تكلم مع الذكاء الاصطناعي
Deck 10: Derivatives
ملء الشاشة (f)
سؤال
Make a sign diagram for the function and determine all x-values at which relative minima occur. 
A)
B)
C)
D)
E) no relative minima
A)
B)
C)
D)
E) no relative minima
سؤال
Use the sign diagram for 
to determine the largest interval on which f (x) increases, where 
and 
. 
A)
B)
C)
D)
E)
to determine the largest interval on which f (x) increases, where and . A)
B)
C)
D)
E)
سؤال
For the given function and graph, determine all critical point(s), where 
. 
A)
B)
C)
D)
and 
E)
and 
and 
. A)
B)
C)
D)
and E)
and and سؤال
Use the sign diagram for 
to determine x-values at which relative minima occur, where 
and 
. 
A)
B)
C)
and 
D)
E) no relative minima
to determine x-values at which relative minima occur, where and . A)
B)
C)
and D)
E) no relative minima
سؤال
Use the graph of 
to identify at which of the indicated points the derivative 
changes from positive to negative. 
A) (5, 6)
B) (-1, 2), (2, 4)
C) (-1, 2)
D) (2, 4)
E) (2, 4), (5, 6)
to identify at which of the indicated points the derivative changes from positive to negative. A) (5, 6)
B) (-1, 2), (2, 4)
C) (-1, 2)
D) (2, 4)
E) (2, 4), (5, 6)
سؤال
For the given function, find all intervals of x-values where the function is increasing. 
A)
B)
C)
or 
D)
E)
A)
B)
C)
or D)
E)
سؤال
For the given function, find the critical points. 
A)
and 
B)
and 
C)
and 
D)
and 
E) (
and 
A)
and B)
and C)
and D)
and E) (
and سؤال
Make a sign diagram for the function and determine all x-values at which relative maxima occur. 
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
سؤال
Use the graph of 
to identify at which of the indicated points the derivative 
changes from negative to positive. 
A) (-1, 2), (2, 4)
B) (4, 1)
C) (-1, 2), (2, 4), (5, 6)
D) (2, 4)
E) (2, 4), (5, 6)
to identify at which of the indicated points the derivative changes from negative to positive. A) (-1, 2), (2, 4)
B) (4, 1)
C) (-1, 2), (2, 4), (5, 6)
D) (2, 4)
E) (2, 4), (5, 6)
سؤال
For the given function, find 
A) 
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Use the sign diagram for 
to determine the largest interval on which f (x) decreases, where 
and 
. 
A)
B)
C)
D)
E)
to determine the largest interval on which f (x) decreases, where and . A)
B)
C)
D)
E)
سؤال
Use the graph of 
to identify at which of the indicated points the derivative 
does not change sign. 
A) (-1, 2), (2, 4), (5, 6)
B) (-1, 2)
C) (-1, 2), (5, 6)
D) (-1, 2), (2, 4)
E) (2, 4)
to identify at which of the indicated points the derivative does not change sign. A) (-1, 2), (2, 4), (5, 6)
B) (-1, 2)
C) (-1, 2), (5, 6)
D) (-1, 2), (2, 4)
E) (2, 4)
سؤال
For the given function, find all critical values. 
A) 
B)
and 
C)
and 
D)
and 
E)
and 
A)
B)
and C)
and D)
and E)
and سؤال
For the given function, find 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
For the given function, find all intervals of x-values where the function is decreasing. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
For the given function and graph, determine all critical value(s), where 
. 
A)
B)
C)
D)
and 
E)
and 
and 
. A)
B)
C)
D)
and E)
and and سؤال
Use the sign diagram for 
to determine x-values at which relative maxima occur, where 
and 
. 
A)
B) 
C)
and 
D)
E) no relative maxima
to determine x-values at which relative maxima occur, where and . A)
B)
C)
and D)
E) no relative maxima
سؤال
For the given function and graph, estimate the coordinates of the relative minima by observing the graph, where 
. 
A)
B)
C)
D)
E) no relative minima
. A)
B)
C)
D)
E) no relative minima
سؤال
For the given function and graph, estimate the coordinates of the relative maxima by observing the graph, where 
. 
A)
B)
C)
D)
E) no relative maxima
. A)
B)
C)
D)
E) no relative maxima
سؤال
Use the sign diagram for 
to determine all critical values of f (x), where 
and 
. 
A)
B)
C)
and 
D)
E)
to determine all critical values of f (x), where and . A)
B)
C)
and D)
E)
سؤال
For the given function, find intervals of x-values where the function is decreasing. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
For the given function, classify the critical points as relative maxima, relative minima, or points of inflection. In each case, you may check your conclusions with a graphing utility. 
A)
is a relative minimum and 
is point of inflection.
B)
is a point of inflection and 
is a relative minimum.
C)
is a point of inflection and 
is a relative maximum.
D)
is a relative minimum and 
is a relative maximum.
E)
is a relative maximum and 
is a relative minimum.
A)
is a relative minimum and is point of inflection.B)
is a point of inflection and is a relative minimum.C)
is a point of inflection and is a relative maximum.D)
is a relative minimum and is a relative maximum.E)
is a relative maximum and is a relative minimum. سؤال
For the given function, find the relative maxima. 
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
سؤال
For the given function, find intervals of x-values where the function is increasing. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
For the given function, use the graph to identify x-values for which 
. You may use the derivative to check your conclusion. 
A)
B)
C)
D)
E)
. You may use the derivative to check your conclusion. A)
B)
C)
D)
E)
سؤال
Use the derivative to locate all critical points. Use a graphing utility if desired. Round your answers to two decimal places. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
For the given function, find the critical points. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Both a function and its derivative are given. Use them to find intervals on which the function is decreasing. 
A)
B)
C)
D)
or 
E)
A)
B)
C)
D)
or E)
سؤال
For the given function, find the horizontal points of inflection. 
A)
B)
C)
D)
E) no horizontal points of inflection
A)
B)
C)
D)
E) no horizontal points of inflection
سؤال
For the given function, find the critical values. 
A)
and 
B)
and 
C)
and 
D)
and 
E)
and 
A)
and B)
and C)
and D)
and E)
and سؤال
Both a function and its derivative are given. Use them to find all critical values. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
For the given function find the relative maxima, and sketch the graph. You may check your graph with a graphing utility. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Both a function and its derivative are given. Use them to find all critical points. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
For the given function, use the graph to identify x-values for which 
. You may use the derivative to check your conclusion. 
A)
B)
C)
D)
E)
. You may use the derivative to check your conclusion. A)
B)
C)
D)
E)
سؤال
For the given function, use the graph to identify the x-value for which 
. You may use the derivative to check your conclusion. 
A)
B)
C)
D)
E)
. You may use the derivative to check your conclusion. A)
B)
C)
D)
E)
سؤال
For the given function, find the relative minima. 
A)
B)
C)
D)
E) no relative minima
A)
B)
C)
D)
E) no relative minima
سؤال
Both a function and its derivative are given. Use them to find intervals on which the function is increasing. 
A)
or 
or 
B)
C)
or 
D)
or 
E)
A)
or or B)
C)
or D)
or E)
سؤال
Both a function and its derivative are given. Use them to find the relative maxima. 
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
سؤال
For the given function find the relative minima, and sketch the graph. You may check your graph with a graphing utility. 
A)
B)
and 
C)
and 
D)
E)
A)
B)
and C)
and D)
E)
سؤال
Both a function and its derivative are given. Use them to find the relative minima. 
A)
B)
C)
D)
E) no relative minima
A)
B)
C)
D)
E) no relative minima
سؤال
Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that its monthly sales volume y (in thousands of dollars) is related to the price p (in dollars) of an oil change by 
. Is y increasing or decreasing for all values of 
A) increasing
B) decreasing
. Is y increasing or decreasing for all values of A) increasing
B) decreasing
سؤال
The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by 
, 
. For what t-values is x increasing? Round answers to two decimal places.
A)
B)
C)
D)
E)
, . For what t-values is x increasing? Round answers to two decimal places. A)
B)
C)
D)
E)
سؤال
Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by 
, 
, where x is the number of machines used. Over what interval in the domain do average costs decrease?
A) 
B)
C)
D)
E)
, , where x is the number of machines used. Over what interval in the domain do average costs decrease? A)
B)
C)
D)
E)
سؤال
Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by 
, 
, where x is the number of machines used. What is the minimum average cost? Round your answer to the nearest dollar.
A)
B)
C)
D)
E)
, , where x is the number of machines used. What is the minimum average cost? Round your answer to the nearest dollar. A)
B)
C)
D)
E)
سؤال
A graph of 
is given. Use the graph to determine where the graph of 
has a relative maximum. 
A)
B)
C)
D)
E) no relative maxima
is given. Use the graph to determine where the graph of has a relative maximum. A)
B)
C)
D)
E) no relative maxima
سؤال
The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by 
, 
. Find the maximum value of x. Round your answer to two decimal places.
A) 2.83 mg
B) 530.33 mg
C) 1166.73 mg
D) 8.77 mg
E) 1125.30 mg
, . Find the maximum value of x. Round your answer to two decimal places. A) 2.83 mg
B) 530.33 mg
C) 1166.73 mg
D) 8.77 mg
E) 1125.30 mg
سؤال
Determine whether the given function is concave up or concave down at the indicated point. 
at 
A) concave down
B) concave up
at A) concave down
B) concave up
سؤال
Use the indicated x-values on the graph of 
to find all the intervals where 
. 
A)
B)
and 
C)
D)
E)
and 
to find all the intervals where . A)
B)
and C)
D)
E)
and سؤال
Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by 
, 
, where x is the number of machines used. Over what interval in the domain do average costs increase?
A)
B)
C)
D)
E)
, , where x is the number of machines used. Over what interval in the domain do average costs increase? A)
B)
C)
D)
E)
سؤال
The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by 
, 
. Find the t-value at which x is maximum. Round your answer to two decimal places.
A) 0 hours
B) 1.41 hours
C) 1767.77 hours
D) 3.41 hours
E) 3.45 hours
, . Find the t-value at which x is maximum. Round your answer to two decimal places. A) 0 hours
B) 1.41 hours
C) 1767.77 hours
D) 3.41 hours
E) 3.45 hours
سؤال
A function and its graph are given. Use the second derivative to locate all x-values of points of inflection on the graph of 
. Check these results against the graph shown. 
A)
B)
C)
D)
, 
E)
, 
, 
. Check these results against the graph shown. A)
B)
C)
D)
, E)
, , سؤال
Use the indicated x-values on the graph of 
to find the x-coordinate of any horizontal point of inflection. 
A) a
B) c
C) f
D) e
E) d
to find the x-coordinate of any horizontal point of inflection. A) a
B) c
C) f
D) e
E) d
سؤال
A graph of 
is given. Use the graph to determine all critical values of 
. 
A)
B)
C)
D)
E)
, 
, 
is given. Use the graph to determine all critical values of . A)
B)
C)
D)
E)
, , سؤال
Use the indicated x-values on the graph of 
to determine intervals over which the graph is concave up. 
A)
and 
B)
C)
D)
E)
and 
to determine intervals over which the graph is concave up. A)
and B)
C)
D)
E)
and سؤال
Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by 
, 
, where x is the number of machines used. How many machines give minimum average costs?
A) Using 25 machines gives the minimum average costs.
B) Using zero machines gives the minimum average costs.
C) Using 35 machines gives the minimum average costs.
D) Using 50 machines gives the minimum average costs.
E) Using 55 machines gives the minimum average costs.
, , where x is the number of machines used. How many machines give minimum average costs? A) Using 25 machines gives the minimum average costs.
B) Using zero machines gives the minimum average costs.
C) Using 35 machines gives the minimum average costs.
D) Using 50 machines gives the minimum average costs.
E) Using 55 machines gives the minimum average costs.
سؤال
A graph of 
is given. Use the graph to determine where the graph of 
has a relative minimum. 
A)
B)
C)
D)
E) no relative minima
is given. Use the graph to determine where the graph of has a relative minimum. A)
B)
C)
D)
E) no relative minima
سؤال
A graph of 
is given. Use the graph to determine where f (x) is decreasing. 
A)
B)
C)
D)
or 
E)
is given. Use the graph to determine where f (x) is decreasing. A)
B)
C)
D)
or E)
سؤال
A function and its graph are given. Use the second derivative to determine intervals on which the function is concave up. Check these results against the graph shown. 
A)
B)
C)
D)
or 
E)
A)
B)
C)
D)
or E)
سؤال
Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by 
, 
, where x is the number of machines used. Find the critical values of 
that lie in the domain of the problem.
A)
B)
C)
D)
E)
, , where x is the number of machines used. Find the critical values of that lie in the domain of the problem.
A)
B)
C)
D)
E)
سؤال
A function and its graph are given. Use the second derivative to determine intervals on which the function is concave down. Check these results against the graph shown. 
A)
B)
C)
D)
or 
E)
A)
B)
C)
D)
or E)
سؤال
A function and its graph are given. From the graph estimate where 
has a relative minimum. 
A)
B)
C)
D)
E) no relative minima
has a relative minimum. A)
B)
C)
D)
E) no relative minima
سؤال
Find all points of inflection of the given function. Round your answer to two decimal places. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
The following figure shows the growth of a population as a function of time. 
Which of A, B, and C correspond(s) to the point(s) at which the growth rate attains its maximum?
A) B
B) A
C) C
D) A and C
E) none of the above
Which of A, B, and C correspond(s) to the point(s) at which the growth rate attains its maximum? A) B
B) A
C) C
D) A and C
E) none of the above
سؤال
Use the graph shown in the figure and identify points from A through I that satisfy the given condition. 
and 
A) B
B) D
C) C
D) F
E) G
and A) B
B) D
C) C
D) F
E) G
سؤال
Use the graph shown in the figure and identify points from A through I that satisfy the given condition. 
and 
A) C
B) F
C) I
D) A
E) B
and A) C
B) F
C) I
D) A
E) B
سؤال
A function and its graph are given. From the graph, estimate where 
. 
A)
B) 
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
Use the graph shown in the figure and identify points from A through I that satisfy the given condition. 
and 
A) D
B) G
C) B
D) C
E) E
and A) D
B) G
C) B
D) C
E) E
سؤال
The following figure shows the growth of a population as a function of time. 
If P represents the population and t represents the time, write a mathematical symbol that represents the rate of change (growth rate) of the population with respect to time.
A) 
B)
C)
D)
E)
If P represents the population and t represents the time, write a mathematical symbol that represents the rate of change (growth rate) of the population with respect to time.A)
B)
C)
D)
E)
سؤال
A function and its first and second derivatives are given. Use these to find all points of inflection. 
A)
B)
C)
D)
E) no points of inflection
A)
B)
C)
D)
E) no points of inflection
سؤال
A function and its graph are given. From the graph estimate where 
has a relative maximum. 
A)
B)
C)
D)
E) no relative maxima
has a relative maximum. A)
B)
C)
D)
E) no relative maxima
سؤال
In this problem, 
and its graph are given. Use the graph of 
to determine where 
has a point of inflection. 
A)
B)
C)
D)
E) no point of inflection
and its graph are given. Use the graph of to determine where has a point of inflection. A)
B)
C)
D)
E) no point of inflection
سؤال
A function and its graph are given. From the graph, estimate where 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
A function and its first and second derivatives are given. Use these to find the relative maxima. 
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
سؤال
In this problem, 
and its graph are given. Use the graph of 
to determine where 
is concave down. 
A)
B)
C)
D)
E)
and its graph are given. Use the graph of to determine where is concave down. A)
B)
C)
D)
E)
سؤال
A function and its first and second derivatives are given. Use these to find all critical values. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
Find all relative minima of the given function. 
A)
B)
C)
D)
E) no relative minima
A)
B)
C)
D)
E) no relative minima
سؤال
Use the graph shown in the figure and identify points from A through I that satisfy the given condition. 
and 
A) I
B) F
C) C
D) D
E) H
and A) I
B) F
C) C
D) D
E) H
سؤال
Find all relative maxima of the given function. 
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
سؤال
In this problem, 
and its graph are given. Use the graph of 
to determine where 
is concave up. 
A)
B)
C)
D)
E)
and its graph are given. Use the graph of to determine where is concave up. A)
B)
C)
D)
E)
سؤال
A function and its first and second derivatives are given. Use these to find all relative minima. 
A)
B)
C)
D)
E) no relative minima
A)
B)
C)
D)
E) no relative minima
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ملء الشاشة (f)
Deck 10: Derivatives
1
Make a sign diagram for the function and determine all x-values at which relative minima occur.
A)
B)
C)
D)
E) no relative minima
A)
B)
C)
D)
E) no relative minima
2
Use the sign diagram for to determine the largest interval on which f (x) increases, where and .
A)
B)
C)
D)
E)
to determine the largest interval on which f (x) increases, where and . A)
B)
C)
D)
E)
3
For the given function and graph, determine all critical point(s), where .
A)
B)
C)
D) and
E) and and
. A)
B)
C)
D)
and E)
and and and 4
Use the sign diagram for to determine x-values at which relative minima occur, where and .
A)
B)
C) and
D)
E) no relative minima
to determine x-values at which relative minima occur, where and . A)
B)
C)
and D)
E) no relative minima
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5
Use the graph of to identify at which of the indicated points the derivative changes from positive to negative.
A) (5, 6)
B) (-1, 2), (2, 4)
C) (-1, 2)
D) (2, 4)
E) (2, 4), (5, 6)
to identify at which of the indicated points the derivative changes from positive to negative. A) (5, 6)
B) (-1, 2), (2, 4)
C) (-1, 2)
D) (2, 4)
E) (2, 4), (5, 6)
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6
For the given function, find all intervals of x-values where the function is increasing.
A)
B)
C) or
D)
E)
A)
B)
C)
or D)
E)
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7
For the given function, find the critical points.
A) and
B) and
C) and
D) and
E) ( and
A)
and B)
and C)
and D)
and E) (
and فتح الحزمة
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8
Make a sign diagram for the function and determine all x-values at which relative maxima occur.
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
فتح الحزمة
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9
Use the graph of to identify at which of the indicated points the derivative changes from negative to positive.
A) (-1, 2), (2, 4)
B) (4, 1)
C) (-1, 2), (2, 4), (5, 6)
D) (2, 4)
E) (2, 4), (5, 6)
to identify at which of the indicated points the derivative changes from negative to positive. A) (-1, 2), (2, 4)
B) (4, 1)
C) (-1, 2), (2, 4), (5, 6)
D) (2, 4)
E) (2, 4), (5, 6)
فتح الحزمة
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10
For the given function, find
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
فتح الحزمة
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11
Use the sign diagram for to determine the largest interval on which f (x) decreases, where and .
A)
B)
C)
D)
E)
to determine the largest interval on which f (x) decreases, where and . A)
B)
C)
D)
E)
فتح الحزمة
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12
Use the graph of to identify at which of the indicated points the derivative does not change sign.
A) (-1, 2), (2, 4), (5, 6)
B) (-1, 2)
C) (-1, 2), (5, 6)
D) (-1, 2), (2, 4)
E) (2, 4)
to identify at which of the indicated points the derivative does not change sign. A) (-1, 2), (2, 4), (5, 6)
B) (-1, 2)
C) (-1, 2), (5, 6)
D) (-1, 2), (2, 4)
E) (2, 4)
فتح الحزمة
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13
For the given function, find all critical values.
A)
B) and
C) and
D) and
E) and
A)
B)
and C)
and D)
and E)
and فتح الحزمة
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14
For the given function, find
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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15
For the given function, find all intervals of x-values where the function is decreasing.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
فتح الحزمة
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16
For the given function and graph, determine all critical value(s), where .
A)
B)
C)
D) and
E) and and
. A)
B)
C)
D)
and E)
and and فتح الحزمة
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17
Use the sign diagram for to determine x-values at which relative maxima occur, where and .
A)
B)
C) and
D)
E) no relative maxima
to determine x-values at which relative maxima occur, where and . A)
B)
C)
and D)
E) no relative maxima
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18
For the given function and graph, estimate the coordinates of the relative minima by observing the graph, where .
A)
B)
C)
D)
E) no relative minima
. A)
B)
C)
D)
E) no relative minima
فتح الحزمة
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19
For the given function and graph, estimate the coordinates of the relative maxima by observing the graph, where .
A)
B)
C)
D)
E) no relative maxima
. A)
B)
C)
D)
E) no relative maxima
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20
Use the sign diagram for to determine all critical values of f (x), where and .
A)
B)
C) and
D)
E)
to determine all critical values of f (x), where and . A)
B)
C)
and D)
E)
فتح الحزمة
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21
For the given function, find intervals of x-values where the function is decreasing.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
فتح الحزمة
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22
For the given function, classify the critical points as relative maxima, relative minima, or points of inflection. In each case, you may check your conclusions with a graphing utility.
A) is a relative minimum and is point of inflection.
B) is a point of inflection and is a relative minimum.
C) is a point of inflection and is a relative maximum.
D) is a relative minimum and is a relative maximum.
E) is a relative maximum and is a relative minimum.
A)
is a relative minimum and is point of inflection.B)
is a point of inflection and is a relative minimum.C)
is a point of inflection and is a relative maximum.D)
is a relative minimum and is a relative maximum.E)
is a relative maximum and is a relative minimum. فتح الحزمة
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23
For the given function, find the relative maxima.
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
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24
For the given function, find intervals of x-values where the function is increasing.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
فتح الحزمة
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25
For the given function, use the graph to identify x-values for which . You may use the derivative to check your conclusion.
A)
B)
C)
D)
E)
. You may use the derivative to check your conclusion. A)
B)
C)
D)
E)
فتح الحزمة
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26
Use the derivative to locate all critical points. Use a graphing utility if desired. Round your answers to two decimal places.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
فتح الحزمة
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27
For the given function, find the critical points.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
فتح الحزمة
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28
Both a function and its derivative are given. Use them to find intervals on which the function is decreasing.
A)
B)
C)
D) or
E)
A)
B)
C)
D)
or E)
فتح الحزمة
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29
For the given function, find the horizontal points of inflection.
A)
B)
C)
D)
E) no horizontal points of inflection
A)
B)
C)
D)
E) no horizontal points of inflection
فتح الحزمة
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30
For the given function, find the critical values.
A) and
B) and
C) and
D) and
E) and
A)
and B)
and C)
and D)
and E)
and فتح الحزمة
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31
Both a function and its derivative are given. Use them to find all critical values.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
فتح الحزمة
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32
For the given function find the relative maxima, and sketch the graph. You may check your graph with a graphing utility.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
فتح الحزمة
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33
Both a function and its derivative are given. Use them to find all critical points.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
فتح الحزمة
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34
For the given function, use the graph to identify x-values for which . You may use the derivative to check your conclusion.
A)
B)
C)
D)
E)
. You may use the derivative to check your conclusion. A)
B)
C)
D)
E)
فتح الحزمة
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35
For the given function, use the graph to identify the x-value for which . You may use the derivative to check your conclusion.
A)
B)
C)
D)
E)
. You may use the derivative to check your conclusion. A)
B)
C)
D)
E)
فتح الحزمة
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36
For the given function, find the relative minima.
A)
B)
C)
D)
E) no relative minima
A)
B)
C)
D)
E) no relative minima
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37
Both a function and its derivative are given. Use them to find intervals on which the function is increasing.
A) or or
B)
C) or
D) or
E)
A)
or or B)
C)
or D)
or E)
فتح الحزمة
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38
Both a function and its derivative are given. Use them to find the relative maxima.
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
فتح الحزمة
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39
For the given function find the relative minima, and sketch the graph. You may check your graph with a graphing utility.
A)
B) and
C) and
D)
E)
A)
B)
and C)
and D)
E)
فتح الحزمة
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40
Both a function and its derivative are given. Use them to find the relative minima.
A)
B)
C)
D)
E) no relative minima
A)
B)
C)
D)
E) no relative minima
فتح الحزمة
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41
Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that its monthly sales volume y (in thousands of dollars) is related to the price p (in dollars) of an oil change by . Is y increasing or decreasing for all values of
A) increasing
B) decreasing
. Is y increasing or decreasing for all values of A) increasing
B) decreasing
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42
The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by , . For what t-values is x increasing? Round answers to two decimal places.
A)
B)
C)
D)
E)
, . For what t-values is x increasing? Round answers to two decimal places. A)
B)
C)
D)
E)
فتح الحزمة
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43
Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Over what interval in the domain do average costs decrease?
A)
B)
C)
D)
E)
, , where x is the number of machines used. Over what interval in the domain do average costs decrease? A)
B)
C)
D)
E)
فتح الحزمة
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44
Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. What is the minimum average cost? Round your answer to the nearest dollar.
A)
B)
C)
D)
E)
, , where x is the number of machines used. What is the minimum average cost? Round your answer to the nearest dollar. A)
B)
C)
D)
E)
فتح الحزمة
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45
A graph of is given. Use the graph to determine where the graph of has a relative maximum.
A)
B)
C)
D)
E) no relative maxima
is given. Use the graph to determine where the graph of has a relative maximum. A)
B)
C)
D)
E) no relative maxima
فتح الحزمة
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46
The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by , . Find the maximum value of x. Round your answer to two decimal places.
A) 2.83 mg
B) 530.33 mg
C) 1166.73 mg
D) 8.77 mg
E) 1125.30 mg
, . Find the maximum value of x. Round your answer to two decimal places. A) 2.83 mg
B) 530.33 mg
C) 1166.73 mg
D) 8.77 mg
E) 1125.30 mg
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47
Determine whether the given function is concave up or concave down at the indicated point. at
A) concave down
B) concave up
at A) concave down
B) concave up
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48
Use the indicated x-values on the graph of to find all the intervals where .
A)
B) and
C)
D)
E) and
to find all the intervals where . A)
B)
and C)
D)
E)
and فتح الحزمة
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49
Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Over what interval in the domain do average costs increase?
A)
B)
C)
D)
E)
, , where x is the number of machines used. Over what interval in the domain do average costs increase? A)
B)
C)
D)
E)
فتح الحزمة
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50
The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by , . Find the t-value at which x is maximum. Round your answer to two decimal places.
A) 0 hours
B) 1.41 hours
C) 1767.77 hours
D) 3.41 hours
E) 3.45 hours
, . Find the t-value at which x is maximum. Round your answer to two decimal places. A) 0 hours
B) 1.41 hours
C) 1767.77 hours
D) 3.41 hours
E) 3.45 hours
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51
A function and its graph are given. Use the second derivative to locate all x-values of points of inflection on the graph of . Check these results against the graph shown.
A)
B)
C)
D) ,
E) , ,
. Check these results against the graph shown. A)
B)
C)
D)
, E)
, , فتح الحزمة
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52
Use the indicated x-values on the graph of to find the x-coordinate of any horizontal point of inflection.
A) a
B) c
C) f
D) e
E) d
to find the x-coordinate of any horizontal point of inflection. A) a
B) c
C) f
D) e
E) d
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53
A graph of is given. Use the graph to determine all critical values of .
A)
B)
C)
D)
E) , ,
is given. Use the graph to determine all critical values of . A)
B)
C)
D)
E)
, , فتح الحزمة
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54
Use the indicated x-values on the graph of to determine intervals over which the graph is concave up.
A) and
B)
C)
D)
E) and
to determine intervals over which the graph is concave up. A)
and B)
C)
D)
E)
and فتح الحزمة
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55
Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. How many machines give minimum average costs?
A) Using 25 machines gives the minimum average costs.
B) Using zero machines gives the minimum average costs.
C) Using 35 machines gives the minimum average costs.
D) Using 50 machines gives the minimum average costs.
E) Using 55 machines gives the minimum average costs.
, , where x is the number of machines used. How many machines give minimum average costs? A) Using 25 machines gives the minimum average costs.
B) Using zero machines gives the minimum average costs.
C) Using 35 machines gives the minimum average costs.
D) Using 50 machines gives the minimum average costs.
E) Using 55 machines gives the minimum average costs.
فتح الحزمة
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56
A graph of is given. Use the graph to determine where the graph of has a relative minimum.
A)
B)
C)
D)
E) no relative minima
is given. Use the graph to determine where the graph of has a relative minimum. A)
B)
C)
D)
E) no relative minima
فتح الحزمة
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57
A graph of is given. Use the graph to determine where f (x) is decreasing.
A)
B)
C)
D) or
E)
is given. Use the graph to determine where f (x) is decreasing. A)
B)
C)
D)
or E)
فتح الحزمة
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58
A function and its graph are given. Use the second derivative to determine intervals on which the function is concave up. Check these results against the graph shown.
A)
B)
C)
D) or
E)
A)
B)
C)
D)
or E)
فتح الحزمة
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59
Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Find the critical values of that lie in the domain of the problem.
A)
B)
C)
D)
E)
, , where x is the number of machines used. Find the critical values of that lie in the domain of the problem.
A)
B)
C)
D)
E)
فتح الحزمة
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60
A function and its graph are given. Use the second derivative to determine intervals on which the function is concave down. Check these results against the graph shown.
A)
B)
C)
D) or
E)
A)
B)
C)
D)
or E)
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61
A function and its graph are given. From the graph estimate where has a relative minimum.
A)
B)
C)
D)
E) no relative minima
has a relative minimum. A)
B)
C)
D)
E) no relative minima
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62
Find all points of inflection of the given function. Round your answer to two decimal places.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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63
The following figure shows the growth of a population as a function of time. Which of A, B, and C correspond(s) to the point(s) at which the growth rate attains its maximum?
A) B
B) A
C) C
D) A and C
E) none of the above
Which of A, B, and C correspond(s) to the point(s) at which the growth rate attains its maximum? A) B
B) A
C) C
D) A and C
E) none of the above
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64
Use the graph shown in the figure and identify points from A through I that satisfy the given condition. and
A) B
B) D
C) C
D) F
E) G
and A) B
B) D
C) C
D) F
E) G
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65
Use the graph shown in the figure and identify points from A through I that satisfy the given condition. and
A) C
B) F
C) I
D) A
E) B
and A) C
B) F
C) I
D) A
E) B
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66
A function and its graph are given. From the graph, estimate where .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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67
Use the graph shown in the figure and identify points from A through I that satisfy the given condition. and
A) D
B) G
C) B
D) C
E) E
and A) D
B) G
C) B
D) C
E) E
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68
The following figure shows the growth of a population as a function of time. If P represents the population and t represents the time, write a mathematical symbol that represents the rate of change (growth rate) of the population with respect to time.
A)
B)
C)
D)
E)
If P represents the population and t represents the time, write a mathematical symbol that represents the rate of change (growth rate) of the population with respect to time.A)
B)
C)
D)
E)
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69
A function and its first and second derivatives are given. Use these to find all points of inflection.
A)
B)
C)
D)
E) no points of inflection
A)
B)
C)
D)
E) no points of inflection
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70
A function and its graph are given. From the graph estimate where has a relative maximum.
A)
B)
C)
D)
E) no relative maxima
has a relative maximum. A)
B)
C)
D)
E) no relative maxima
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71
In this problem, and its graph are given. Use the graph of to determine where has a point of inflection.
A)
B)
C)
D)
E) no point of inflection
and its graph are given. Use the graph of to determine where has a point of inflection. A)
B)
C)
D)
E) no point of inflection
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72
A function and its graph are given. From the graph, estimate where .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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73
A function and its first and second derivatives are given. Use these to find the relative maxima.
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
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74
In this problem, and its graph are given. Use the graph of to determine where is concave down.
A)
B)
C)
D)
E)
and its graph are given. Use the graph of to determine where is concave down. A)
B)
C)
D)
E)
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75
A function and its first and second derivatives are given. Use these to find all critical values.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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76
Find all relative minima of the given function.
A)
B)
C)
D)
E) no relative minima
A)
B)
C)
D)
E) no relative minima
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77
Use the graph shown in the figure and identify points from A through I that satisfy the given condition. and
A) I
B) F
C) C
D) D
E) H
and A) I
B) F
C) C
D) D
E) H
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78
Find all relative maxima of the given function.
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
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79
In this problem, and its graph are given. Use the graph of to determine where is concave up.
A)
B)
C)
D)
E)
and its graph are given. Use the graph of to determine where is concave up. A)
B)
C)
D)
E)
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80
A function and its first and second derivatives are given. Use these to find all relative minima.
A)
B)
C)
D)
E) no relative minima
A)
B)
C)
D)
E) no relative minima
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