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AP CALCULUS AB :: 2006-2007 :: QUIZ TWO
Name______________________ Page 1
Problem 1 Source: AP CALCULUS AB EXAM 2004 – SECTION II PART A PROBLEM 1 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:
F (t ) = 82 + 4sin ( 2 t ) for 0 £ t £ 30 , where F (t ) is measured in cars/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 t = 7 ? Give a reason for your answer. c) What is the average value of the traffic flow over the time interval 10 £ t £ 15 ? Indicate units of measure. d) What is the average rate of change of the traffic flow over the time interval 10 £ t £ 15 ? Indicate units of measure. Problem 2 Source: AP CALCULUS AB EXAM 2005 – SECTION II PART A PROBLEM 2 The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by p t R(t ) = 2 + 5sin ( 4 25 ) .
A pumping station adds sand to the beach at a rate modeled by the function S, given by
S (t ) =
15 t . 1 + 3 t
Both R (t ) and S (t ) have units of cubic yards per hour and t is measured in hour for 0 £ t £ 6 . At time
t = 0, 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 the 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 t = 4. d) For 0 £ t £ 6 , at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.
TURN OVER
AP CALCULUS AB :: 2006-2007 :: QUIZ TWO
Name______________________ Page 2
SHORT QUESTIONS (Show work for credit!) x 5
3. Solve for x :
2
ò t dt = 54
x 2
4. Find the average value of f ( x) = 4 x 3 + 2 on the closed interval [0, 2]. b 3
2
b
5. If f ( x ) = x - 2 x + 1 and g ( x ) = x - 2 x + 1 , for what values of a and b is f ( x ) dx < g ( x) dx ?
ò a
k
6. If 0 < k < p , find the value of k that makes cos 2 x dx = 1 2 .
ò
0
"I was x years old in the year x 2 ." -- Augustus De Morgan (when asked about his age)
ò
a
Name______________________ Page 1
Problem 1 Source: AP CALCULUS AB EXAM 2004 – SECTION II PART A PROBLEM 1 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:
F (t ) = 82 + 4sin ( 2 t ) for 0 £ t £ 30 , where F (t ) is measured in cars/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 t = 7 ? Give a reason for your answer. c) What is the average value of the traffic flow over the time interval 10 £ t £ 15 ? Indicate units of measure. d) What is the average rate of change of the traffic flow over the time interval 10 £ t £ 15 ? Indicate units of measure. Problem 2 Source: AP CALCULUS AB EXAM 2005 – SECTION II PART A PROBLEM 2 The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by p t R(t ) = 2 + 5sin ( 4 25 ) .
A pumping station adds sand to the beach at a rate modeled by the function S, given by
S (t ) =
15 t . 1 + 3 t
Both R (t ) and S (t ) have units of cubic yards per hour and t is measured in hour for 0 £ t £ 6 . At time
t = 0, 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 the 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 t = 4. d) For 0 £ t £ 6 , at what time t is the amount of sand on the beach a minimum? What is the minimum value? Justify your answers.
TURN OVER
AP CALCULUS AB :: 2006-2007 :: QUIZ TWO
Name______________________ Page 2
SHORT QUESTIONS (Show work for credit!) x 5
3. Solve for x :
2
ò t dt = 54
x 2
4. Find the average value of f ( x) = 4 x 3 + 2 on the closed interval [0, 2]. b 3
2
b
5. If f ( x ) = x - 2 x + 1 and g ( x ) = x - 2 x + 1 , for what values of a and b is f ( x ) dx < g ( x) dx ?
ò a
k
6. If 0 < k < p , find the value of k that makes cos 2 x dx = 1 2 .
ò
0
"I was x years old in the year x 2 ." -- Augustus De Morgan (when asked about his age)
ò
a
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