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Deck 11: Applications of Derivatives
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Question
For the given function, find all intervals of x-values where the function is decreasing. 
A)
B)
C)
Or
D)
E)
A)
B)
C)
Or
D)
E)
Question
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
Question
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
Question
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
Question
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)
Question
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
Question
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)
Question
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
Question
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)
Question
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
Question
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
Question
Use the graph of 
to identify at which of the indicated points the derivative 
changes from positive to negative. 
A)(5, 6)
B)(2, 4)
C)(-1, 2), (5, 6)
D)(-1, 2), (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)(2, 4)
C)(-1, 2), (5, 6)
D)(-1, 2), (2, 4)
E)(2, 4), (5, 6)
Question
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)
Question
For the given function, find 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
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
Question
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)
Question
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
Question
For the given function, find 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
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)
Question
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
Question
For the given function, find the critical points. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
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)
Question
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)
Question
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.
Question
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
Question
For the given function, find the relative maxima. 
A)
B)
C)
D)
E)no relative maxima
A)
B)
C)
D)
E)no relative maxima
Question
For the given function, find the relative minima. 
A)
B)
C)
D)
E)no relative minima
A)
B)
C)
D)
E)no relative minima
Question
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)
Question
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)
Question
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)
,
,
Question
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)
Question
For the given function, find intervals of x-values where the function is decreasing. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
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
Question
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
Question
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
Question
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)
Question
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)
Question
For the given function, find intervals of x-values where the function is increasing. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
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)
Question
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)
Question
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)
Question
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
Question
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)
Question
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)
Question
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)
,
,
Question
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
Question
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
Question
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
Question
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
Question
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)
Question
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.
Question
Use the indicated 
-values on the graph of 
to find all the intervals where 
. 
A)
B)
And
C)
D)
E)
And
-values on the graph of to find all the intervals where . A)
B)
And
C)
D)
E)
And
Question
The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by 
, 
. For what 
-values is 
increasing? Round answers to two decimal places.
A)
B)
C)
D)
E)
, . For what -values is increasing? Round answers to two decimal places.A)
B)
C)
D)
E)
Question
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)$0
B)$35
C)$490
D)$245
E)$8,582
, , where x is the number of machines used. What is the minimum average cost? Round your answer to the nearest dollar. A)$0
B)$35
C)$490
D)$245
E)$8,582
Question
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)
,
,
Question
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
Question
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)
Question
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)
Question
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
Question
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
Question
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)
Question
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)
Question
Find all relative minima of the given function. 
A)
B)
C)
D)
,
E)no relative minima
A)
B)
C)
D)
,
E)no relative minima
Question
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
Question
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)
Question
A function and its graph are given. From the graph, estimate where 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
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)
Question
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
Question
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
Question
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
Question
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
Question
Find all relative maxima of the given function. 
A)
B)
C)
D)
E)no relative maxima
A)
B)
C)
D)
E)no relative maxima
Question
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
Question
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)
Question
Use the graph shown in the figure and identify points from A through I that satisfy the given condition. 
and 
A)B
B)E
C)A
D)D
E)H
and A)B
B)E
C)A
D)D
E)H
Question
A function and its graph are given. From the graph, estimate where 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
Question
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
Question
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
Question
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
Question
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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Deck 11: Applications of Derivatives
1
For the given function, find all intervals of x-values where the function is decreasing.
A)
B)
C)
Or
D)
E)
A)
B)
C)
Or
D)
E)
2
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
And
3
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
no relative minima
4
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
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5
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)
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6
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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7
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)
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8
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
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k this deck
9
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)
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Unlock for access to all 172 flashcards in this deck.
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k this deck
10
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
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Unlock Deck
k this deck
11
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
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k this deck
12
Use the graph of to identify at which of the indicated points the derivative changes from positive to negative.
A)(5, 6)
B)(2, 4)
C)(-1, 2), (5, 6)
D)(-1, 2), (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)(2, 4)
C)(-1, 2), (5, 6)
D)(-1, 2), (2, 4)
E)(2, 4), (5, 6)
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k this deck
13
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)
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k this deck
14
For the given function, find
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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k this deck
15
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
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k this deck
16
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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k this deck
17
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
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k this deck
18
For the given function, find .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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Unlock Deck
k this deck
19
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)
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k this deck
20
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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k this deck
21
For the given function, find the critical points.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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k this deck
22
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)
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k this deck
23
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)
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k this deck
24
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.
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25
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
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k this deck
26
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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Unlock for access to all 172 flashcards in this deck.
Unlock Deck
k this deck
27
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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Unlock Deck
k this deck
28
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)
Unlock Deck
Unlock for access to all 172 flashcards in this deck.
Unlock Deck
k this deck
29
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)
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Unlock for access to all 172 flashcards in this deck.
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k this deck
30
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)
,
,
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k this deck
31
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)
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Unlock Deck
k this deck
32
For the given function, find intervals of x-values where the function is decreasing.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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k this deck
33
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
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Unlock Deck
k this deck
34
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
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k this deck
35
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
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Unlock Deck
k this deck
36
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)
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k this deck
37
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)
Unlock Deck
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Unlock Deck
k this deck
38
For the given function, find intervals of x-values where the function is increasing.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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Unlock Deck
k this deck
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)
Unlock Deck
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Unlock Deck
k this deck
40
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)
Unlock Deck
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k this deck
41
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)
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k this deck
42
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
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k this deck
43
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)
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k this deck
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. 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)
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k this deck
45
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)
,
,
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k this deck
46
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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47
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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k this deck
48
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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k this deck
49
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
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k this deck
50
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)
Unlock Deck
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Unlock Deck
k this deck
51
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.
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k this deck
52
Use the indicated -values on the graph of to find all the intervals where .
A)
B)
And
C)
D)
E)
And
-values on the graph of to find all the intervals where . A)
B)
And
C)
D)
E)
And
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k this deck
53
The number of milligrams x of a medication in the bloodstream t hours after a dose is taken can be modeled by , . For what -values is increasing? Round answers to two decimal places.
A)
B)
C)
D)
E)
, . For what -values is increasing? Round answers to two decimal places.A)
B)
C)
D)
E)
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k this deck
54
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)$0
B)$35
C)$490
D)$245
E)$8,582
, , where x is the number of machines used. What is the minimum average cost? Round your answer to the nearest dollar. A)$0
B)$35
C)$490
D)$245
E)$8,582
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k this deck
55
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)
,
,
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k this deck
56
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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k this deck
57
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)
Unlock Deck
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k this deck
58
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)
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k this deck
59
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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k this deck
60
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
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k this deck
61
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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k this deck
62
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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k this deck
63
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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k this deck
64
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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k this deck
65
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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k this deck
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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k this deck
67
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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k this deck
68
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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k this deck
69
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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k this deck
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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k this deck
71
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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Unlock Deck
k this deck
72
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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Unlock Deck
k this deck
73
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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Unlock Deck
k this deck
74
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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k this deck
75
Use the graph shown in the figure and identify points from A through I that satisfy the given condition. and
A)B
B)E
C)A
D)D
E)H
and A)B
B)E
C)A
D)D
E)H
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k this deck
76
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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k this deck
77
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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k this deck
78
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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79
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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Unlock for access to all 172 flashcards in this deck.
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k this deck
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
Unlock Deck
Unlock for access to all 172 flashcards in this deck.
Unlock Deck
k this deck
Unlock Deck
Unlock for access to all 172 flashcards in this deck.
