Final Value = Starting Value + Accumulated Change. Final Position = Initial Position + Displacement

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1 Accumulation, Particle Motion Big Ideas Fundamental Theorem of Calculus and Accumulation AP Calculus Course Description Goals page 6 Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change, and should be able to use integrals to solve a variety of problems. Accumulation of rate of change is an important part of the AP Calculus syllabus. An alternative statement of the usual FTC may be used on the many accumulation problems that appear on the AP Calculus exam. FTC: f ( t ) dt = f ( b ) f ( a ) a Alternate statement of the FTC: f ( b) = f ( a) + f ( t) dt b b a x or f ( x) = f ( a) + f ( t) dt a These problems involve both accumulation of rate of change, particle motion problems, and antiderivatives with initial conditions. Consider the following. Final Value = Starting Value + Accumulated Change Final Position = Initial Position + Displacement Example: dy 1 Let y = f( x) be a function with derivative = tan ( x) that satisfies the condition f(3) = 2. Find f (6). dx 6 1 (6) = (3) + tan ( ) 3 f f x dx f (6) = = Example: t A particle moves along the x-axis with velocity given by vt ( ) = 2 e for time t. If the particle is at position x = 5 at time t =, what is the position of the particle at time t = 2? 2 t x(2) = x() + 2e dt x(2) = = Motion Additional Notes Questions based on numerical or graphical prompts Communicate understanding of concepts using correct mathematical language and notation Can be introduced early in the school year distance, velocity, acceleration Can be revisited throughout the year Mike Koehler 5-1 Accumulation, Particle Motion

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3 AP Multiple Choice Questions 28 AB Multiple Choice An object traveling in a straight line has position xt () at time t. If the initial position is x () = 2 and the 3 2 velocity of the object is vt () = 1+ t, what is the position of the object at time t = 3? A).431 B) C) D) E) BC Multiple Choice 77 t. Water is pumped out of a lake at the rate of Rt ( ) = 12 cubic meters per minute, where t is measured t + 1 in minutes. How much water is pumped from time t = to t = 5? A) cubic meters B) cubic meters C) cubic meters D) cubic meters E) cubic meters 23 AB Multiple Choice 22. The graph of f, the derivative of f, is the line shown in the figure on the right. If f () = 5, then f (1) = A) B) 3 C) 6 D) 8 E) 11 Mike Koehler 5-3 Accumulation, Particle Motion

4 77. The regions A, B, and C in the figure on the right are bounded by the graph of the function f and the x- axis. If the area of each region is 2, what is the value 3 of ( f ( x) + 1) dx? 3 A) -2 B) -1 C) 4 D) 7 E) The rate of change of the altitude of a hot-air balloon is given by rt ( ) = t 4t + 6 for t 8. Which of the following expressions gives the change in altitude of the balloon during the time the altitude is decreasing? A) r() t dt B) r() t dt C) r() t dt D) r () t dt E) r () t dt 84. A pizza, heated to a temperature of 35 degrees Fahrenheit ( F) is taken out of an oven and placed in a.4t 75 F room at time t = minutes. The temperature of the pizza is changing at a rate of 11e degrees Fahrenheit per minute. To the nearest degree, what is the temperature of the pizza at time t = 5 minutes? A) 112 o F B) 119 o F C) 147 o F D) 238 o F E) 335 o F 91. A particle moves along the x-axis so that at any time t>, its acceleration is given by at ( ) = ln ( 1+ 2 t ). If the velocity of the particle is 2 at time t = 1 then the velocity of the particle at time t = 2 is A).462 B) 1.69 C) D) E) BC Multiple Choice 8..1t 1e Insects destroyed a crop at the rate of tons per day, where time t is measured in days. To the 3t 2 e nearest ton, how many tons did the insects destroy during the time interval 7 t 14? A) 125 B) 1 C) 88 D) 5 E) 12 Mike Koehler 5-4 Accumulation, Particle Motion

5 87. A particle moves along the x-axis so that at any time 2 t, its velocity is given by vt ( ) cos( 2 t ) =. The position of the particle is 3 at time t =. What is the position of the particle when its velocity is first equal to? A).411 B) 1.31 C) D) 3.91 E) AB Multiple Choice 2. The graph of a piecewise-linear function f, for 1 x 4, is shown on the right. What is the 4 value of f ( x ) dx? 1 A) 1 B) 2.5 C) 4 D) 5.5 E) AB Multiple Choice 78. The graph of f is shown in the figure of the right. 3 If f( x) dx = 2.3 and F ( x) = f( x), then 1 F(3) F() = A).3 B) 1.3 C) 3.3 D) 4.3 E) 5.3 Mike Koehler 5-5 Accumulation, Particle Motion

6 88. x Let f ( x) = h( t) dt, where h has the graph shown a on the right. Which of the following could be the graph of f? A) B) C) D) E) Mike Koehler 5-6 Accumulation, Particle Motion

7 AP Free Response Questions 212 AB1 211 AB2/BC2 As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for t 1, where time t is measured in minutes and temperature Ht () is measured in degrees Celsius. Values of Ht () at selected values of time t are shown in the table to the right. a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time t = 3.5. Show the computations that lead to your answer. b) 1 1 Using correct units, explain the meaning of () 1 H t dt in the context of this problem. Use a trapezoidal 1 1 sum with the four subintervals indicated by the table to estimate () 1 H t dt. c) 1 Evaluate H () t dt. Using correct units, explain the meaning of the expression in the context of this problem. d) At time t =, biscuits with temperature 1 o C were removed from an oven. The temperature of the biscuits at.173t time t is modeled by a differentiable function B for which it is known that B ( t) = 13.84e. Using the given models, at time t = 1, how much cooler are the biscuits than the tea? 21 AB1 There is no snow on Janet s driveway when snow begins to fall at midnight. From midnight to 9 A.M., snow cos t accumulates on the driveway at a rate modeled by f() t = 7e cubic feet per hour, where t is measured in hours since midnight. Janet starts removing snow at 6 A.M. (t = 6). The rate g(t), in cubic feet per hour, at which Janet removes snow from the driveway at time t hours after midnight is modeled by for t < 6 gt ( ) = 125 for 6 t< for 7 t 9 a) How many cubic feet of snow have accumulated on the driveway by 6 A.M.? b) Find the rate of change of the volume of snow on the driveway at 8 A.M. c) Let h(t) represent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time t hours after midnight. Express h as a piecewise-defined function with domain t 9. d) How many cubic feet of snow are on the driveway at 9 A.M.? Mike Koehler 5-7 Accumulation, Particle Motion

8 21 AB2 A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon (t = ) and 8 P.M. (t = 8). The number of entries in the box t hours after noon is modeled by a differentiable function E for t 8. Values of Et (), in hundreds of entries, at various times t are shown in the table above. a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time t = 6. Show the computations that lead to your answer. b) 1 8 Use a trapezoidal sum with the four subintervals given by the table to approximate the value of () 8 E t dt 1 8 Using correct units, explain the meaning of () 8 E t dt in terms of the number of entries. c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the 3 2 function P, where Pt ( ) = t 3t + 298t 976 hundreds of entries per hour for 8 t 12. According to the model, how many entries had not yet been processed by midnight (t = 12)? d) According to the model from part (c), at what time were the entries being processed most quickly? Justify your answer. 21 AB3 There are 7 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 move onto the ride at a rate of 8 people per hour. The graph above shows the rate, rt (), at which people arrive at the ride throughout the day. Time t is measured in hours from the time the ride begins operation. a) How many people arrive at the ride between t = and t = 3? Show the computations that lead to your answer. b) Is the number of people waiting in line to get on the ride increasing or decreasing between t = 2 and t = 3? Justify your answer. c) At what time t is the line for the ride the longest? How many people are in line at that time? Justify your answers. d) Write, but do not solve, an equation involving an integral expression of r whose solution gives the earliest time t at which there is no longer a line for the ride. Mike Koehler 5-8 Accumulation, Particle Motion

9 29 AB1 Caren rides her bicycle along a straight road from home to school, starting at home at time t = minutes and arriving at school at time t = 12 minutes. During the time interval t 12 minutes, her velocity v(t), in miles per minute, is modeled by the piecewise-linear function whose graph is shown above. a) Find the acceleration of Caren s bicycle at time t = 7.5 minutes. Indicate units of measure. b) Using correct units, explain the meaning of v() t dt in terms of Caren s trip. Find the value of v() t dt. c) Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer. d) Larry also rides his bicycle along a straight road from home to school in 12 minutes. His velocity is modeled π π by the function wgiven by wt ( ) = sin t where wt () is in miles per minute for t minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer. 29 AB2 The rate at which people enter an auditorium for a rock concert is modeled by the function R given by 2 3 Rt ( ) = 138t 675 t for t 2 hours; Rt () is measured in people per hour. No one is in the auditorium at time t =, when the doors open. The doors close and the concert begins at time t = 2. a) How many people are in the auditorium when the concert begins? b) Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer. c) The total wait time for all the people in the auditorium is found by adding the time each person waits, starting at the time the person enters the auditorium and ending when the concert begins. The function w models the total wait time for all the people who enter the auditorium before time t. The derivative of w is given by w () t = ( 2 t) Rt (). Find w(2) w(1), the total wait time for those who enter the auditorium after time t = 1. d) On average, how long does a person wait in the auditorium for the concert to begin? Consider all people who enter the auditorium after the doors open, and use the model for total wait time from part (c). Mike Koehler 5-9 Accumulation, Particle Motion

10 29 AB3 Mighty Cable Company manufactures cable that sells for $12 per meter. For a cable of fixed length, the cost of producing a portion of the cable varies with its distance from the beginning of the cable. Mighty reports that the cost to produce a portion of a cable that is x meters from the beginning of the cable is 6 x dollars per meter. (Note: Profit is defined to be the difference between the amount of money received by the company for selling the cable and the company s cost of producing the cable.) a) Find Mighty s profit on the sale of a 25-meter cable. b) 3 Using correct units, explain the meaning of 6 x dx in the context of this problem. 25 c) Write an expression, involving an integral that represents Mighty s profit on the sale of a cable that is k meters long. d) Find the maximum profit that Mighty could earn on the sale of one cable. Justify your answer. 27 AB2/BC2 The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval t 7, where t is measured in hours. In this model, rates are given as follows: The rate at which water enter the tank is f( t) 1t ( ) 2 sin t = gallons per hour for t for t < 3 The rate at which water leaves the tank is gt () = gallons per hour. 2 for 3 < t 7 The graphs of f and g, which intersect at t = and t = 5.76, are shown in the figure. At time t =, the amount of water in the tank is 5 gallons a) How many gallons of water enter the tank during the time interval t 7? Round your answer to the nearest gallon. b) For t 7, find the time intervals during which the amount of water in the tank is decreasing. Give a reason for your answer. c) For t 7, at what time t is the amount of water in the tank greatest? To the nearest gallon, compute the amount of water at this time. Justify your answer. Mike Koehler 5-1 Accumulation, Particle Motion

11 26 AB2/BC2 At an intersection in Thomasville, Oregon, cars turn left at 2 t the rate Lt ( ) = 6 tsin cars per hour over the time 3 interval t 18 hours. The graph of y= Lt () is shown. a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval t 18 hours. b) Traffic engineers will consider turn restrictions when Lt ( ) 15 cars per hour. Find all values of t for which Lt ( ) 15 and compute the average value of L over this time interval. Indicate units of measure. 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 2,. In every two-hour time interval, 5 oncoming cars travel straight through the intersection. Does the intersection require a traffic signal? Explain the reasoning that leads to your conclusion. 25 AB2 The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by 4π t Rt ( ) = 2 + 5sin t A pumping station adds sand to the beach at a rate modeled by the function S, given by St () = t Both Rt ( ) and St ( ) have units of cubic yards per hour and t is measured in hours for t 6. At time t =, the beach contains 25 cubic yards of sand. a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure. b) Write an expression for Yt (), the total number of cubic yards of sand on the beach at time t. c) Find the rate at which the total amount of sand on the beach is changing at time t = 4. d) For t 6, at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justify your answer. Mike Koehler 5-11 Accumulation, Particle Motion

12 24 AB1/BC1 Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The t traffic flow at a particular intersection is modeled by the function F defined by Ft ( ) = sin for 3 2 t, where Ft () is measured in cars per minute and t is measured in minutes. a) To the nearest whole number, how many cars pass through the intersection over the 3-minute period? b) Is the traffic flow increasing or decreasing at t = 7? Give a reason for your answer. c) What is the average value of the traffic flow over the time interval 1 t 15? Indicate units of measure. d) What is the average rate of change of the traffic flow over the time interval 1 t 15? Indicate units of measure. 24 AB3 1 t A particle moves along the y-axis so that its velocity v at time t is given by vt ( ) = 1 tan ( e). At time t =, the particle is at y = 1. a) Find the acceleration of the particle at time t = 2. b) Is the speed of the particle increasing or decreasing at time t = 2? Give a reason for your answer. c) Find the time t at which the particle reaches its highest point. Justify your answer. d) Find the position of the particle at time t = 2. Is the particle moving toward the origin or away from the origin at time t = 2? Justify your answer. 24 AB5 The graph of the function f shown on the right consists of a semicircle and three line segments. Let g be the function given x by g( x) = f () t dt. 3 a) Find g() and g (). b) Find all values of x in the open interval ( 5, 4) at which g attains a relative minimum. Justify your answer. c) Find the absolute minimum value of g on the closed interval [ 5, 4]. Justify your answer. d) Find all values of x in the open interval ( 5, 4) at which the graph of g has a point of inflection. Mike Koehler 5-12 Accumulation, Particle Motion

13 23 AB3 The rate of fuel consumption 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 Rt () for the time interval t 9 minutes are shown above. a) Use data from the table to find an approximation for R (45). Show the computations that lead to your answer. Indicate units of measure. b) The rate of fuel consumption is increasing fastest at time t = 45 minutes. What is the value of R (45)? Explain your reasoning. c) 9 Approximate the value of R() t dt using a left Riemann sum with the five subintervals indicated by the data d) in the table. Is this numerical approximation less that the value of For < b 9 minutes, explain the meaning of the meaning of answers. 9 R() t dt? Explain your reasoning. b R () t dt in terms of fuel consumption for the plane. Explain 1 b R () t dt b in terms of fuel consumption for the plane. Indicate units of measure in both Mike Koehler 5-13 Accumulation, Particle Motion

14 22 AB2/BC2 The rate at which people enter an amusement park on a given day is modeled by the function E defined by 156 Et () =. 2 t 24t+ 16 The rate at which people leave the same amusement park on the same day is modeled by the function L defined by 989 Lt 9) =. 2 t 38t+ 37 Both Et ( ) and Lt ( ) are measured in people per hour, and time t is measured in hours after midnight. These functions are valid for 9 t 23, which are the hours that the park is open. At time t = 9, there are no people in the park. a) How many people have entered the park by 5: P.M. ( t = 17)? Round your answer to the nearest whole number. b) The price of admission to the park is $15 until 5: P.M. ( t = 17). After 5: P.M., the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number. c) t Let H ( t) = ( E( x) L( x) ) dx for 9 t 23. The value of H (17) to the nearest whole number is Find 9 the value of H (17) and explain the meaning of H (17) and H (17) in the context of the park. d) At what time t, for 9 t 23, does the model predict that the number of people in the park is a maximum? 22 AB3 An object moves along the x-axis with initial position x () = 2. The velocity of the object at time t is given by π vt ( ) = sin t 3. a) What is the acceleration of the object at time t = 4? b) Consider the following two statements. Statement I: For 3 < t < 4.5, the velocity of the object is decreasing. Statement II: For 3 < t < 4.5, the speed of the object is increasing. Are either or both of these statements correct? For each statement provide a reason why it is correct or not correct. c) What is the total distance traveled by the object over the time interval t 4? d) What is the position of the object at time t = 4? Mike Koehler 5-14 Accumulation, Particle Motion

15 22 AB4/BC4 The graph of the function f shown to the right consists of two line segments. Let g be the function given by g( x) = f () t dt. x a) Find g( 1), g ( 1), and g ( 1) b) For what values of x in the open interval ( 2, 2) is g increasing? Explain your reasoning. c) For what values of x in the open interval ( 2, 2) is the graph of g concave down? Explain your reasoning. d) Sketch a graph of g on the closed interval [ 2, 2]. 21 AB 3 / BC 3 A car is traveling on a straight road with velocity 55 ft sec at time t =. For t 18 seconds, the car's 2 acceleration, in ft sec, is the piecewise linear function defined by the graph. a) Is the velocity of the car increasing at t = 2 seconds? Why or why not? b) At what time in the interval t 18, other than t =, is the velocity of the car 55 ft sec? c) On the time interval t 18, what is the car's absolute maximum velocity, in ft sec, and at what time does it occur? Justify your answer. d) At what time in the interval t 18, if any, is the car's velocity equal to zero? Justify your answer. Mike Koehler 5-15 Accumulation, Particle Motion

16 2 AB 2 / BC 2 Two runners, A and B, run on a straight racetrack for t 1 seconds. The graph, which consists of two line segments, shows the velocity, in meters per second, of Runner A. The velocity, in meters per second, of Runner B is 24t given by the function v defined by vt () = 2t + 3. a) Find the velocity of Runner A and the velocity of Runner B at time t = 2 seconds. Indicate units of measure. b) Find the acceleration of Runner A and the acceleration of Runner B at time t = 2 seconds. Indicate units of measure. c) Find the total distance run by Runner A and the total distance run by Runner B over the time interval t 1 seconds. Indicate units of measure. 2 AB4 Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at the rate of t + 1 gallons per minute, for t 12 minutes. At time t =, the tank contains 3 gallons of water. a) How many gallons leak out of the tank from time t = to t = 3 minutes? b) How many gallons of water are in the tank at t = minutes? c) Write an expression for At (), the total number of gallons of water in the tank at time t. d) At what time t, for t 12, is the amount of water in the tank a maximum? Justify your answer. Mike Koehler 5-16 Accumulation, Particle Motion

17 1999 AB3 The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function R of time t. The table shows the rate as measured every 3 hours for a 24-hour period. a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate t Rt () (hours) (gallons per hour) R() t dt. Using correct units, explain the meaning of your answer in terms of water flow. b) Is there some time t, < t < 24, such that R () t >? Justify your answer. c) 1 2 The rate of water flow Rt () can be approximated by Qt ( ) = ( t t ). Use Qt () to approximate the 79 average rate of water flow during the 24-hour time period. Indicate units of measure. Mike Koehler 5-17 Accumulation, Particle Motion

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19 Textbook Problems Calculus, Finney, Demanna, Waits, Kennedy; Prentice Hal, l212 Section Questions R Handouts Mike Koehler 5-19 Accumulation, Particle Motion

20 Mike Koehler 5-2 Accumulation, Particle Motion

21 AP Calculus Chapter 7 Section 1 1. The audience at an outdoor show fills a semicircular strip composed of two concentric circles with the stage at the center, as shown in the figure on the right. A semicircular barricade with the stage at the center keeps the audience at least 3 meters from the stage. The density of the people at a distance x x meters from the stage is 3 people per 1 square meter, for 3 x 3. a) How fast is the density of the people changing at a distance of 1 meters from the stage? Using appropriate units, interpret your answer. b) How many people are in the audience between 3 and 15 meters of the stage? Express the answer as a Riemann sum and as an integral. c) What is the size of the audience? d) Write an equation that could be solved to determine the radius within which half the audience is contained. Solve the equation. (calculator) e) If instead of the density given in the problem, if the density of the people in the audience was uniform, and the number of people was the same as your answer to part (c), what would the density be? Mike Koehler 5-21 Accumulation, Particle Motion

22 AP Calculus Chapter 7 Section 1 Answers 1. a) d x 1 person per square meter 3 = dx 1 1 meter b) x people 2π x 2 3 meter 2 x 1 meter 2 15 x 3 ( π x) dx = people 3 1 c) 3 x 3 ( π x) dx = people 3 1 d) r x ( π x) dx 687 r meters 3 = = = 1 2 e) 1374 people = or π 2 π meter 2 2 Mike Koehler 5-22 Accumulation, Particle Motion

23 AP Calculus Chapter 7 Section 1 Problems Water Tank A water tank at Camp Newton holds 12 gallons of water at time t =. During the time interval t 18 hours, 2 t water is pumped into the tank at the rate Wt ( ) = 95 tsin gallons per hour. 6 2 t During the same time interval, water is removed from the tank at the rate of Rt ( ) = 275sin gallons per hour. 3 a. Is the amount of water in the tank increasing at time t = 15? Why or why not? b. To the nearest whole number, how many gallons of water are in the tank at time t = 18? c. At what time t, for t 18, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion. d. For t > 18, no water is pumped into the tank, but water continues to be removes at the rate Rt () until the tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of k. Mosquitoes For t 31, the rate of change of the number of mosquitoes on a tropical island at time t days is modeled by t Rt ( ) = 5 tcos mosquitoes per day. There are 1 mosquitoes on the island at time 5 t =. a. Show the number of mosquitoes is increasing at time t = 6. b. At time t = 6, is the number of mosquitoes increasing at an increasing rate, or is the number of mosquitoes increasing at a decreasing rate. Give a reason for your answer. c. According t the model, how many mosquitoes will be on the island at time t = 31? Round you answer to the nearest whole number. d. To the nearest whole number, what is the maximum number of mosquitoes for t 31? Show the analysis that leads to your conclusion. Heating Oil A tank contains 125 gallons of heating oil at time t =. During the time interval t 12 hours, heating oil is 1 pumped into the tank at the rate Ht () = 3 + gallons per hour. During the same time interval, heating oil (1 + ln( t + 1)) 2 t is removed from the tank at the rate of Rt ( ) = 12sin gallons per hour. 47 a. How many gallons of heating oil are pumped into the tank during the time interval t 12 hours? b. Is the level of heating oil in the tank rising or falling at time t = 6 hours? Give a reason for your answer. c. How many gallons of heating oil are in the tank at time t = 12 hours? d. At what time t, for t 12, is the volume of heating oil in the tan the least? Show the analysis that leads to your conclusion. Mike Koehler 5-23 Accumulation, Particle Motion

24 AP Calculus Chapter 7 Section 1 Answers Water Tank a. No. W(15) R(15) = < b. 131 gallons c. t = or Check values at critical points and endpoints. d. k R ( t ) dt = Mosquitoes a. Increasing, R (6) = > b. Increasing at a decreasing rate. R (6) = <. c. 964 mosquitoes. d. The maximum number of mosquitoes is 139 at time t = 31. Heating Oil a or gallons b. The level of heating oil is falling at time t = 6. c or gallons d. The volume is least at t = hours. Mike Koehler 5-24 Accumulation, Particle Motion

25 AP Calculus Chapter 7 Section 1 Lake Newton Rivers and streams flow into Lake Newton and heavy rains can cause flooding. Officials monitor the level of water in the lake and open floodgates on the Lake Newton dam to allow water to flow downstream. Let Et () = t+ sin() t be the rate of water entering Lake Newton at a hundred cubic feet hour. 2 1 t 6 Let Rt () = be the rate of water being released from the floodgates at a hundred cubic feet hour < t 12 Note that Rt ( ) = when t< Graph Et ( ) and Rt ( ). Label axes and units. 2. At what time t is the rate of the water entering the lake equal to the rate of the water being released? π? Explain why there must be a value of t for which the 3. What is the average rate of change of Et ( ) on [, ] average rate of change of Et () is equal to the instantaneous rate of change of Et ( ) on [, ] exact value of t. π and find that 4. How much water entered the lake in the first hour of monitoring? 5, Find the average value of Rt ( ) on [ 1,12]. 6. From t = 2 to t = 12 hours, give the time intervals when more water was entering the lake than was being released. Determine how much more water entered than was released in those time intervals. 7. If A is the amount of cubic feet of water in Lake Newton at time t =, what is the amount of water Q in the lake at t = 7? 8. At what time t will the amount of water in the lake return to the amount at time t =? Mike Koehler 5-25 Accumulation, Particle Motion

26 Lake Newton Answers 1 2 t = 1.16, hours 3 Et () is continuous and differentiable on [,π ], therefore the Mean Value Theorem holds and the average E( π) E() π π π E () t = 1= 1+ cos() t π t = 2 rate of change of Et ( ), = = 1 = E ( t) for some tin [, π ] hundred cubic feet dt + 1.5dt = hundred cubic feet hour t 6, t hundred cubic feet 7 A hundred cubic feet hours Mike Koehler 5-26 Accumulation, Particle Motion

27 AP Calculus Chapter 7 Section 1 The function St () given below can be used to estimate the number of hours of daylight in Kansas City, MO for a given day of the year for t 365 ( t is in days and t = is midnight on January 1). π St ( ) = 2.657sin ( x ) hours day Use the given formula to find the following: a. What is the total number of daylight hours from the April 1 (day 91) to June 3 (day 181)? b. Is the length of the day increasing or decreasing on the 1th day? Explain. c. What is the average number of hours of daylight from day 3 to day 15? d. What is the average number of hours of daylight in Kansas City for the year? e. On what day is the Summer Solstice (longest day of the year)? Explain. Mike Koehler 5-27 Accumulation, Particle Motion

28 Data Date Day Number Hours of Daylight 1/ :28:34 2/ :13:47 3/ :19:25 4/ :38:4 5/ :5:41 6/ :43:49 7/ :52:12 8/ :11:39 9/ :2:39 1/ :47:26 11/ :32:7 12/ :37:47 Source: Mike Koehler 5-28 Accumulation, Particle Motion