒕≤𝟏𝟐𝟎 minutes. At time 𝒕 = 𝟎, the tank contains 30 gallons of water.
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§7.5A: Accumulation

19. Example 3 – Calculator (2006 AB2)
At an intersection in Thomasville, Oregon, cars left at the rate of 𝑳 𝒕 =𝟔𝟎 𝒕 𝐬𝐢𝐧 𝟐 𝒕 𝟑 cars per hour over the time interval of 𝟎≤𝒕≤𝟏𝟖 hours. The graph of 𝒚=𝑳 𝒕 is shown.
To the nearest whole number, find the total number of cars turning left at the intersection over the time interval, 𝟎≤𝒕≤𝟏𝟖 hours.
Traffic engineers will consider turn restrictions when 𝑳(𝒕)≥𝟏𝟓𝟎 cars per hour. Find all values of 𝒕 for which 𝑳(𝒕)≥𝟏𝟓𝟎 and compute the average value of L over this time interval. Indicate the units of measurement.
Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to the conclusion.
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§7.5A: Accumulation

20. Example 3a – Calculator (2006 AB2)
At an intersection in Thomasville, Oregon, cars left at the rate of 𝑳 𝒕 =𝟔𝟎 𝒕 𝐬𝐢𝐧 𝟐 𝒕 𝟑 cars per hour over the time interval of 𝟎≤𝒕≤𝟏𝟖 hours. The graph of 𝒚=𝑳 𝒕 is shown.
To the nearest whole number, find the total number of cars turning left at the intersection over the time interval, 𝟎≤𝒕≤𝟏𝟖 hours.
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§7.5A: Accumulation

21. Example 3b – Calculator (2006 AB2)
At an intersection in Thomasville, Oregon, cars left at the rate of 𝑳 𝒕 =𝟔𝟎 𝒕 𝐬𝐢𝐧 𝟐 𝒕 𝟑 cars per hour over the time interval of 𝟎≤𝒕≤𝟏𝟖 hours. The graph of 𝒚=𝑳 𝒕 is shown. b) Traffic engineers will consider turn restrictions when 𝑳 𝒕 ≥𝟏𝟓𝟎 cars per hour. Find all values of 𝒕 for which 𝑳 𝒕 ≥𝟏𝟓𝟎 and compute the average value of 𝑳 over this time interval. Indicate the units of measurement.
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§7.5A: Accumulation

22. Example 3c – Calculator (2006 AB2)
At an intersection in Thomasville, Oregon, cars left at the rate of 𝑳 𝒕 =𝟔𝟎 𝒕 𝐬𝐢𝐧 𝟐 𝒕 𝟑 cars per hour over the time interval of 𝟎≤𝒕≤𝟏𝟖 hours. The graph of 𝒚=𝑳 𝒕 is shown.
c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to the conclusion.
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§7.5A: Accumulation

23. Example 3 – Score Sheet
At an intersection in Thomasville, Oregon, cars left at the rate of 𝑳 𝒕 =𝟔𝟎 𝒕 𝐬𝐢𝐧 𝟐 𝒕 𝟑 cars per hour over the time interval of 𝟎≤𝒕≤𝟏𝟖 hours. The graph of 𝒚 = 𝑳 𝒕 is shown.
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§7.5A: Accumulation

24. Your Turn w/Calc – 15 Minutes (2005 AB2)
The tide removes sand from Sandy Point Beach at a rate modeled by the function 𝑹, given by: 𝑹 𝒕 =𝟐+𝟓𝐬𝐢𝐧 𝟒𝝅𝒕 𝟐𝟓 .
A pumping station adds sand to the beach at a rate modeled by the function 𝑺, given by 𝑺 𝒕 = 𝟏𝟓𝒕 𝟏+𝟑𝒕 .
Both 𝑹 𝒕 and 𝑺 𝒕 have units of cubic yards per hour and t is measured in hours for 𝟎≤𝒕≤𝟔. At time 𝒕=𝟎, the beach contains 2500 cubic yards of sand.
How much sand will the tide remove from the beach during the 6-hour period? Indicate units of measure.
Write an expression for 𝒀(𝒕), the total number of cubic yards of sand on the beach at time, 𝒕.
Find the rate at which the total amount of sand on the beach is changing at time 𝒕=𝟒.
For 𝟎≤𝒕≤𝟔, at what time t is the amount of sand on the beach the minimum? What is the minimum value? Justify.
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§7.5A: Accumulation

25. 11/17/ :54 PM
§7.5A: Accumulation

26. Example 4a – Calculator
The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by: 𝑹 𝒕 =𝟐+𝟓𝐬𝐢𝐧 𝟒𝝅𝒕 𝟐𝟓 .
A pumping station adds sand to the beach at a rate modeled by the function 𝑺, given by 𝑺 𝒕 = 𝟏𝟓𝒕 𝟏+𝟑𝒕 .
Both 𝑹 𝒕 and 𝑺 𝒕 have units of cubic yards per hour and 𝒕 is measured in hours for 𝟎≤𝒕≤𝟔. At time 𝒕=𝟎, the beach contains 2500 cubic yards of sand.
How much sand will the tide remove from the beach during the 6-hour period? Indicate units of measure.
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§7.5A: Accumulation

27. Example 4b – Calculator
The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by: 𝑹 𝒕 =𝟐+𝟓𝐬𝐢𝐧 𝟒𝝅𝒕 𝟐𝟓 . A pumping station adds sand to the beach at a rate modeled by the function 𝑺, given by 𝑺 𝒕 = 𝟏𝟓𝒕 𝟏+𝟑𝒕 . Both 𝑹 𝒕 and 𝑺 𝒕 have units of cubic yards per hour and 𝒕 is measured in hours for 𝟎≤𝒕≤𝟔. At time 𝒕=𝟎, the beach contains 2500 cubic yards of sand. B) Write an expression for 𝒀 𝒕 , the total number of cubic yards of sand on the beach at time, 𝒕.
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§7.5A: Accumulation

28. Example 4c – Calculator
The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by: 𝑹 𝒕 =𝟐+𝟓𝐬𝐢𝐧 𝟒𝝅𝒕 𝟐𝟓 . A pumping station adds sand to the beach at a rate modeled by the function 𝑺, given by 𝑺 𝒕 = 𝟏𝟓𝒕 𝟏+𝟑𝒕 . Both 𝑹 𝒕 and 𝑺 𝒕 have units of cubic yards per hour and 𝒕 is measured in hours for 𝟎≤𝒕≤𝟔. At time 𝒕=𝟎, the beach contains 2500 cubic yards of sand. C) Find the rate at which the total amount of sand on the beach is changing at time 𝒕=𝟒.
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§7.5A: Accumulation

29. Example 4d – Calculator
The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by: 𝑹 𝒕 =𝟐+𝟓𝐬𝐢𝐧 𝟒𝝅𝒕 𝟐𝟓 . A pumping station adds sand to the beach at a rate modeled by the function 𝑺, given by 𝑺 𝒕 = 𝟏𝟓𝒕 𝟏+𝟑𝒕 . Both 𝑹 𝒕 and 𝑺 𝒕 have units of cubic yards per hour and 𝒕 is measured in hours for 𝟎≤𝒕≤𝟔. At time 𝒕=𝟎, the beach contains 2500 cubic yards of sand. D) For 𝟎≤𝒕≤𝟔, at what time 𝒕 is the amount of sand on the beach the minimum? What is the minimum value? Justify. 𝒕
𝒀(𝒕) 𝟎
𝟐𝟓𝟎𝟎
𝟓.𝟏𝟏𝟕𝟖
𝟐𝟒𝟗𝟐.𝟑𝟔𝟗 𝟔
𝟐𝟒𝟗𝟑.𝟐𝟕𝟔𝟔 t
Y(t) 𝟎
𝟐𝟓𝟎𝟎
𝟓.𝟏𝟏𝟕𝟖 𝟔
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§7.5A: Accumulation

30. Example 4 – Score Sheet
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§7.5A: Accumulation

31. Assignment
Worksheet
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§7.5A: Accumulation