Accumulation, Particle Motion - Thornton...

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: ( ) ( ) ( )

b

af t dt f b f a= −′∫

Alternate statement of the FTC: ( ) ( ) + ( ) b

af b f a f t dt= ′∫

or ( ) ( ) + ( ) x

af x f a f t dt= ′∫

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:

Let 1( ) be a function with derivative tan ( ) that satisfies the condition (3) 2dyy f x x fdx

−= = = . Find (6)f .

6 1

3(6) (3) tan ( )

(6) 2 3.4777 5.4777

f f x dx

f

−= +

= + =∫

Example: A particle moves along the x-axis with velocity given by ( ) 2 for time 0tv t e t−= ≥ . If the particle is at position

5 at time 0x t= = , what is the position of the particle at time 2t = ?

2

0(2) (0) 2

(2) 5 1.6522 6.652

tx x e dt

x

−= +

= + =

∫

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

AP Multiple Choice Questions 2008 AB Multiple Choice 7 9 81 86 92 87. An object traveling in a straight line has position ( )x t at time t . If the initial position is (0) 2x = and the

velocity of the object is 3 2( ) 1v t t= + , what is the position of the object at time 3t = ? A) 0.431 B) 2.154 C) 4.512 D) 6.512 E) 17.408

2008 BC Multiple Choice 77 . Water is pumped out of a lake at the rate of ( ) 12

1tR t

t=

+cubic meters per minute, where t is measured

in minutes. How much water is pumped from time 0 to 5t t= = ? A) 9.439 cubic meters B) 10.954 cubic meters C) 43.816 cubic meters D) 47.193 cubic meters E) 54.772 cubic meters

2003 AB Multiple Choice 22. The graph of f ′ , the derivative of f , is the line shown in

the figure on the right. If (0) 5f = , then (1)f =

A) 0 B) 3 C) 6 D) 8 E) 11


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

of ( )3

3( ) 1f x dx

−+∫ ?

A) -2 B) -1 C) 4 D) 7 E) 12

82. The rate of change of the altitude of a hot-air balloon is given by 3 2( ) 4 6 for 0 t 8r t t t= − + ≤ ≤ .

Which of the following expressions gives the change in altitude of the balloon during the time the altitude is decreasing? A)

3.514

1.572( )r t dt∫ B)

8

0( )r t dt∫ C)

2.667

0( )r t dt∫

D) 3.514

1.572( )r t dt′∫ E)

2.667

0( )r t dt′∫

84. A pizza, heated to a temperature of 350 degrees Fahrenheit (° F) is taken out of an oven and placed in a

75° F room at time t = 0 minutes. The temperature of the pizza is changing at a rate of 0.4110 te−− degrees Fahrenheit per minute. To the nearest degree, what is the temperature of the pizza at time t = 5 minutes? A) 112oF B) 119oF C) 147oF D) 238oF E) 335oF

91. A particle moves along the x-axis so that at any time t> 0, its acceleration is given by ( )( ) ln 1 2ta 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) 0.462 B) 1.609 C) 2.555 D) 5.886 E) 3.346

2003 BC Multiple Choice 80.

Insects destroyed a crop at the rate of0.1

3

1002

t

t

ee

−

−−tons per day, where time t is measured in days. To the

nearest ton, how many tons did the insects destroy during the time interval 7 14t≤ ≤ ? A) 125 B) 100 C) 88 D) 50 E) 12


87. A particle moves along the x-axis so that at any time 0t ≥ , its velocity is given by ( )2( ) cos 2v t t= − . The

position of the particle is 3 at time t = 0. What is the position of the particle when its velocity is first equal to 0? A) 0.411 B) 1.310 C) 2.816 D) 3.091 E) 3.411

1998 AB Multiple Choice 2. The graph of a piecewise-linear function f , for

1 4x− ≤ ≤ , is shown on the right. What is the

value of4

1( )f x dx

−∫ ?

A) 1 B) 2.5 C) 4 D) 5.5 E) 8

1997 AB Multiple Choice 78. The graph of f is shown in the figure of the right.

If3

1( ) 2.3 and ( ) ( ),f x dx F x f x′= =∫ then

(3) (0)F F− =

A) 0.3 B) 1.3 C) 3.3 D) 4.3 E) 5.3


88. Let ( ) ( ) , where x

af x h t dt h= ∫ has the graph shown

on the right. Which of the following could be the graph of f ?

A)

B)

C)

D)

E)


AP Free Response Questions 2012 AB1 2011 AB2/BC2 As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for 0 10t≤ ≤ , where time t is measured in minutes and temperature ( )H t is measured in degrees Celsius. Values of ( )H t 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) Using correct units, explain the meaning of

10

0

1 ( )10

H t dt∫ in the context of this problem. Use a trapezoidal

sum with the four subintervals indicated by the table to estimate10

0

1 ( )10

H t dt∫ .

c) Evaluate10

0( )H t dt′∫ . Using correct units, explain the meaning of the expression in the context of this

problem. d) At time t = 0, biscuits with temperature 100oC were removed from an oven. The temperature of the biscuits at

time t is modeled by a differentiable function B for which it is known that 0.173( ) 13.84 tB t e−′ = − . Using the given models, at time t = 10, how much cooler are the biscuits than the tea?

2010 AB1 There is no snow on Janet’s driveway when snow begins to fall at midnight. From midnight to 9 A.M., snow accumulates on the driveway at a rate modeled by cos( ) 7 tf t e= 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

0 for 0 6( ) 125 for 6 7

108 for 7 9

tg t t

t

≤ <= ≤ < ≤ ≤

.

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 0 9t≤ ≤ . d) How many cubic feet of snow are on the driveway at 9 A.M.?


2010 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 = 0) 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 0 8t≤ ≤ . Values of ( )E t , 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)

Use a trapezoidal sum with the four subintervals given by the table to approximate the value of8

0

1 ( )8

E t dt∫

Using correct units, explain the meaning of 8

0

1 ( )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 function P, where 3 2( ) 30 298 976P t t t t= − + − hundreds of entries per hour for 8 12t≤ ≤ . 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.

2010 AB3 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 move 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 the ride begins operation.

a) How many people arrive at the ride between t = 0 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.


2009 AB1

Caren rides her bicycle along a straight road from home to school, starting at home at time t = 0 minutes and arriving at school at time t = 12 minutes. During the time interval 0 12t≤ ≤ 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

12

0( )v t dt∫ in terms of Caren’s trip. Find the value of

12

0( )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

by the function given by ( ) sin15 12

w w t tπ π =

where ( )w t is in miles per minute for 0 12t≤ ≤ minutes.

Who lives closer to school: Caren or Larry? Show the work that leads to your answer. 2009 AB2 The rate at which people enter an auditorium for a rock concert is modeled by the function R given by

2 3( ) 1380 675 for 0 2R t t t t= − ≤ ≤ hours; ( )R t is measured in people per hour. No one is in the auditorium at time0t = , when the doors open. The doors close and the concert begins at time 2t = .

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

( )( ) 2 ( )w t t R t′ = − . Find (2) (1)w w− , 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).


2009 AB3 Mighty Cable Company manufactures cable that sells for $120 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) Using correct units, explain the meaning of

30

256 x dx∫ in the context of this problem.

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. 2007 AB2/BC2

The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval0 7,t≤ ≤ where t is measured in hours. In this model, rates are given as follows:

The rate at which water enter the tank is ( )2( ) 100 sinf t t t= gallons per hour for 0 7t≤ ≤ .

The rate at which water leaves the tank is 250 for 0 3

( )2000 for 3 7

tg t

t≤ <

= < ≤gallons per hour.

The graphs of and f g , which intersect at 1.617 and 5.076t t= = , are shown in the figure. At time 0t = , the amount of water in the tank is 5000 gallons a) How many gallons of water enter the tank during the time interval 0 7t≤ ≤ ? Round your answer to the

nearest gallon. b) For 0 7t≤ ≤ , find the time intervals during which the amount of water in the tank is decreasing. Give a

reason for your answer. c) For 0 7t≤ ≤ , 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.


2006 AB2/BC2 At an intersection in Thomasville, Oregon, cars turn left at

the rate 2( ) 60 sin3tL t t =

cars per hour over the time

interval 0 18t≤ ≤ hours. The graph of ( )y L t= is shown.

a) To the nearest whole number, find the total number of cars turning left at the intersection over the time

interval 0 18t≤ ≤ hours. b) Traffic engineers will consider turn restrictions when ( ) 150L t ≥ cars per hour. Find all values of t for which

( ) 150L t ≥ 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 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does the intersection require a traffic signal? Explain the reasoning that leads to your conclusion.

2005 AB2 The tide removes sand from Sandy Point Beach at a rate modeled by the function R , given by

4( ) 2 5sin25

tR t π = +

.

A pumping station adds sand to the beach at a rate modeled by the function S , given by 15( )1 3

tS tt

=+

.

Both ( ) and ( )R t S t have units of cubic yards per hour and t is measured in hours for 0 6t≤ ≤ . At time 0t = , the beach contains 2500 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 ( )Y t , 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 4t = . d) For 0 6t≤ ≤ , at what time t is the amount of sand on the beach a minimum? What is the minimum value?

Justify your answer.


2004 AB1/BC1 Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The

traffic flow at a particular intersection is modeled by the function F defined by ( ) 82 4sin2tF t = +

for 0 30t≤ ≤ ,

where ( )F t 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 30-minute period? b) Is the traffic flow increasing or decreasing at 7t = ? Give a reason for your answer. c) What is the average value of the traffic flow over the time interval 10 15t≤ ≤ ? Indicate units of measure. d) What is the average rate of change of the traffic flow over the time interval 10 15t≤ ≤ ? Indicate units of

measure. 2004 AB3 A particle moves along the y-axis so that its velocity v at time 0t ≥ is given by ( )1( ) 1 tan tv t e−= − . At time 0t = , the particle is at 1y = − . a) Find the acceleration of the particle at time 2t = . b) Is the speed of the particle increasing or decreasing at time 2t = ? Give a reason for your answer. c) Find the time 0t ≥ at which the particle reaches its highest point. Justify your answer. d) Find the position of the particle at time 2t = . Is the particle moving toward the origin or away from the

origin at time 2t = ? Justify your answer. 2004 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

by3

( ) ( )x

g x f t dt−

= ∫ .

a) Find (0) and (0).g 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.


2003 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 ( )R t for the time interval 0 90t≤ ≤ minutes are shown above. a) Use data from the table to find an approximation for (45).R′ Show the computations that lead to your answer.

Indicate units of measure. b) The rate of fuel consumption is increasing fastest at time 45t = minutes. What is the value of (45)R′′ ?

Explain your reasoning. c) Approximate the value of

90

0( )R t dt∫ using a left Riemann sum with the five subintervals indicated by the data

in the table. Is this numerical approximation less that the value of 90

0( )R t dt∫ ? Explain your reasoning.

d) For 0 90b< ≤ minutes, explain the meaning of 0

( )bR t dt∫ in terms of fuel consumption for the plane. Explain

the meaning of 0

1 ( )bR t dt

b ∫ in terms of fuel consumption for the plane. Indicate units of measure in both

answers.


2002 AB2/BC2 The rate at which people enter an amusement park on a given day is modeled by the function E defined by

2

15600( )24 160

E tt t

=− +

.

The rate at which people leave the same amusement park on the same day is modeled by the function L defined by

2

98909 )38 370

L tt t

=− +

.

Both ( ) and ( )E t L t are measured in people per hour, and time t is measured in hours after midnight. These functions are valid for 9 23t≤ ≤ , which are the hours that the park is open. At time 9t = , there are no people in the park. a) How many people have entered the park by 5:00 P.M. ( 17)t = ? Round your answer to the nearest whole

number. b) The price of admission to the park is $15 until 5:00 P.M. ( 17)t = . After 5:00 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) Let ( )9

( ) ( ) ( ) for 9 23t

H t E x L x dx t= − ≤ ≤∫ . The value of (17)H to the nearest whole number is 3725. Find

the value of (17)H ′ and explain the meaning of (17)H and (17)H ′ in the context of the park. d) At what time t, for 9 23t≤ ≤ , does the model predict that the number of people in the park is a maximum?

2002 AB3 An object moves along the x-axis with initial position (0) 2x = . The velocity of the object at time 0t ≥ is given by

( ) sin3

v t tπ =

.

a) What is the acceleration of the object at time 4t = ? b) Consider the following two statements.

Statement I: For 3 4.5t< < , the velocity of the object is decreasing. Statement II: For 3 4.5t< < , 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 0 4t≤ ≤ ? d) What is the position of the object at time 4t = ?


2002 AB4/BC4 The graph of the function f shown to the right consists of two line

segments. Let g be the function given by0

( ) ( )x

g x f t dt= ∫ .

a) Find ( 1), ( 1), and ( 1)g g g′ ′′− − − 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− .

2001 AB 3 / BC 3 A car is traveling on a straight road with velocity 55 ft sec at time 0t = . For 0 18t≤ ≤ seconds, the car's acceleration, in 2ft sec , is the piecewise linear function defined by the graph.

a) Is the velocity of the car increasing at 2t = seconds? Why or why not? b) At what time in the interval 0 18t≤ ≤ , other than 0t = , is the velocity of the car 55 ft sec ? c) On the time interval 0 18t≤ ≤ , 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 0 18t≤ ≤ , if any, is the car's velocity equal to zero? Justify your answer.


2000 AB 2 / BC 2 Two runners, A and B, run on a straight racetrack for 0 10t≤ ≤ 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

given by the function v defined by 24( )2 3

tv tt

=+

.

a) Find the velocity of Runner A and the velocity of Runner B at time 2t = seconds. Indicate units of measure. b) Find the acceleration of Runner A and the acceleration of Runner B at time 2t = 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

0 10t≤ ≤ seconds. Indicate units of measure.

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 the rate of 1t + gallons per minute, for 0 120t≤ ≤ minutes. At time 0t = , the tank contains 30 gallons of water. a) How many gallons leak out of the tank from time 0 to 3t t= = minutes? b) How many gallons of water are in the tank at 0t = minutes? c) Write an expression for ( )A t , the total number of gallons of water in the tank at time t . d) At what time t , for 0 120t≤ ≤ , is the amount of water in the tank a maximum? Justify your answer.


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.

t (hours)

( )R t (gallons per hour)

0 9.6 3 10.4 6 10.8 9 11.2

12 11.4 15 11.3 18 10.7 21 10.2 24 9.6

a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate 24

0( )R t dt∫ . Using correct

units, explain the meaning of your answer in terms of water flow. b) Is there some time , 0 24,t t< < such that ( ) 0R t′ > ? Justify your answer. c)

The rate of water flow ( )R t can be approximated by ( )21( ) 768 2379

Q t t t= + − . Use ( )Q t to approximate the

average rate of water flow during the 24-hour time period. Indicate units of measure.


Textbook Problems Calculus, Finney, Demanna, Waits, Kennedy; Prentice Hal, l2012

Section Questions 8.1 12-16 17 18 21 22 23 24 8.R 2 3 5 54 55

Handouts


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

meters from the stage is 310x

− people per

square meter, for 3 30x≤ ≤ .

a) How fast is the density of the people changing at a distance of 10 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?


AP Calculus Chapter 7 Section 1 Answers 1. a) 1 person per square meter3

10 10 meterd xdx

− = −

b)

( )

22

15

3

people 23 meter10 meter 2

3 667.274 667people10

x x x

x x dx

π

π

− ⋅ ∆ − =

∑

∫

c) ( )

30

33 1374.132 1374people

10x x dxπ − =

∫

d) ( )

3

1374.1323 687 15.279 meters10 2

r x x dx rπ − = = = ∫

e) 2

2 2

1374 people.9817 0.981 or 0.982 meter30 3

2 2π π =

−


AP Calculus

Chapter 7 Section 1 Problems Water Tank A water tank at Camp Newton holds 1200 gallons of water at time 0t = . During the time interval 0 18t≤ ≤ hours,

water is pumped into the tank at the rate 2( ) 95 sin6tW t t =

gallons per hour.

During the same time interval, water is removed from the tank at the rate of 2( ) 275sin3tR t =

gallons per hour.

a. Is the amount of water in the tank increasing at time 15t = ? Why or why not? b. To the nearest whole number, how many gallons of water are in the tank at time 18t = ? c. At what time , for 0 18t t≤ ≤ , is the amount of water in the tank at an absolute minimum? Show the work

that leads to your conclusion. d. For 18t > , no water is pumped into the tank, but water continues to be removes at the rate ( )R t 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 0 31t≤ ≤ , the rate of change of the number of mosquitoes on a tropical island at time t days is modeled by

( ) 5 cos5tR t t =

mosquitoes per day. There are 1000 mosquitoes on the island at time 0t = .

a. Show the number of mosquitoes is increasing at time 6.t = b. At time 6t = , 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 31t = ? Round you answer to the

nearest whole number. d. To the nearest whole number, what is the maximum number of mosquitoes for 0 31t≤ ≤ ? Show the analysis

that leads to your conclusion. Heating Oil A tank contains 125 gallons of heating oil at time 0t = . During the time interval 0 12t≤ ≤ hours, heating oil is

pumped into the tank at the rate 10( ) 3(1 ln( 1))

H tt

= ++ +

gallons per hour. During the same time interval, heating oil

is removed from the tank at the rate of 2

( ) 12sin47tR t

=

gallons per hour.

a. How many gallons of heating oil are pumped into the tank during the time interval 0 12t≤ ≤ hours? b. Is the level of heating oil in the tank rising or falling at time 6t = hours? Give a reason for your answer. c. How many gallons of heating oil are in the tank at time 12t = hours? d. At what time , for 0 12t t≤ ≤ , is the volume of heating oil in the tan the least? Show the analysis that leads to

your conclusion.


AP Calculus Chapter 7 Section 1 Answers

Water Tank

a. No. (15) (15) 121.09 0W R− = − <

b. 1310 gallons

c. 6.494 or 6.495t = . Check values at critical points and endpoints.

d. 18

( ) 1310k

R t dt =∫

Mosquitoes

a. Increasing, (6) 4.438 0R = >

b. Increasing at a decreasing rate. (6) 1.913 0R′ = − < .

c. 964 mosquitoes.

d. The maximum number of mosquitoes is 1039 at time 31t = .

Heating Oil

a. 70.570 or 70.571 gallons

b. The level of heating oil is falling at time 6t = .

c. 122.025 or 122026 gallons

d. The volume is least at 11.318t = hours.


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 ( ) sin( )E t t t= + be the rate of water entering Lake Newton at a hundred cubic feet hour .

Let 2 1 6

( )10.5 6 12

tR t

t≤ ≤

= < ≤be the rate of water being released from the floodgates at a hundred cubic feet hour .

Note that ( ) 0 when 0 1R t t= ≤ < . 1. Graph ( ) and ( )E t R t . 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? 3. What is the average rate of change of [ ]( ) on 0,E t π ? Explain why there must be a value of t for which the

average rate of change of ( )E t is equal to the instantaneous rate of change of [ ]( ) on 0,E t π and find that exact value of t .

4. How much water entered the lake in the first hour of monitoring? 5, Find the average value of [ ]( ) on 1,12R t . 6. From 2 to 12t t= = 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 0t = , what is the amount of water Q in

the lake at 7t = ? 8. At what time t will the amount of water in the lake return to the amount at time 0t = ?


Lake Newton Answers

1

2 1.106, 11.414 hourst =

3

( )E t is continuous and differentiable on [ ]0,π , therefore the Mean Value Theorem holds and the average

rate of change of [ ]( ) (0) 0( ), 1 ( ) for some in 0,0 0

E EE t E t tπ π ππ π− − ′= = =− −

.

( ) 1 1 cos( )2

E t t t π′ = = + =

4 0.95969 hundred cubic feet

5 6 12

1 62 10.5

6.636 hundred cubic feet hour12 1

dt dt+=

−∫ ∫

6 2 6, 11.414 12 6.893 hundred cubic feett t≤ ≤ ≤ ≤

7 4.246 hundred cubic feetA+

8 9.84736 hours


AP Calculus Chapter 7 Section 1 The function ( )S t 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 0 365t≤ ≤ ( t is in days and 0t = is midnight on January 1).

( ) hours( ) 2.657sin 78.5 12.11 day186S t xπ = − +

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 30 (day 181)? b. Is the length of the day increasing or decreasing on the 100th day? Explain. c. What is the average number of hours of daylight from day 30 to day 150? 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.


Data

Date Day Number Hours of Daylight 1/1 1 9.5 09:28:34 2/1 32 10.3 10:13:47 3/1 60 11.3 11:19:25 4/1 91 12.6 12:38:40 5/1 121 13.8 13:50:41 6/1 152 14.6 14:43:49 7/1 182 14.8 14:52:12 8/1 213 14.2 14:11:39 9/1 243 13.0 13:02:39 10/1 274 11.8 11:47:26 11/1 305 10.5 10:32:07 12/1 335 9.6 09:37:47

Source: http://www.davidcmartin.com/Nav/Sunrise/sunrise.htm


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