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Deck 3: Applications of the Derivative
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Question
Find the x-values of all relative maxima of the given function.
A) 
B) 
C) 
D) 
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
Question
For the function 
(a)Find the critical numbers of f (if any);
(b)Find the open intervals where the function is increasing or decreasing; and
(c)Apply the First Derivative Test to identify all relative extrema.
A) (a)
(b) increasing:
; decreasing: 
(c) relative max:
; relative min: 
B) (a)
(b) decreasing:
; increasing: 
(c) relative min:
; relative max: 
C) (a)
(b) increasing:
; decreasing: 
(c) relative max:
; relative min: 
D) (a)
(b) decreasing:
; increasing: 
(c) relative min:
; relative max: 
E) (a)
(b)increasing:
; decreasing: 
(c)relative max:
; relative min: no relative min.
(a)Find the critical numbers of f (if any);(b)Find the open intervals where the function is increasing or decreasing; and
(c)Apply the First Derivative Test to identify all relative extrema.
A) (a)
(b) increasing:
; decreasing: (c) relative max:
; relative min:
B) (a)
(b) decreasing:
; increasing: (c) relative min:
; relative max:
C) (a)
(b) increasing:
; decreasing: (c) relative max:
; relative min:
D) (a)
(b) decreasing:
; increasing: (c) relative min:
; relative max:
E) (a)
(b)increasing:
; decreasing: (c)relative max:
; relative min: no relative min.
Question
Both a function and its derivative are given.Use them to find all critical numbers.
A) 
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Find the open intervals on which the function 
is increasing or decreasing.
A) The function is increasing on the interval
, and decreasing on the intervals 
and 
B) The function is increasing on the interval
, and decreasing on the intervals 
and 
C) The function is increasing on the interval
, and decreasing on the intervals 
and 
D) The function is decreasing on the interval
, and increasing on the intervals 
and 
E) The function is decreasing on the interval
,and increasing on the intervals 
and 
is increasing or decreasing. A) The function is increasing on the interval
, and decreasing on the intervals and B) The function is increasing on the interval
, and decreasing on the intervals and C) The function is increasing on the interval
, and decreasing on the intervals and D) The function is decreasing on the interval
, and increasing on the intervals and E) The function is decreasing on the interval
,and increasing on the intervals and Question
Identify the open intervals where the function 
is increasing or decreasing.
A) decreasing:
; increasing: 
B) increasing:
; decreasing: 
C) increasing on
D) decreasing on
E) none of the above
is increasing or decreasing. A) decreasing:
; increasing: B) increasing:
; decreasing: C) increasing on
D) decreasing on
E) none of the above
Question
Suppose the number y of medical degrees conferred in the United States can be modeled by 
,for 
,where t is the time in years,with 
corresponding to 1972.Use the test for increasing and decreasing functions to estimate the years during which the number of medical degrees is increasing and the years during which it is decreasing.
A) The number of medical degrees is increasing from 1972 to 1989 and 1997 to 2005, and decreasing during 1989 to 1997.
B) The number of medical degrees is increasing from 1972 to 1988 and 1996 to 2005, and decreasing during 1988 to 1996.
C) The number of medical degrees is increasing from 1972 to 1989 and 2001 to 2005, and decreasing during 1989 to 2001.
D) The number of medical degrees is increasing from 1972 to 1990 and 1996 to 2005, and decreasing during 1990 to 2001.
E) The number of medical degrees is increasing from 1972 to 1989 and 1995 to 2005,and decreasing during 1989 to 1995.
,for ,where t is the time in years,with corresponding to 1972.Use the test for increasing and decreasing functions to estimate the years during which the number of medical degrees is increasing and the years during which it is decreasing. A) The number of medical degrees is increasing from 1972 to 1989 and 1997 to 2005, and decreasing during 1989 to 1997.
B) The number of medical degrees is increasing from 1972 to 1988 and 1996 to 2005, and decreasing during 1988 to 1996.
C) The number of medical degrees is increasing from 1972 to 1989 and 2001 to 2005, and decreasing during 1989 to 2001.
D) The number of medical degrees is increasing from 1972 to 1990 and 1996 to 2005, and decreasing during 1990 to 2001.
E) The number of medical degrees is increasing from 1972 to 1989 and 1995 to 2005,and decreasing during 1989 to 1995.
Question
Locate the absolute extrema of the function 
on the closed interval 
A) absolute max:
; absolute min: 
B) absolute max:
; absolute min: 
C) absolute max:
; absolute min: no absolute min
D) no absolute max; absolute min:
E) no absolute max or min
on the closed interval A) absolute max:
; absolute min: B) absolute max:
; absolute min: C) absolute max:
; absolute min: no absolute minD) no absolute max; absolute min:
E) no absolute max or min
Question
Use the graph of 
to identify at which of the indicated points the derivative 
changes from negative to positive. 
A) (2,4), (5,6)
B) (-1,2)
C) (-1,2), (2,4)
D) (2,4)
E) (-1,2),(5,6)
to identify at which of the indicated points the derivative changes from negative to positive.
A) (2,4), (5,6)
B) (-1,2)
C) (-1,2), (2,4)
D) (2,4)
E) (-1,2),(5,6)
Question
Locate the absolute extrema of the function 
on the closed interval 
A) no absolute max; absolute min:
B) absolute max:
; absolute min: 
C) absolute max:
; no absolute min
D) absolute max:
; 
E) no absolute max or min
on the closed interval A) no absolute max; absolute min:
B) absolute max:
; absolute min: C) absolute max:
; no absolute minD) absolute max:
; E) no absolute max or min
Question
For the given function,find all critical numbers.
A) 
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Identify the open intervals where the function 
is increasing or decreasing.
A) decreasing:
; increasing: 
B) increasing:
; decreasing: 
C) increasing on
D) decreasing on
E) none of the above
is increasing or decreasing. A) decreasing:
; increasing: B) increasing:
; decreasing: C) increasing on
D) decreasing on
E) none of the above
Question
Find all relative minima of the given function.
A) 
B) 
C) 
D) 
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
Question
Identify the open intervals where the function 
is increasing or decreasing.
A) decreasing:
; increasing: 
B) increasing:
; decreasing: 
C) increasing:
; decreasing: 
D) increasing:
; decreasing: 
E) decreasing for all x
is increasing or decreasing. A) decreasing:
; increasing: B) increasing:
; decreasing: C) increasing:
; decreasing: D) increasing:
; decreasing: E) decreasing for all x
Question
Find the open intervals on which the function 
is increasing or decreasing.
A) The function is increasing on the interval
and decreasing on the interval 
B) The function is increasing on the interval
and decreasing on the interval 
C) The function is increasing on the interval
and decreasing on the interval 
D) The function is increasing on the interval
and decreasing on the interval 
E) The function is increasing on the interval
and decreasing on the interval 
is increasing or decreasing. A) The function is increasing on the interval
and decreasing on the interval B) The function is increasing on the interval
and decreasing on the interval C) The function is increasing on the interval
and decreasing on the interval D) The function is increasing on the interval
and decreasing on the interval E) The function is increasing on the interval
and decreasing on the interval Question
Find any critical numbers of the function 
.
A) 0
B) 
C) 
D) both A and B
E) both A and C
. A) 0
B)
C)
D) both A and B
E) both A and C
Question
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) (2, 4), (5, 6)
D) (2, 4)
E) (-1,2)
to identify at which of the indicated points the derivative changes from positive to negative.
A) (5, 6)
B) (-1, 2), (2, 4)
C) (2, 4), (5, 6)
D) (2, 4)
E) (-1,2)
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
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 fast-food restaurant determines the cost model, 
and revenue model, 
for 
where x is the number of hamburgers sold.Determine the intervals on which the profit function is increasing and on which it is decreasing.
A) The profit function is increasing on the interval
and decreasing on the interval 
B) The profit function is increasing on the interval
and decreasing on the interval 
C) The profit function is increasing on the interval
and decreasing on the interval 
D) The profit function is increasing on the interval
and decreasing on the interval 
E) The profit function is increasing on the interval
and decreasing on the interval 
and revenue model, for where x is the number of hamburgers sold.Determine the intervals on which the profit function is increasing and on which it is decreasing.
A) The profit function is increasing on the interval
and decreasing on the interval B) The profit function is increasing on the interval
and decreasing on the interval C) The profit function is increasing on the interval
and decreasing on the interval D) The profit function is increasing on the interval
and decreasing on the interval E) The profit function is increasing on the interval
and decreasing on the interval Question
For the given function,find the critical numbers.
A) 
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
Sketch a graph of a function f having the following characteristics.
A) 
B) 
C)
D) 
E) 
A)
B)
C)
D)
E)
Question
Determine the open intervals on which the graph of 
is concave downward or concave upward.
A) concave upward on
; concave downward on 
B) concave downward on
C) concave upward on
D) concave downward on
; concave upward 
E) concave upward on
; concave downward on 
is concave downward or concave upward.A) concave upward on
; concave downward on B) concave downward on
C) concave upward on
D) concave downward on
; concave upward E) concave upward on
; concave downward on Question
Medication.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) 2.83 hours
C) 707.11 hours
D) 4.83 hours
E) 6.91 hours
Find the t-value at which x is maximum.Round your answer to two decimal places. A) 0 hours
B) 2.83 hours
C) 707.11 hours
D) 4.83 hours
E) 6.91 hours
Question
Find all relative extrema of the function 
Use the Second Derivative Test where applicable.
A) relative max:
; no relative min
B) relative max:
; no relative min
C) no relative max or min
D) relative min:
; no relative max
E) relative min:
; no relative max
Use the Second Derivative Test where applicable.
A) relative max:
; no relative minB) relative max:
; no relative minC) no relative max or min
D) relative min:
; no relative maxE) relative min:
; no relative max Question
Suppose the resident population P(in millions)of the United States can be modeled by 
, 
,where 
corresponds to 1800.Analytically find the minimum and maximum populations in the U.S.for 
A) The population is minimum at
and maximum at 
B) The population is minimum at
and maximum at 
C) The population is minimum at
and maximum at 
D) The population is minimum at
and maximum at 
E) The population is minimum at
and maximum at 
, ,where corresponds to 1800.Analytically find the minimum and maximum populations in the U.S.for A) The population is minimum at
and maximum at B) The population is minimum at
and maximum at C) The population is minimum at
and maximum at D) The population is minimum at
and maximum at E) The population is minimum at
and maximum at Question
Find all relative extrema of the function 
Use the Second-Derivative Test when applicable.
A) The relative minimum is
and the relative maximum is 
B) The relative maximum is
C) The relative minimum is
D) The relative maximum is
and the relative minima are 
and 
E) The relative minimum is
and the relative maximum is 
Use the Second-Derivative Test when applicable. A) The relative minimum is
and the relative maximum is B) The relative maximum is
C) The relative minimum is
D) The relative maximum is
and the relative minima are and E) The relative minimum is
and the relative maximum is Question
State the signs of 
and 
on the interval (0,2).
A) f ' < 0
f '' = 0
B) f ' < 0
f '' > 0
C) f ' > 0
f '' < 0
D) f ' > 0
f '' > 0
E) f ' < 0
f '' < 0
and on the interval (0,2).
A) f ' < 0
f '' = 0
B) f ' < 0
f '' > 0
C) f ' > 0
f '' < 0
D) f ' > 0
f '' > 0
E) f ' < 0
f '' < 0
Question
The graph of f is shown.Graph f,f ' and f '' on the same set of coordinate axes.
A) 
B) 
C) 
D) 
E) none of the above
A)
B)
C)
D)
E) none of the above
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
Find the points of inflection and discuss the concavity of the function.
A) inflection point at
; concave downward on 
; concave upward on 
B) inflection point at
; concave upward on 
; concave downward on 
C) inflection point at
; concave downward on 
; concave upward on 
D) inflection point at
; concave upward on 
; concave downward on 
E) none of the above
A) inflection point at
; concave downward on ; concave upward on B) inflection point at
; concave upward on ; concave downward on C) inflection point at
; concave downward on ; concave upward on D) inflection point at
; concave upward on ; concave downward on E) none of the above
Question
Find the absolute extrema of the function 
on the interval 
A) The maximum of the function is 1 and the minimum of the function is 0..
B) The maximum of the function is 0 and the minimum of the function is -1..
C) The maximum of the function is -1 and the minimum of the function is 0.
D) The maximum of the function is 1 and the minimum of the function is 0.
E) The maximum of the function is 0 and the minimum of the function is 1.
on the interval A) The maximum of the function is 1 and the minimum of the function is 0..
B) The maximum of the function is 0 and the minimum of the function is -1..
C) The maximum of the function is -1 and the minimum of the function is 0.
D) The maximum of the function is 1 and the minimum of the function is 0.
E) The maximum of the function is 0 and the minimum of the function is 1.
Question
Find all relative extrema of the function 
Use the Second Derivative Test where applicable.
A) relative max:
B) relative min:
C) no relative max or min
D) both A and B
E) none of the above
Use the Second Derivative Test where applicable. A) relative max:
B) relative min:
C) no relative max or min
D) both A and B
E) none of the above
Question
Find the x-value at which the given function has a point of inflection.
A) 
B) 
C) 
D) 
E) no point of inflection
A)
B)
C)
D)
E) no point of inflection
Question
Find all relative extrema of the function 
Use the Second-Derivative Test when applicable.
A) The relative maximum is
B) The relative minimum is
C) The relative maximum is
D) The relative minimum is
E) The relative maximum is
Use the Second-Derivative Test when applicable.A) The relative maximum is
B) The relative minimum is
C) The relative maximum is
D) The relative minimum is
E) The relative maximum is
Question
Graph a function on the interval 
having the following characteristics. Absolute maximum at 
Absolute minimum at 
Relative minimum at 
Relative maximum at 
A) 
B) 
C) 
D) 
E) 
having the following characteristics. Absolute maximum at Absolute minimum at Relative minimum at Relative maximum at A)
B)
C)
D)
E)
Question
Find the absolute extrema of the function 
on the closed interval 
Round your answer to two decimal places.
A) The maximum of the function is 1 and the minimum of the function is 0.
B) The maximum of the function is 2.92 and the minimum of the function is 1.
C) The maximum of the function is 2.92 and the minimum of the function is 0.
D) The maximum of the function is1 and the minimum of the function is 2.52.
E) The maximum of the function is 0 and the minimum of the function is 2.52.
on the closed interval Round your answer to two decimal places. A) The maximum of the function is 1 and the minimum of the function is 0.
B) The maximum of the function is 2.92 and the minimum of the function is 1.
C) The maximum of the function is 2.92 and the minimum of the function is 0.
D) The maximum of the function is1 and the minimum of the function is 2.52.
E) The maximum of the function is 0 and the minimum of the function is 2.52.
Question
Medication.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) 1.73 hours.
B) 1443.37 hours
C) 3.73 hours
D) 4.22 hours
E) 1.73 hours
Find the maximum value of x.Round your answer to two decimal places. A) 1.73 hours.
B) 1443.37 hours
C) 3.73 hours
D) 4.22 hours
E) 1.73 hours
Question
Approximate the critical numbers of the function shown in the graph and determine whether the function has a relative maximum,a relative minimum,an absolute maximum,an absolute minimum,or none of these at each critical number on the interval shown. 
A) The critical number 
yields an absolute maximum and the critical number 
yields an absolute minimum.
B) Both the critical numbers
& 
yield an absolute maximum.
C) The critical number
yields an absolute minimum and the critical number 
yields an absolute maximum.
D) Both the critical numbers
and 
yield an absolute minimum.
E) The critical number
yields a relative minimum and the critical number 
yields a relative maximum.
A) The critical number
yields an absolute maximum and the critical number yields an absolute minimum.B) Both the critical numbers
& yield an absolute maximum.C) The critical number
yields an absolute minimum and the critical number yields an absolute maximum.D) Both the critical numbers
and yield an absolute minimum.E) The critical number
yields a relative minimum and the critical number yields a relative maximum. Question
Determine the open intervals on which the graph of 
is concave downward or concave upward.
A) concave downward on
B) concave downward on
; concave upward on 
C) concave upward on
; concave downward on 
D) concave downward on
; concave upward on 
E) concave upward on
; concave downward on 
is concave downward or concave upward. A) concave downward on
B) concave downward on
; concave upward on C) concave upward on
; concave downward on D) concave downward on
; concave upward on E) concave upward on
; concave downward on Question
Find the x-value at which the absolute minimum of f(x)occurs on the interval [a,b].
![<strong>Find the x-value at which the absolute minimum of f(x)occurs on the interval [a,b]. </strong> A) B) C) D) E) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a28_c2d2_861b_11e8b722a524_TB1301_11.jpg)
A) ![<strong>Find the x-value at which the absolute minimum of f(x)occurs on the interval [a,b]. </strong> A) B) C) D) E) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a28_e9e3_861b_41b2fa8d7b94_TB1301_11.jpg)
B) ![<strong>Find the x-value at which the absolute minimum of f(x)occurs on the interval [a,b]. </strong> A) B) C) D) E) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a28_e9e4_861b_4faf7d96be92_TB1301_11.jpg)
C) ![<strong>Find the x-value at which the absolute minimum of f(x)occurs on the interval [a,b]. </strong> A) B) C) D) E) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a28_e9e5_861b_8bd7ea811539_TB1301_11.jpg)
D) ![<strong>Find the x-value at which the absolute minimum of f(x)occurs on the interval [a,b]. </strong> A) B) C) D) E) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a28_e9e6_861b_ffd11e05b186_TB1301_11.jpg)
E) ![<strong>Find the x-value at which the absolute minimum of f(x)occurs on the interval [a,b]. </strong> A) B) C) D) E) <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a29_10f7_861b_e91db55c8201_TB1301_11.jpg)
A)
B)
C)
D)
E)
Question
If the total cost function for a product is 
dollars.Find the minimum average cost.
A) $14.25 per unit
B) $18 per unit
C) $14.36 per unit
D) $14 per unit
E) $13 per unit
dollars.Find the minimum average cost. A) $14.25 per unit
B) $18 per unit
C) $14.36 per unit
D) $14 per unit
E) $13 per unit
Question
A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: ![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_26ff_861b_8d445c2fc44e_TB1301_11.jpg)
where k > 0 is the proportionality constant.] ![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_2700_861b_4f11f8d2b4c7_TB1301_00.jpg)
A)![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_2701_861b_91398ba9b0c2_TB1301_11.jpg)
inches and ![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_2702_861b_e974f7349586_TB1301_11.jpg)
inches
B)![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_4e13_861b_1b3c3d464f1a_TB1301_11.jpg)
inches and ![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_4e14_861b_2f49d292c854_TB1301_11.jpg)
inches
C)![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_4e15_861b_7d139805eca6_TB1301_11.jpg)
inches and ![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_4e16_861b_f1d7dda4b0a6_TB1301_11.jpg)
inches
D)![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_4e17_861b_c3ffa7e8e9ac_TB1301_11.jpg)
inches and ![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_7528_861b_15645d055f4b_TB1301_11.jpg)
inches
E)![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_7529_861b_a31a1ac4439b_TB1301_11.jpg)
inches and ![<strong>A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.] </strong> A) inches and inches B) inches and inches C) inches and inches D) inches and inches E) inches and inches <div style=padding-top: 35px>](https://storage.examlex.com/TB1301/11ea8970_6a31_752a_861b_71c9fec96556_TB1301_11.jpg)
inches
where k > 0 is the proportionality constant.] A)
inches and inchesB)
inches and inchesC)
inches and inchesD)
inches and inchesE)
inches and inches Question
Question
Average costs.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 14 machines gives the minimum average costs.
B) Using zero machines gives the minimum average costs.
C) Using 24 machines gives the minimum average costs.
D) Using 28 machines gives the minimum average costs.
E) Using 33 machines gives the minimum average costs.
, ,where x is the number of machines used.How many machines give minimum average costs? A) Using 14 machines gives the minimum average costs.
B) Using zero machines gives the minimum average costs.
C) Using 24 machines gives the minimum average costs.
D) Using 28 machines gives the minimum average costs.
E) Using 33 machines gives the minimum average costs.
Question
If the total revenue function for a blender is 
,find the maximum revenue.
A) $100
B) $2,500
C) $40
D) $250
E) $2,000
,find the maximum revenue. A) $100
B) $2,500
C) $40
D) $250
E) $2,000
Question
Determine the dimensions of a rectangular solid (with a square base)with maximum volume if its surface area is 441 square meters.
A) square base side
; height 
B) square base side
; height 
C) square base side
; height 
D) square base side
; height 
E) square base side
; height 
A) square base side
; height B) square base side
; height C) square base side
; height D) square base side
; height E) square base side
; height Question
Question
A firm has total revenue given by 
for x units of a product.Find the maximum revenue from sales of that product.
A) $2400
B) $2432
C) $404
D) $3300
E) $307
for x units of a product.Find the maximum revenue from sales of that product. A) $2400
B) $2432
C) $404
D) $3300
E) $307
Question
Volume.A rectangular box with a square base is to be formed from a square piece of metal with 54-inch sides.If a square piece with side x is cut from each corner of the metal and the sides are folded up to form an open box,the volume of the box is 
What value of x will maximize the volume of the box? 
A) 27
B) 2
C) 9
D) 14
E) 1
What value of x will maximize the volume of the box? A) 27
B) 2
C) 9
D) 14
E) 1
Question
Find the point on the graph of 
that is closest to the point (8,0.5).Round your answer to two decimal places.
A) (1.59, 2.53)
B) (2, 4)
C) (2.52, 6.35)
D) (1.39, 1.93)
E) (1.26,1.59)
that is closest to the point (8,0.5).Round your answer to two decimal places. A) (1.59, 2.53)
B) (2, 4)
C) (2.52, 6.35)
D) (1.39, 1.93)
E) (1.26,1.59)
Question
If the total revenue function for a blender is 
,determine how many units x must be sold to provide the maximum total revenue in dollars.
A) 625
B) 15,000
C) 25
D) 300
E) 50
,determine how many units x must be sold to provide the maximum total revenue in dollars. A) 625
B) 15,000
C) 25
D) 300
E) 50
Question
A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure).Find the dimensions of a Norman window of maximum area if the total perimeter is 22 feet.Round yours answers to two decimal places. 
A) 
feet and 
feet
B) 
feet and 
feet
C) 
feet and 
feet
D) 
feet and 
feet
E) 
feet and 
feet
A)
feet and feetB)
feet and feetC)
feet and feetD)
feet and feetE)
feet and feet Question
Production.Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function 
Find the number of hours before the rate of production is maximized.That is,find the point of diminishing returns.
A) 
B) 
C) 
D) 
E) 
Find the number of hours before the rate of production is maximized.That is,find the point of diminishing returns. A)
B)
C)
D)
E)
Question
Question
You are in a boat 2 miles from the nearest point on the coast.You are to go to point Q located 3 miles down the coast and 1 mile inland (see figure).You can row at a rate of 1 miles per hour and you can walk at a rate of 2 miles per hour.Toward what point on the coast should you row in order to reach point Q in the least time? 
A) 3 miles
B) 8 miles
C) 2 miles
D) 1 mile
E) 5 miles
A) 3 miles
B) 8 miles
C) 2 miles
D) 1 mile
E) 5 miles
Question
The profit P (in thousands of dollars)for a company spending an amount s (in thousands of dollars)on advertising is 
The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline.Find the point of diminishing returns.Round your answer to the nearest thousand dollars.
A) 43 thousand dollars
B) 130 thousand dollars
C) 104 thousand dollars
D) 173 thousand dollars
E) 87 thousand dollars
The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline.Find the point of diminishing returns.Round your answer to the nearest thousand dollars. A) 43 thousand dollars
B) 130 thousand dollars
C) 104 thousand dollars
D) 173 thousand dollars
E) 87 thousand dollars
Question
Find the dimensions of the rectangle of maximum area bounded by the x-axis and y-axis and the graph of 
A) length 1.5; width 1.25
B) length 2; width 1
C) length 0.5; width 1.75
D) length 1; width 1.5
E) none of the above
A) length 1.5; width 1.25
B) length 2; width 1
C) length 0.5; width 1.75
D) length 1; width 1.5
E) none of the above
Question
The number of people who donated to a certain organization between 1975 and 1992 can be modeled by the equation 
donors,where t is the number of years after 1975.Find the inflection point(s)from 
through 
,if any exist.
A) There are no inflection points from
B) There is one inflection point at
C) There are inflection points at
D) There is one inflection point at
E) There are inflection points at
, 
, 
donors,where t is the number of years after 1975.Find the inflection point(s)from through ,if any exist. A) There are no inflection points from
B) There is one inflection point at
C) There are inflection points at
D) There is one inflection point at
E) There are inflection points at
, , Question
A rancher has 440 feet of fencing to enclose two adjacent rectangular corrals (see figure).What dimensions should be used so that the enclosed area will be a maximum?
A) 
B) 
C) 
D) 
E) 
A)
B)
C)
D)
E)
Question
If the total cost function for a product is 
dollars,determine how many units x should be produced to minimize the average cost per unit?
A) 450 units
B) 400 units
C) 141 units
D) 200 units
E) 97 units
dollars,determine how many units x should be produced to minimize the average cost per unit? A) 450 units
B) 400 units
C) 141 units
D) 200 units
E) 97 units
Question
Find any vertical asymptotes for the given function.
A) 
B) 
C) 
D) 
E) no vertical asymptotes
A)
B)
C)
D)
E) no vertical asymptotes
Question
p is in dollars and q is the number of units.Find the elasticity of the demand function 
at the price 
A) -4.04
B) 1.00
C) 4.04
D) -0.78
E) 0.78
at the price A) -4.04
B) 1.00
C) 4.04
D) -0.78
E) 0.78
Question
A power station is on one side of a river that is 0.5 mile wide,and a factory is 6 miles downstream on the other side of the river.It costs $14 per foot to run overland power lines and $20 per foot to run underwater power lines.Estimate the value of x that minimizes the cost.The value of 
is the length of the overland piece of power line.
A) 2.56
B) 5.51
C) 3
D) 5.49
E) 0.87
is the length of the overland piece of power line. A) 2.56
B) 5.51
C) 3
D) 5.49
E) 0.87
Question
Question
Find any vertical asymptotes for the given function.
A) 
B)
C) 
D) 
E) no vertical asymptotes
A)
B)
C)
D)
E) no vertical asymptotes
Question
Average costs.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?
A) $0
B) $35
C) $490
D) $245
E) $8582
, ,where x is the number of machines used.What is the minimum average cost? A) $0
B) $35
C) $490
D) $245
E) $8582
Question
A function and its graph are given.Use the graph to find the horizontal asymptotes,if they exist,where 
Confirm your results analytically.
A) 
B) 
C) 
D) 
E) no horizontal asymptotes
Confirm your results analytically.
A)
B)
C)
D)
E) no horizontal asymptotes
Question
Find the speed v,in miles per hour,that will minimize costs on a 125-mile delivery trip.The cost per hour for fuel is 
dollars,and the driver is paid 
dollars per hour.(Assume there are no costs other than wages and fuel.)
A) 75 mph
B) 125 mph
C) 200 mph
D) 80 mph
E) 40 mph
dollars,and the driver is paid dollars per hour.(Assume there are no costs other than wages and fuel.) A) 75 mph
B) 125 mph
C) 200 mph
D) 80 mph
E) 40 mph
Question
Analytically determine the location(s)of any horizontal asymptote(s).
A) 
B) 
C) 
, 
D) 
E) no horizontal asymptotes
A)
B)
C)
, D)
E) no horizontal asymptotes
Question
Find any horizontal asymptotes for the given function.
A) 
B) 
C) 
D) 
E) no horizontal asymptotes
A)
B)
C)
D)
E) no horizontal asymptotes
Question
A function and its graph are given.Use the graph to find the vertical asymptotes,if they exist.Confirm your results analytically. 
A) 
B) 
C) 
D) 
E) no vertical asymptotes
A)
B)
C)
D)
E) no vertical asymptotes
Question
This problem contains a function and its graph,where 
Use the graph to determine,as well as you can,the vertical asymptote.Check your conclusion by using the function to determine the vertical asymptote analytically.
A) 
B) 
C) 
D) 
E) no vertical asymptote
Use the graph to determine,as well as you can,the vertical asymptote.Check your conclusion by using the function to determine the vertical asymptote analytically.
A)
B)
C)
D)
E) no vertical asymptote
Question
Suppose the sales S (in billions of dollars per year)for Proctor & Gamble for the years 1997 through 2002 can be modeled by 
, 
where t represents the year.During which year were the sales increasing at the lowest rate?
A) Sales are increasing at the lowest rate in the year 2002.
B) Sales are increasing at the lowest rate in the year 1997.
C) Sales are increasing at the lowest rate in the year 1998.
D) Sales are increasing at the lowest rate in the year 2000.
E) Sales are increasing at the lowest rate in the year 1999.
, where t represents the year.During which year were the sales increasing at the lowest rate? A) Sales are increasing at the lowest rate in the year 2002.
B) Sales are increasing at the lowest rate in the year 1997.
C) Sales are increasing at the lowest rate in the year 1998.
D) Sales are increasing at the lowest rate in the year 2000.
E) Sales are increasing at the lowest rate in the year 1999.
Question
Analytically determine the location(s)of any vertical asymptote(s).
A) 
B) 
C) 
, 
D) 
E) no vertical asymptotes
A)
B)
C)
, D)
E) no vertical asymptotes
Question
A function and its graph are given.Use the graph to find the horizontal asymptotes,if they exist.Confirm your results analytically. 
A) 
B) 
C) 
D) 
E) no horizontal asymptotes
A)
B)
C)
D)
E) no horizontal asymptotes
Question
Find any horizontal asymptotes for the given function. 
A)
B)
C)
D)
E) no horizontal asymptotes
A)
B)
C)
D)
E) no horizontal asymptotes
Question
A firm can produce 100 units per week.If its total cost function is 
dollars,and its total revenue function is 
dollars,find the maximum profit.
A) $3,679
B) $1,600
C) $7,031
D) $1,265
E) $1,941
dollars,and its total revenue function is dollars,find the maximum profit. A) $3,679
B) $1,600
C) $7,031
D) $1,265
E) $1,941
Question
This problem contains a function and its graph,where 
Use the graph to determine,as well as you can,the horizontal asymptote.Check your conclusion by using the function to determine the horizontal asymptote analytically.
A) 
B) 
C) 
D) 
E) 
Use the graph to determine,as well as you can,the horizontal asymptote.Check your conclusion by using the function to determine the horizontal asymptote analytically.
A)
B)
C)
D)
E)
Question
A function and its graph are given.Use the graph to find the vertical asymptotes,if they exist,where 
Confirm your results analytically.
A) 
B) 
C) 
D) 
E) no vertical asymptotes
Confirm your results analytically.
A)
B)
C)
D)
E) no vertical asymptotes
Question
A firm can produce 100 units per week.If its total cost function is 
dollars,and its total revenue function is 
dollars,how many units x should it produce to maximize its profit?
A) 1250 units
B) 650 units
C) 97 units
D) 50 units
E) 100 units
dollars,and its total revenue function is dollars,how many units x should it produce to maximize its profit? A) 1250 units
B) 650 units
C) 97 units
D) 50 units
E) 100 units
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Deck 3: Applications of the Derivative
1
Find the x-values of all relative maxima of the given function.
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
2
For the function (a)Find the critical numbers of f (if any);
(b)Find the open intervals where the function is increasing or decreasing; and
(c)Apply the First Derivative Test to identify all relative extrema.
A) (a)
(b) increasing: ; decreasing:
(c) relative max: ; relative min:
B) (a)
(b) decreasing: ; increasing:
(c) relative min: ; relative max:
C) (a)
(b) increasing: ; decreasing:
(c) relative max: ; relative min:
D) (a)
(b) decreasing: ; increasing:
(c) relative min: ; relative max:
E) (a)
(b)increasing: ; decreasing:
(c)relative max: ; relative min: no relative min.
(a)Find the critical numbers of f (if any);(b)Find the open intervals where the function is increasing or decreasing; and
(c)Apply the First Derivative Test to identify all relative extrema.
A) (a)
(b) increasing:
; decreasing: (c) relative max:
; relative min:
B) (a)
(b) decreasing:
; increasing: (c) relative min:
; relative max:
C) (a)
(b) increasing:
; decreasing: (c) relative max:
; relative min:
D) (a)
(b) decreasing:
; increasing: (c) relative min:
; relative max:
E) (a)
(b)increasing:
; decreasing: (c)relative max:
; relative min: no relative min.
(a)
(b) increasing: ; decreasing:
(c) relative max: ; relative min:
(b) increasing:
; decreasing: (c) relative max:
; relative min:
3
Both a function and its derivative are given.Use them to find all critical numbers.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
4
Find the open intervals on which the function is increasing or decreasing.
A) The function is increasing on the interval , and decreasing on the intervals and
B) The function is increasing on the interval , and decreasing on the intervals and
C) The function is increasing on the interval , and decreasing on the intervals and
D) The function is decreasing on the interval , and increasing on the intervals and
E) The function is decreasing on the interval ,and increasing on the intervals and
is increasing or decreasing. A) The function is increasing on the interval
, and decreasing on the intervals and B) The function is increasing on the interval
, and decreasing on the intervals and C) The function is increasing on the interval
, and decreasing on the intervals and D) The function is decreasing on the interval
, and increasing on the intervals and E) The function is decreasing on the interval
,and increasing on the intervals and Unlock Deck
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5
Identify the open intervals where the function is increasing or decreasing.
A) decreasing: ; increasing:
B) increasing: ; decreasing:
C) increasing on
D) decreasing on
E) none of the above
is increasing or decreasing. A) decreasing:
; increasing: B) increasing:
; decreasing: C) increasing on
D) decreasing on
E) none of the above
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6
Suppose the number y of medical degrees conferred in the United States can be modeled by ,for ,where t is the time in years,with corresponding to 1972.Use the test for increasing and decreasing functions to estimate the years during which the number of medical degrees is increasing and the years during which it is decreasing.
A) The number of medical degrees is increasing from 1972 to 1989 and 1997 to 2005, and decreasing during 1989 to 1997.
B) The number of medical degrees is increasing from 1972 to 1988 and 1996 to 2005, and decreasing during 1988 to 1996.
C) The number of medical degrees is increasing from 1972 to 1989 and 2001 to 2005, and decreasing during 1989 to 2001.
D) The number of medical degrees is increasing from 1972 to 1990 and 1996 to 2005, and decreasing during 1990 to 2001.
E) The number of medical degrees is increasing from 1972 to 1989 and 1995 to 2005,and decreasing during 1989 to 1995.
,for ,where t is the time in years,with corresponding to 1972.Use the test for increasing and decreasing functions to estimate the years during which the number of medical degrees is increasing and the years during which it is decreasing. A) The number of medical degrees is increasing from 1972 to 1989 and 1997 to 2005, and decreasing during 1989 to 1997.
B) The number of medical degrees is increasing from 1972 to 1988 and 1996 to 2005, and decreasing during 1988 to 1996.
C) The number of medical degrees is increasing from 1972 to 1989 and 2001 to 2005, and decreasing during 1989 to 2001.
D) The number of medical degrees is increasing from 1972 to 1990 and 1996 to 2005, and decreasing during 1990 to 2001.
E) The number of medical degrees is increasing from 1972 to 1989 and 1995 to 2005,and decreasing during 1989 to 1995.
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7
Locate the absolute extrema of the function on the closed interval
A) absolute max: ; absolute min:
B) absolute max: ; absolute min:
C) absolute max: ; absolute min: no absolute min
D) no absolute max; absolute min:
E) no absolute max or min
on the closed interval A) absolute max:
; absolute min: B) absolute max:
; absolute min: C) absolute max:
; absolute min: no absolute minD) no absolute max; absolute min:
E) no absolute max or min
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8
Use the graph of to identify at which of the indicated points the derivative changes from negative to positive.
A) (2,4), (5,6)
B) (-1,2)
C) (-1,2), (2,4)
D) (2,4)
E) (-1,2),(5,6)
to identify at which of the indicated points the derivative changes from negative to positive.
A) (2,4), (5,6)
B) (-1,2)
C) (-1,2), (2,4)
D) (2,4)
E) (-1,2),(5,6)
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9
Locate the absolute extrema of the function on the closed interval
A) no absolute max; absolute min:
B) absolute max: ; absolute min:
C) absolute max: ; no absolute min
D) absolute max: ;
E) no absolute max or min
on the closed interval A) no absolute max; absolute min:
B) absolute max:
; absolute min: C) absolute max:
; no absolute minD) absolute max:
; E) no absolute max or min
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10
For the given function,find all critical numbers.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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11
Identify the open intervals where the function is increasing or decreasing.
A) decreasing: ; increasing:
B) increasing: ; decreasing:
C) increasing on
D) decreasing on
E) none of the above
is increasing or decreasing. A) decreasing:
; increasing: B) increasing:
; decreasing: C) increasing on
D) decreasing on
E) none of the above
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12
Find all relative minima of the given function.
A)
B)
C)
D)
E) no relative maxima
A)
B)
C)
D)
E) no relative maxima
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13
Identify the open intervals where the function is increasing or decreasing.
A) decreasing: ; increasing:
B) increasing: ; decreasing:
C) increasing: ; decreasing:
D) increasing: ; decreasing:
E) decreasing for all x
is increasing or decreasing. A) decreasing:
; increasing: B) increasing:
; decreasing: C) increasing:
; decreasing: D) increasing:
; decreasing: E) decreasing for all x
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14
Find the open intervals on which the function is increasing or decreasing.
A) The function is increasing on the interval and decreasing on the interval
B) The function is increasing on the interval and decreasing on the interval
C) The function is increasing on the interval and decreasing on the interval
D) The function is increasing on the interval and decreasing on the interval
E) The function is increasing on the interval and decreasing on the interval
is increasing or decreasing. A) The function is increasing on the interval
and decreasing on the interval B) The function is increasing on the interval
and decreasing on the interval C) The function is increasing on the interval
and decreasing on the interval D) The function is increasing on the interval
and decreasing on the interval E) The function is increasing on the interval
and decreasing on the interval Unlock Deck
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15
Find any critical numbers of the function .
A) 0
B)
C)
D) both A and B
E) both A and C
. A) 0
B)
C)
D) both A and B
E) both A and C
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16
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) (2, 4), (5, 6)
D) (2, 4)
E) (-1,2)
to identify at which of the indicated points the derivative changes from positive to negative.
A) (5, 6)
B) (-1, 2), (2, 4)
C) (2, 4), (5, 6)
D) (2, 4)
E) (-1,2)
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17
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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18
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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19
A fast-food restaurant determines the cost model, and revenue model, for where x is the number of hamburgers sold.Determine the intervals on which the profit function is increasing and on which it is decreasing.
A) The profit function is increasing on the interval and decreasing on the interval
B) The profit function is increasing on the interval and decreasing on the interval
C) The profit function is increasing on the interval and decreasing on the interval
D) The profit function is increasing on the interval and decreasing on the interval
E) The profit function is increasing on the interval and decreasing on the interval
and revenue model, for where x is the number of hamburgers sold.Determine the intervals on which the profit function is increasing and on which it is decreasing.
A) The profit function is increasing on the interval
and decreasing on the interval B) The profit function is increasing on the interval
and decreasing on the interval C) The profit function is increasing on the interval
and decreasing on the interval D) The profit function is increasing on the interval
and decreasing on the interval E) The profit function is increasing on the interval
and decreasing on the interval Unlock Deck
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20
For the given function,find the critical numbers.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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21
Sketch a graph of a function f having the following characteristics.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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22
Determine the open intervals on which the graph of is concave downward or concave upward.
A) concave upward on ; concave downward on
B) concave downward on
C) concave upward on
D) concave downward on ; concave upward
E) concave upward on ; concave downward on
is concave downward or concave upward.A) concave upward on
; concave downward on B) concave downward on
C) concave upward on
D) concave downward on
; concave upward E) concave upward on
; concave downward on Unlock Deck
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23
Medication.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) 2.83 hours
C) 707.11 hours
D) 4.83 hours
E) 6.91 hours
Find the t-value at which x is maximum.Round your answer to two decimal places. A) 0 hours
B) 2.83 hours
C) 707.11 hours
D) 4.83 hours
E) 6.91 hours
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24
Find all relative extrema of the function Use the Second Derivative Test where applicable.
A) relative max: ; no relative min
B) relative max: ; no relative min
C) no relative max or min
D) relative min: ; no relative max
E) relative min: ; no relative max
Use the Second Derivative Test where applicable.
A) relative max:
; no relative minB) relative max:
; no relative minC) no relative max or min
D) relative min:
; no relative maxE) relative min:
; no relative max Unlock Deck
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25
Suppose the resident population P(in millions)of the United States can be modeled by , ,where corresponds to 1800.Analytically find the minimum and maximum populations in the U.S.for
A) The population is minimum at and maximum at
B) The population is minimum at and maximum at
C) The population is minimum at and maximum at
D) The population is minimum at and maximum at
E) The population is minimum at and maximum at
, ,where corresponds to 1800.Analytically find the minimum and maximum populations in the U.S.for A) The population is minimum at
and maximum at B) The population is minimum at
and maximum at C) The population is minimum at
and maximum at D) The population is minimum at
and maximum at E) The population is minimum at
and maximum at Unlock Deck
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26
Find all relative extrema of the function Use the Second-Derivative Test when applicable.
A) The relative minimum is and the relative maximum is
B) The relative maximum is
C) The relative minimum is
D) The relative maximum is and the relative minima are and
E) The relative minimum is and the relative maximum is
Use the Second-Derivative Test when applicable. A) The relative minimum is
and the relative maximum is B) The relative maximum is
C) The relative minimum is
D) The relative maximum is
and the relative minima are and E) The relative minimum is
and the relative maximum is Unlock Deck
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27
State the signs of and on the interval (0,2).
A) f ' < 0
f '' = 0
B) f ' < 0
f '' > 0
C) f ' > 0
f '' < 0
D) f ' > 0
f '' > 0
E) f ' < 0
f '' < 0
and on the interval (0,2).
A) f ' < 0
f '' = 0
B) f ' < 0
f '' > 0
C) f ' > 0
f '' < 0
D) f ' > 0
f '' > 0
E) f ' < 0
f '' < 0
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28
The graph of f is shown.Graph f,f ' and f '' on the same set of coordinate axes.
A)
B)
C)
D)
E) none of the above
A)
B)
C)
D)
E) none of the above
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29
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)
, , Unlock Deck
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30
Find the points of inflection and discuss the concavity of the function.
A) inflection point at ; concave downward on ; concave upward on
B) inflection point at ; concave upward on ; concave downward on
C) inflection point at ; concave downward on ; concave upward on
D) inflection point at ; concave upward on ; concave downward on
E) none of the above
A) inflection point at
; concave downward on ; concave upward on B) inflection point at
; concave upward on ; concave downward on C) inflection point at
; concave downward on ; concave upward on D) inflection point at
; concave upward on ; concave downward on E) none of the above
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31
Find the absolute extrema of the function on the interval
A) The maximum of the function is 1 and the minimum of the function is 0..
B) The maximum of the function is 0 and the minimum of the function is -1..
C) The maximum of the function is -1 and the minimum of the function is 0.
D) The maximum of the function is 1 and the minimum of the function is 0.
E) The maximum of the function is 0 and the minimum of the function is 1.
on the interval A) The maximum of the function is 1 and the minimum of the function is 0..
B) The maximum of the function is 0 and the minimum of the function is -1..
C) The maximum of the function is -1 and the minimum of the function is 0.
D) The maximum of the function is 1 and the minimum of the function is 0.
E) The maximum of the function is 0 and the minimum of the function is 1.
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32
Find all relative extrema of the function Use the Second Derivative Test where applicable.
A) relative max:
B) relative min:
C) no relative max or min
D) both A and B
E) none of the above
Use the Second Derivative Test where applicable. A) relative max:
B) relative min:
C) no relative max or min
D) both A and B
E) none of the above
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33
Find the x-value at which the given function has a point of inflection.
A)
B)
C)
D)
E) no point of inflection
A)
B)
C)
D)
E) no point of inflection
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34
Find all relative extrema of the function Use the Second-Derivative Test when applicable.
A) The relative maximum is
B) The relative minimum is
C) The relative maximum is
D) The relative minimum is
E) The relative maximum is
Use the Second-Derivative Test when applicable.A) The relative maximum is
B) The relative minimum is
C) The relative maximum is
D) The relative minimum is
E) The relative maximum is
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35
Graph a function on the interval having the following characteristics. Absolute maximum at Absolute minimum at Relative minimum at Relative maximum at
A)
B)
C)
D)
E)
having the following characteristics. Absolute maximum at Absolute minimum at Relative minimum at Relative maximum at A)
B)
C)
D)
E)
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36
Find the absolute extrema of the function on the closed interval Round your answer to two decimal places.
A) The maximum of the function is 1 and the minimum of the function is 0.
B) The maximum of the function is 2.92 and the minimum of the function is 1.
C) The maximum of the function is 2.92 and the minimum of the function is 0.
D) The maximum of the function is1 and the minimum of the function is 2.52.
E) The maximum of the function is 0 and the minimum of the function is 2.52.
on the closed interval Round your answer to two decimal places. A) The maximum of the function is 1 and the minimum of the function is 0.
B) The maximum of the function is 2.92 and the minimum of the function is 1.
C) The maximum of the function is 2.92 and the minimum of the function is 0.
D) The maximum of the function is1 and the minimum of the function is 2.52.
E) The maximum of the function is 0 and the minimum of the function is 2.52.
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37
Medication.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) 1.73 hours.
B) 1443.37 hours
C) 3.73 hours
D) 4.22 hours
E) 1.73 hours
Find the maximum value of x.Round your answer to two decimal places. A) 1.73 hours.
B) 1443.37 hours
C) 3.73 hours
D) 4.22 hours
E) 1.73 hours
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38
Approximate the critical numbers of the function shown in the graph and determine whether the function has a relative maximum,a relative minimum,an absolute maximum,an absolute minimum,or none of these at each critical number on the interval shown.
A) The critical number yields an absolute maximum and the critical number yields an absolute minimum.
B) Both the critical numbers & yield an absolute maximum.
C) The critical number yields an absolute minimum and the critical number yields an absolute maximum.
D) Both the critical numbers and yield an absolute minimum.
E) The critical number yields a relative minimum and the critical number yields a relative maximum.
A) The critical number
yields an absolute maximum and the critical number yields an absolute minimum.B) Both the critical numbers
& yield an absolute maximum.C) The critical number
yields an absolute minimum and the critical number yields an absolute maximum.D) Both the critical numbers
and yield an absolute minimum.E) The critical number
yields a relative minimum and the critical number yields a relative maximum. Unlock Deck
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39
Determine the open intervals on which the graph of is concave downward or concave upward.
A) concave downward on
B) concave downward on ; concave upward on
C) concave upward on ; concave downward on
D) concave downward on ; concave upward on
E) concave upward on ; concave downward on
is concave downward or concave upward. A) concave downward on
B) concave downward on
; concave upward on C) concave upward on
; concave downward on D) concave downward on
; concave upward on E) concave upward on
; concave downward on Unlock Deck
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40
Find the x-value at which the absolute minimum of f(x)occurs on the interval [a,b].
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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41
If the total cost function for a product is dollars.Find the minimum average cost.
A) $14.25 per unit
B) $18 per unit
C) $14.36 per unit
D) $14 per unit
E) $13 per unit
dollars.Find the minimum average cost. A) $14.25 per unit
B) $18 per unit
C) $14.36 per unit
D) $14 per unit
E) $13 per unit
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42
A wooden beam has a rectangular cross section of height h and width w (see figure).The strength S of the beam is directly proportional to the width and the square of the height.What are the dimensions of the strongest beam that can be cut from a round log of diameter d = 26 inches? Round your answers to two decimal places.[Hint: where k > 0 is the proportionality constant.]
A) inches and inches
B) inches and inches
C) inches and inches
D) inches and inches
E) inches and inches
where k > 0 is the proportionality constant.] A)
inches and inchesB)
inches and inchesC)
inches and inchesD)
inches and inchesE)
inches and inches Unlock Deck
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43
A rectangular page is to contain 144 square inches of print.The margins on each side are 1 inch.Find the dimensions of the page such that the least amount of paper is used.
A) 14, 14
B) 12, 12
C) 10, 10
D) 13, 13
E) 11,11
A) 14, 14
B) 12, 12
C) 10, 10
D) 13, 13
E) 11,11
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44
Average costs.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 14 machines gives the minimum average costs.
B) Using zero machines gives the minimum average costs.
C) Using 24 machines gives the minimum average costs.
D) Using 28 machines gives the minimum average costs.
E) Using 33 machines gives the minimum average costs.
, ,where x is the number of machines used.How many machines give minimum average costs? A) Using 14 machines gives the minimum average costs.
B) Using zero machines gives the minimum average costs.
C) Using 24 machines gives the minimum average costs.
D) Using 28 machines gives the minimum average costs.
E) Using 33 machines gives the minimum average costs.
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45
If the total revenue function for a blender is ,find the maximum revenue.
A) $100
B) $2,500
C) $40
D) $250
E) $2,000
,find the maximum revenue. A) $100
B) $2,500
C) $40
D) $250
E) $2,000
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46
Determine the dimensions of a rectangular solid (with a square base)with maximum volume if its surface area is 441 square meters.
A) square base side ; height
B) square base side ; height
C) square base side ; height
D) square base side ; height
E) square base side ; height
A) square base side
; height B) square base side
; height C) square base side
; height D) square base side
; height E) square base side
; height Unlock Deck
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47
Minimum cost.From a tract of land,a developer plans to fence a rectangular region and then divide it into two identical rectangular lots by putting a fence down the middle.Suppose that the fence for the outside boundary costs $7 per foot and the fence for the middle costs $4 per foot.If each lot contains 5000 square feet,find the dimensions of each lot that yield the minimum cost for the fence.
A) Dimensions are 51.59 ft for the side parallel to the divider and 96.92 ft for the other side.
B) Dimensions are 96.92 ft for the side parallel to the divider and 51.59 ft for the other side.
C) Dimensions are 70.71 ft for the side parallel to the divider and 70.71 ft for the other side.
D) Dimensions are 88.19 ft for the side parallel to the divider and 56.7 ft for the other side.
E) Dimensions are 56.7 ft for the side parallel to the divider and 88.19 ft for the other side.
A) Dimensions are 51.59 ft for the side parallel to the divider and 96.92 ft for the other side.
B) Dimensions are 96.92 ft for the side parallel to the divider and 51.59 ft for the other side.
C) Dimensions are 70.71 ft for the side parallel to the divider and 70.71 ft for the other side.
D) Dimensions are 88.19 ft for the side parallel to the divider and 56.7 ft for the other side.
E) Dimensions are 56.7 ft for the side parallel to the divider and 88.19 ft for the other side.
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48
A firm has total revenue given by for x units of a product.Find the maximum revenue from sales of that product.
A) $2400
B) $2432
C) $404
D) $3300
E) $307
for x units of a product.Find the maximum revenue from sales of that product. A) $2400
B) $2432
C) $404
D) $3300
E) $307
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49
Volume.A rectangular box with a square base is to be formed from a square piece of metal with 54-inch sides.If a square piece with side x is cut from each corner of the metal and the sides are folded up to form an open box,the volume of the box is What value of x will maximize the volume of the box?
A) 27
B) 2
C) 9
D) 14
E) 1
What value of x will maximize the volume of the box? A) 27
B) 2
C) 9
D) 14
E) 1
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50
Find the point on the graph of that is closest to the point (8,0.5).Round your answer to two decimal places.
A) (1.59, 2.53)
B) (2, 4)
C) (2.52, 6.35)
D) (1.39, 1.93)
E) (1.26,1.59)
that is closest to the point (8,0.5).Round your answer to two decimal places. A) (1.59, 2.53)
B) (2, 4)
C) (2.52, 6.35)
D) (1.39, 1.93)
E) (1.26,1.59)
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51
If the total revenue function for a blender is ,determine how many units x must be sold to provide the maximum total revenue in dollars.
A) 625
B) 15,000
C) 25
D) 300
E) 50
,determine how many units x must be sold to provide the maximum total revenue in dollars. A) 625
B) 15,000
C) 25
D) 300
E) 50
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52
A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure).Find the dimensions of a Norman window of maximum area if the total perimeter is 22 feet.Round yours answers to two decimal places.
A) feet and feet
B) feet and feet
C) feet and feet
D) feet and feet
E) feet and feet
A)
feet and feetB)
feet and feetC)
feet and feetD)
feet and feetE)
feet and feet Unlock Deck
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53
Production.Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function Find the number of hours before the rate of production is maximized.That is,find the point of diminishing returns.
A)
B)
C)
D)
E)
Find the number of hours before the rate of production is maximized.That is,find the point of diminishing returns. A)
B)
C)
D)
E)
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54
Find the length and width of a rectangle that has perimeter 56 meters and a maximum area.
A) 7, 21
B) 1, 27
C) 14, 14
D) 15, 13
E) 18,10
A) 7, 21
B) 1, 27
C) 14, 14
D) 15, 13
E) 18,10
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55
You are in a boat 2 miles from the nearest point on the coast.You are to go to point Q located 3 miles down the coast and 1 mile inland (see figure).You can row at a rate of 1 miles per hour and you can walk at a rate of 2 miles per hour.Toward what point on the coast should you row in order to reach point Q in the least time?
A) 3 miles
B) 8 miles
C) 2 miles
D) 1 mile
E) 5 miles
A) 3 miles
B) 8 miles
C) 2 miles
D) 1 mile
E) 5 miles
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56
The profit P (in thousands of dollars)for a company spending an amount s (in thousands of dollars)on advertising is The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline.Find the point of diminishing returns.Round your answer to the nearest thousand dollars.
A) 43 thousand dollars
B) 130 thousand dollars
C) 104 thousand dollars
D) 173 thousand dollars
E) 87 thousand dollars
The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline.Find the point of diminishing returns.Round your answer to the nearest thousand dollars. A) 43 thousand dollars
B) 130 thousand dollars
C) 104 thousand dollars
D) 173 thousand dollars
E) 87 thousand dollars
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57
Find the dimensions of the rectangle of maximum area bounded by the x-axis and y-axis and the graph of
A) length 1.5; width 1.25
B) length 2; width 1
C) length 0.5; width 1.75
D) length 1; width 1.5
E) none of the above
A) length 1.5; width 1.25
B) length 2; width 1
C) length 0.5; width 1.75
D) length 1; width 1.5
E) none of the above
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58
The number of people who donated to a certain organization between 1975 and 1992 can be modeled by the equation donors,where t is the number of years after 1975.Find the inflection point(s)from through ,if any exist.
A) There are no inflection points from
B) There is one inflection point at
C) There are inflection points at
D) There is one inflection point at
E) There are inflection points at , ,
donors,where t is the number of years after 1975.Find the inflection point(s)from through ,if any exist. A) There are no inflection points from
B) There is one inflection point at
C) There are inflection points at
D) There is one inflection point at
E) There are inflection points at
, , Unlock Deck
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59
A rancher has 440 feet of fencing to enclose two adjacent rectangular corrals (see figure).What dimensions should be used so that the enclosed area will be a maximum?
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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60
If the total cost function for a product is dollars,determine how many units x should be produced to minimize the average cost per unit?
A) 450 units
B) 400 units
C) 141 units
D) 200 units
E) 97 units
dollars,determine how many units x should be produced to minimize the average cost per unit? A) 450 units
B) 400 units
C) 141 units
D) 200 units
E) 97 units
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61
Find any vertical asymptotes for the given function.
A)
B)
C)
D)
E) no vertical asymptotes
A)
B)
C)
D)
E) no vertical asymptotes
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62
p is in dollars and q is the number of units.Find the elasticity of the demand function at the price
A) -4.04
B) 1.00
C) 4.04
D) -0.78
E) 0.78
at the price A) -4.04
B) 1.00
C) 4.04
D) -0.78
E) 0.78
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63
A power station is on one side of a river that is 0.5 mile wide,and a factory is 6 miles downstream on the other side of the river.It costs $14 per foot to run overland power lines and $20 per foot to run underwater power lines.Estimate the value of x that minimizes the cost.The value of is the length of the overland piece of power line.
A) 2.56
B) 5.51
C) 3
D) 5.49
E) 0.87
is the length of the overland piece of power line. A) 2.56
B) 5.51
C) 3
D) 5.49
E) 0.87
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64
A travel agency will plan a tour for groups of size 53 or larger.If the group contains exactly 53 people,the price is $800 per person.However,each person's price is reduced by $10 for each additional person above the 53.If the travel agency incurs a price of $150 per person for the tour,what size group will give the agency the maximum profit?
A) 6
B) 91
C) 59
D) 70
E) 32
A) 6
B) 91
C) 59
D) 70
E) 32
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65
Find any vertical asymptotes for the given function.
A)
B)
C)
D)
E) no vertical asymptotes
A)
B)
C)
D)
E) no vertical asymptotes
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66
Average costs.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?
A) $0
B) $35
C) $490
D) $245
E) $8582
, ,where x is the number of machines used.What is the minimum average cost? A) $0
B) $35
C) $490
D) $245
E) $8582
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67
A function and its graph are given.Use the graph to find the horizontal asymptotes,if they exist,where Confirm your results analytically.
A)
B)
C)
D)
E) no horizontal asymptotes
Confirm your results analytically.
A)
B)
C)
D)
E) no horizontal asymptotes
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68
Find the speed v,in miles per hour,that will minimize costs on a 125-mile delivery trip.The cost per hour for fuel is dollars,and the driver is paid dollars per hour.(Assume there are no costs other than wages and fuel.)
A) 75 mph
B) 125 mph
C) 200 mph
D) 80 mph
E) 40 mph
dollars,and the driver is paid dollars per hour.(Assume there are no costs other than wages and fuel.) A) 75 mph
B) 125 mph
C) 200 mph
D) 80 mph
E) 40 mph
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69
Analytically determine the location(s)of any horizontal asymptote(s).
A)
B)
C) ,
D)
E) no horizontal asymptotes
A)
B)
C)
, D)
E) no horizontal asymptotes
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Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
70
Find any horizontal asymptotes for the given function.
A)
B)
C)
D)
E) no horizontal asymptotes
A)
B)
C)
D)
E) no horizontal asymptotes
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
71
A function and its graph are given.Use the graph to find the vertical asymptotes,if they exist.Confirm your results analytically.
A)
B)
C)
D)
E) no vertical asymptotes
A)
B)
C)
D)
E) no vertical asymptotes
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
72
This problem contains a function and its graph,where Use the graph to determine,as well as you can,the vertical asymptote.Check your conclusion by using the function to determine the vertical asymptote analytically.
A)
B)
C)
D)
E) no vertical asymptote
Use the graph to determine,as well as you can,the vertical asymptote.Check your conclusion by using the function to determine the vertical asymptote analytically.
A)
B)
C)
D)
E) no vertical asymptote
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
73
Suppose the sales S (in billions of dollars per year)for Proctor & Gamble for the years 1997 through 2002 can be modeled by , where t represents the year.During which year were the sales increasing at the lowest rate?
A) Sales are increasing at the lowest rate in the year 2002.
B) Sales are increasing at the lowest rate in the year 1997.
C) Sales are increasing at the lowest rate in the year 1998.
D) Sales are increasing at the lowest rate in the year 2000.
E) Sales are increasing at the lowest rate in the year 1999.
, where t represents the year.During which year were the sales increasing at the lowest rate? A) Sales are increasing at the lowest rate in the year 2002.
B) Sales are increasing at the lowest rate in the year 1997.
C) Sales are increasing at the lowest rate in the year 1998.
D) Sales are increasing at the lowest rate in the year 2000.
E) Sales are increasing at the lowest rate in the year 1999.
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
74
Analytically determine the location(s)of any vertical asymptote(s).
A)
B)
C) ,
D)
E) no vertical asymptotes
A)
B)
C)
, D)
E) no vertical asymptotes
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
75
A function and its graph are given.Use the graph to find the horizontal asymptotes,if they exist.Confirm your results analytically.
A)
B)
C)
D)
E) no horizontal asymptotes
A)
B)
C)
D)
E) no horizontal asymptotes
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
76
Find any horizontal asymptotes for the given function.
A)
B)
C)
D)
E) no horizontal asymptotes
A)
B)
C)
D)
E) no horizontal asymptotes
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
77
A firm can produce 100 units per week.If its total cost function is dollars,and its total revenue function is dollars,find the maximum profit.
A) $3,679
B) $1,600
C) $7,031
D) $1,265
E) $1,941
dollars,and its total revenue function is dollars,find the maximum profit. A) $3,679
B) $1,600
C) $7,031
D) $1,265
E) $1,941
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
78
This problem contains a function and its graph,where Use the graph to determine,as well as you can,the horizontal asymptote.Check your conclusion by using the function to determine the horizontal asymptote analytically.
A)
B)
C)
D)
E)
Use the graph to determine,as well as you can,the horizontal asymptote.Check your conclusion by using the function to determine the horizontal asymptote analytically.
A)
B)
C)
D)
E)
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
79
A function and its graph are given.Use the graph to find the vertical asymptotes,if they exist,where Confirm your results analytically.
A)
B)
C)
D)
E) no vertical asymptotes
Confirm your results analytically.
A)
B)
C)
D)
E) no vertical asymptotes
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
80
A firm can produce 100 units per week.If its total cost function is dollars,and its total revenue function is dollars,how many units x should it produce to maximize its profit?
A) 1250 units
B) 650 units
C) 97 units
D) 50 units
E) 100 units
dollars,and its total revenue function is dollars,how many units x should it produce to maximize its profit? A) 1250 units
B) 650 units
C) 97 units
D) 50 units
E) 100 units
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
Unlock Deck
k this deck
Unlock Deck
Unlock for access to all 122 flashcards in this deck.
