تكلم مع الذكاء الاصطناعي
Deck 4: Applications of Differentiation
ملء الشاشة (f)
سؤال
For time 
,a particle moves along the x-axis with position given by 
.During what time intervals is the speed of the particle increasing?
A)
< t ≤ 10 only
B)0 ≤ t < 1 and
< t <
C)0 ≤ t < 1 and
< t ≤ 10
D)1 < t <
and
< t ≤ 10
,a particle moves along the x-axis with position given by .During what time intervals is the speed of the particle increasing? A)
< t ≤ 10 onlyB)0 ≤ t < 1 and
< t <C)0 ≤ t < 1 and
< t ≤ 10D)1 < t <
and < t ≤ 10
سؤال
For the function 
,how many of the inflection points of the function are also extrema?
A)3
B)2
C)1
D)None
,how many of the inflection points of the function are also extrema? A)3
B)2
C)1
D)None
سؤال
Let 
be defined as 
in the open interval (-3,3).Determine the coordinate(s)of the relative maxima of 
in the open interval.Explain your reasoning.
be defined as in the open interval (-3,3).Determine the coordinate(s)of the relative maxima of in the open interval.Explain your reasoning. سؤال
Let M represent the absolute maximum of f(x)in an interval. Let R represent a root of f(x)in the given interval.
Let m represent the absolute minimum of f(x)in the interval.
If
,then which of the following is true over the closed interval 
?
A)M and R occur at a critical point and m occurs at an endpoint.
B)M and m occur at critical points.
C)M,m,and R occur at endpoints of the given interval.
D)M occurs at an endpoint,whereas m and R occur at a critical point.
Let m represent the absolute minimum of f(x)in the interval.
If
,then which of the following is true over the closed interval ?
A)M and R occur at a critical point and m occurs at an endpoint.
B)M and m occur at critical points.
C)M,m,and R occur at endpoints of the given interval.
D)M occurs at an endpoint,whereas m and R occur at a critical point.
سؤال
Find all values of x for which 
is concave down.Explain your reasoning.
is concave down.Explain your reasoning. سؤال
The function 
is defined as 
on the open interval (-3,3)as illustrated in the graph shown.
Determine the coordinates of the relative extrema of
in the open interval (-3,3).
is defined as on the open interval (-3,3)as illustrated in the graph shown.
Determine the coordinates of the relative extrema of
in the open interval (-3,3). سؤال
The points of inflection for 
are at 
and 
.Which of the following is (are)true?
I.The points of inflection for
are at 
and 
.
II.The points of inflection for
are at 
and 
.
III.The points of inflection for
are at 
and 
.
A)I only
B)II only
C)I and II only
D)I and III only
are at and .Which of the following is (are)true? I.The points of inflection for
are at and .II.The points of inflection for
are at and .III.The points of inflection for
are at and .
A)I only
B)II only
C)I and II only
D)I and III only
سؤال
The graph of f ′(x)is given below for 
.On which interval(s)is the function f(x)both increasing and concave up? 
A)
B)
C)
D)
.On which interval(s)is the function f(x)both increasing and concave up? A)
B)
C)
D)
سؤال
A rectangle is drawn in the first quadrant so that it has two adjacent sides on the coordinate axes and one vertex on the curve 
.Find the x coordinate of the vertex for which the area of the rectangle is a maximum.
A)
B)
C)
D)
.Find the x coordinate of the vertex for which the area of the rectangle is a maximum. A)
B)
C)
D)
سؤال
Given the following conditions for 
,which graph best illustrates 
? 
: The domain of the function is the real numbers,but 
;
; 
; 
.
for all x where 
,and 
does not exist at 
. 
for 
, 
for 
,and 
does not exist at 
.
A)
B)
C)
D)
,which graph best illustrates ? : The domain of the function is the real numbers,but ;
; ; .
for all x where ,and does not exist at . for , for ,and does not exist at .
A)
B)
C)
D)
سؤال
A calculator may be used for the following question.Let 
be a function such that 
.In the interval 
,the graph of 
has a point of inflection nearest 
.
A)0.37
B)1.233
C)2.141
D)2.268
be a function such that .In the interval ,the graph of has a point of inflection nearest . A)0.37
B)1.233
C)2.141
D)2.268
سؤال
If 
,then on which interval(s)is the continuous function f(x)increasing?
A)
B)
C)
D)
,then on which interval(s)is the continuous function f(x)increasing? A)
B)
C)
D)
سؤال
For time 
,a particle moves along the x-axis with position given by 
.What is the position of the particle when it is farthest to the left?
A)
B)-11
C)-16
D)
,a particle moves along the x-axis with position given by .What is the position of the particle when it is farthest to the left? A)
B)-11
C)-16
D)
سؤال
A farmer has 225 yards of fencing to form two identical rectangular pens and a third pen that is twice as long as the other two pens,as shown in the diagram below.All three pens have the same width,x.Which value of y produces the maximum total fenced area?
A)
B)15
C)
D)
A)
B)15
C)
D)
سؤال
Find the equation of the tangent line T to the graph of 
at the given point 
.
A)
B)
C)
D)
E)
at the given point . A)
B)
C)
D)
E)
سؤال
Find the equation of the tangent line T to the graph of 
at the given point 
.
A)
B)
C)
D)
E)
at the given point . A)
B)
C)
D)
E)
سؤال
For what values of x is 
not defined? Explain your reasoning.
not defined? Explain your reasoning. سؤال
What value of c in the open interval (0,16)satisfies the Mean Value Theorem for 
?
A)0
B)
C)
D)4
? A)0
B)
C)
D)4
سؤال
The position of an object moving along a straight line for 
is given by 
,and the position of a second object moving along the same line is given by 
.If both objects begin at 
,at what time is the distance between the objects a minimum?
A)2
B)
C)
D)0
is given by ,and the position of a second object moving along the same line is given by .If both objects begin at ,at what time is the distance between the objects a minimum? A)2
B)
C)
D)0
سؤال
Based on the graph of 
pictured above,how many points of inflection exist for the twice differentiable function 
on the interval 
?
A)4
B)3
C)2
D)1
pictured above,how many points of inflection exist for the twice differentiable function on the interval ?
A)4
B)3
C)2
D)1
سؤال
Compare 
and 
for 
at 
with 
Give your answers to four decimal places.
A)
B)
C)
D)
E)
and for at with Give your answers to four decimal places. A)
B)
C)
D)
E)
سؤال
The measurement of the side of a square floor tile is 22 inches,with a possible error of 
inch.Use differentials to approximate the possible propagated error in computing the area of the square.
A)
square inches
B)
square inches
C)
square inches
D)
square inches
E)
square inches
inch.Use differentials to approximate the possible propagated error in computing the area of the square. A)
square inchesB)
square inchesC)
square inchesD)
square inchesE)
square inches سؤال
Find the differential 
of the function 
.
A)
B)
C)
D)
E)
of the function .A)
B)
C)
D)
E)
سؤال
سؤال
Use differentials to approximate the value of 
.Round your answer to four decimal places.
A)1.8650
B)1.8750
C)1.9050
D)1.8850
E)1.8950
.Round your answer to four decimal places. A)1.8650
B)1.8750
C)1.9050
D)1.8850
E)1.8950
سؤال
Find the length and width of a rectangle that has an area of 1,152 square feet and whose perimeter is a minimum.
A)
feet; 
feet
B)
feet; 
feet
C)
feet
D)
feet; 
feet
E)
feet; 
feet
A)
feet; feetB)
feet; feetC)
feetD)
feet; feetE)
feet; feet سؤال
سؤال
Compare 
and 
for 
at 
with 
Give your answers to four decimal places.
A)
B)
C)
D)
E)
and for at with Give your answers to four decimal places. A)
B)
C)
D)
E)
سؤال
Find the point on the graph of the function 
that is closest to the point 
.
A)
B)
C)
D)
E)
that is closest to the point . A)
B)
C)
D)
E)
سؤال
سؤال
Use differentials to approximate the value of 
.Round your answer to four decimal places.
A)3.0470
B)3.0270
C)3.0170
D)3.0370
E)3.0570
.Round your answer to four decimal places. A)3.0470
B)3.0270
C)3.0170
D)3.0370
E)3.0570
سؤال
The range R of a projectile is 
where ν0 is the initial velocity in feet per second and θ is the angle of elevation.If ν0 = 2300 feet per second and θ is changed from 13° to 14° use differentials to approximate the change in the range.Round your answer to the nearest integer.
A)2593 ft
B)4568 ft
C)5623 ft
D)2811 ft
E)5186 ft
where ν0 is the initial velocity in feet per second and θ is the angle of elevation.If ν0 = 2300 feet per second and θ is changed from 13° to 14° use differentials to approximate the change in the range.Round your answer to the nearest integer. A)2593 ft
B)4568 ft
C)5623 ft
D)2811 ft
E)5186 ft
سؤال
The measurement of the radius of the end of a log is found to be 10 inches,with a possible error of 
inch.Use differentials to approximate the possible propagated error in computing the area of the end of the log.
A)±12π square inches
B)±16π square inches
C)±5π square inches
D)±20π square inches
E)±8π square inches
inch.Use differentials to approximate the possible propagated error in computing the area of the end of the log. A)±12π square inches
B)±16π square inches
C)±5π square inches
D)±20π square inches
E)±8π square inches
سؤال
Find the point on the graph of the function 
that is closest to the point 
.Round all numerical values in your answer to four decimal places.
A)
B)
C)
D)
E)
that is closest to the point .Round all numerical values in your answer to four decimal places. A)
B)
C)
D)
E)
سؤال
Find the differential 
of the function 
.
A)
B)
C)
D)
E)
of the function . A)
B)
C)
D)
E)
سؤال
The total stopping distance T of a vehicle is 
where T is in feet and x is the speed in miles per hour.Use differentials to approximate the percent change in total stopping distance as speed changes from x = 22 to x = 27 miles per hour.Round your answer to one decimal place.
A)51.8%
B)22.1%
C)42.7%
D)78.0%
E)122.7%
where T is in feet and x is the speed in miles per hour.Use differentials to approximate the percent change in total stopping distance as speed changes from x = 22 to x = 27 miles per hour.Round your answer to one decimal place. A)51.8%
B)22.1%
C)42.7%
D)78.0%
E)122.7%
سؤال
سؤال
Find the differential 
of the function 
.
A)
B)
C)
D)
E)
of the function . A)
B)
C)
D)
E)
سؤال
Find two positive numbers whose product is 186 and whose sum is a minimum.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
سؤال
سؤال
The graph of a function f is is shown below.Sketch the graph of the derivative 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
Analyze and sketch a graph of the function 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
Analyze and sketch a graph of the function 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
The graph of a function f is is shown below.Sketch the graph of the derivative 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder.The total volume of the solid is 28 cubic centimeters.Find the radius,r,of the cylinder that produces the minimum surface area.Round your answer to two decimal places.
A)
cm
B)
cm
C)
cm
D)
cm
E)
cm
A)
cmB)
cmC)
cmD)
cmE)
cm سؤال
Determine the slant asymptote of the graph of 
.
A)
B)
C)
D)
E)no slant asymptotes
. A)
B)
C)
D)
E)no slant asymptotes
سؤال
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 38 feet. 
A)
feet; 
feet
B)
feet; 
feet
C)
feet; 
feet
D)
feet; 
feet
E)
feet; 
feet
A)
feet; feetB)
feet; feetC)
feet; feetD)
feet; feetE)
feet; feet سؤال
The sum of the perimeters of an equilateral triangle and a square is 8.Find the dimensions of the triangle and the square that produce a minimum total area.
A)side of square:
;side of triangle:
B)side of square:
;side of triangle:
C)side of square:
;side of triangle:
D)side of square:
;side of triangle:
E)side of square:
;side of triangle:
A)side of square:
;side of triangle:B)side of square:
;side of triangle:C)side of square:
;side of triangle:D)side of square:
;side of triangle:E)side of square:
;side of triangle: سؤال
The graph of a function f is is shown below.Sketch the graph of the derivative 
. 
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
Analyze and sketch a graph of the function 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
Analyze and sketch a graph of the function 
.
A)
B)
C)
D)
E)none of the above.
. A)
B)
C)
D)
E)none of the above.
سؤال
سؤال
Analyze and sketch a graph of the function 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
Determine the dimensions of a rectangular solid (with a square base)with maximum volume if its surface area is 121 square meters.
A)Dimensions:
B)Dimensions:
C)Dimensions:
D)Dimensions:
E)Dimensions:
A)Dimensions:
B)Dimensions:
C)Dimensions:
D)Dimensions:
E)Dimensions:
سؤال
Analyze and sketch a graph of the function 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
A sector with central angle 
is cut from a circle of radius 4 inches,and the edges of the sector are brought together to form a cone.Find the magnitude of 
such that the volume of the cone is a maximum.
A)
radians
B)
radians
C)
radians
D)
radians
E)
radians
is cut from a circle of radius 4 inches,and the edges of the sector are brought together to form a cone.Find the magnitude of such that the volume of the cone is a maximum. A)
radiansB)
radiansC)
radiansD)
radiansE)
radians سؤال
On a given day,the flow rate F (cars per hour)on a congested roadway is given by 
,where v is the speed of the traffic in miles per hour.What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour.
A)19 miles per hour
B)17 miles per hour
C)18 miles per hour
D)10 miles per hour
E)20 miles per hour
,where v is the speed of the traffic in miles per hour.What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour. A)19 miles per hour
B)17 miles per hour
C)18 miles per hour
D)10 miles per hour
E)20 miles per hour
سؤال
A rectangle is bounded by the x- and y-axes and the graph of 
(see figure).What length and width should the rectangle have so that its area is a maximum? 
A)
B)
C)
D)
E)
(see figure).What length and width should the rectangle have so that its area is a maximum? A)
B)
C)
D)
E)
سؤال
Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on this money.Furthermore,the bank can reinvest this money at 
.Find the interest rate the bank should pay to maximize profit.(Use the simple interest formula. )
A)
B)
C)
D)
E)
.Find the interest rate the bank should pay to maximize profit.(Use the simple interest formula. ) A)
B)
C)
D)
E)
سؤال
The graph of a function f is is shown below. 
Sketch the graph of the derivative
.
A)
B)
C)
D)
E)
Sketch the graph of the derivative
.
A)
B)
C)
D)
E)
سؤال
Find the points of inflection and discuss the concavity of the function 
on the interval 
.
A)no inflection points.concave up on
B)concave downward on
;concave upward on
;inflection point at
C)no inflection points.concave down on
D)concave upward on
;concave downward on
;inflection point at
E)none of the above
on the interval . A)no inflection points.concave up on
B)concave downward on
;concave upward on ;inflection point at
C)no inflection points.concave down on
D)concave upward on
;concave downward on ;inflection point atE)none of the above
سؤال
Sketch a graph of the function 
over the interval 
.
A)
B)
C)
D)
E)
over the interval . A)
B)
C)
D)
E)
سؤال
Find the point of inflection of the graph of the function 
on the interval 
.
A)
B)
C)
D)
E)
on the interval . A)
B)
C)
D)
E)
سؤال
Determine the open intervals on which the graph of 
is concave downward or concave upward.
A)concave downward on
;concave upward 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
;concave upward onB)concave downward on
;concave upward onC)concave upward on
;concave downward on
D)concave downward on
;concave upward onE)concave upward on
;concave downward on سؤال
Sketch a graph of the function 
over the interval 
.
A)
B)
C)
D)
E)
over the interval . A)
B)
C)
D)
E)
سؤال
Find all points of inflection on the graph of the function 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
The graph of f is shown below.For which values of x is 
zero? 
A)
B)
C)
D)
E)
zero? A)
B)
C)
D)
E)
سؤال
Determine the open intervals on which the graph of the function 
is concave upward or concave downward.
A)concave upward:
;concave downward:
B)concave upward:
;concave downward:
C)concave upward:
;concave downward:
D)concave upward:
;concave downward:
E)concave upward:
;concave downward:
is concave upward or concave downward. A)concave upward:
;concave downward:B)concave upward:
;concave downward:C)concave upward:
;concave downward:D)concave upward:
;concave downward:E)concave upward:
;concave downward: سؤال
Determine the open intervals on which the graph of 
is concave downward or concave upward.
A)concave downward on
B)concave upward on
;concave downward on
C)concave upward on
;concave downward on
D)concave downward on
;concave upward on
E)concave downward on
;concave upward on
is concave downward or concave upward. A)concave downward on
B)concave upward on
;concave downward onC)concave upward on
;concave downward onD)concave downward on
;concave upward onE)concave downward on
;concave upward on سؤال
Analyze the graph of the function 
.Determine any intercepts,relative extrema,points of inflection and asymptotes.Also determine where the graph is increasing or decreasing and concave up or concave down.Then identify the graph from the choices below.
A)
B)
C)
D)
E)
.Determine any intercepts,relative extrema,points of inflection and asymptotes.Also determine where the graph is increasing or decreasing and concave up or concave down.Then identify the graph from the choices below. A)
B)
C)
D)
E)
سؤال
Determine the open intervals on which the graph of the function 
is concave upward or concave downward.
A)concave upward:
;concave downward:
B)concave upward:
;concave downward:
C)concave upward:
;concave downward:
D)concave upward:
;concave downward:
E)concave upward:
;concave downward:
is concave upward or concave downward. A)concave upward:
;concave downward:B)concave upward:
;concave downward:C)concave upward:
;concave downward:D)concave upward:
;concave downward:E)concave upward:
;concave downward: سؤال
A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process.The flow rate R (in liters per hour)at time t (in hours)is given by the linear model 
.Write the linear model in exponential form.Round to four decimal places.
A)
B)
C)
D)
E)
.Write the linear model in exponential form.Round to four decimal places. A)
B)
C)
D)
E)
سؤال
The graph of f is shown below.On what interval is 
an increasing function? 
A)
B)
C)
D)
E)
an increasing function? A)
B)
C)
D)
E)
سؤال
Determine the open intervals on which the graph of 
is concave downward or concave upward.
A)concave upward on
B)concave downward on
;concave upward on
C)concave downward on
;concave upward on
D)concave downward on
E)concave upward on
;concave downward on
is concave downward or concave upward. A)concave upward on
B)concave downward on
;concave upward onC)concave downward on
;concave upward onD)concave downward on
E)concave upward on
;concave downward on سؤال
Find all points of inflection,if any exist,of the graph of the function 
.Round your answers to two decimal places.
A)
B)
C)
D)
E)no points of inflection
.Round your answers to two decimal places. A)
B)
C)
D)
E)no points of inflection
سؤال
Analyze and sketch a graph of the function 
.
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
سؤال
Find the points of inflection and discuss the concavity of the function. 
A)inflection point at
;concave upward on
;concave downward on
B)inflection point at
;concave downward on
;concave upward on
C)inflection point at
;concave downward on
;concave upward on
D)inflection point at
;concave downward on
;concave upward on
E)inflection point at
;concave upward on
;concave downward on
A)inflection point at
;concave upward on ;concave downward onB)inflection point at
;concave downward on ;concave upward onC)inflection point at
;concave downward on ;concave upward onD)inflection point at
;concave downward on ;concave upward onE)inflection point at
;concave upward on ;concave downward on سؤال
The graph of f is shown below.For which value of x is 
zero? 
A)
B)
C)
D)
E)
zero? A)
B)
C)
D)
E)
سؤال
Find the points of inflection and discuss the concavity of the function 
.
A)no inflection points;concave up on
B)no inflection points;concave down on
C)inflection point at
;concave up on
D)inflection point at
;concave up on
;concave down on
E)inflection point at
;concave down on
. A)no inflection points;concave up on
B)no inflection points;concave down on
C)inflection point at
;concave up onD)inflection point at
;concave up on ;concave down onE)inflection point at
;concave down on سؤال
The graph of f is shown below.For which value of x is 
minimum? 
A)
B)
C)
D)
E)
minimum? A)
B)
C)
D)
E)
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ملء الشاشة (f)
Deck 4: Applications of Differentiation
1
For time ,a particle moves along the x-axis with position given by .During what time intervals is the speed of the particle increasing?
A) < t ≤ 10 only
B)0 ≤ t < 1 and < t <
C)0 ≤ t < 1 and < t ≤ 10
D)1 < t < and
< t ≤ 10
,a particle moves along the x-axis with position given by .During what time intervals is the speed of the particle increasing? A)
< t ≤ 10 onlyB)0 ≤ t < 1 and
< t <C)0 ≤ t < 1 and
< t ≤ 10D)1 < t <
and < t ≤ 100 ≤ t < 1 and < t ≤ 10
< t ≤ 10 2
For the function ,how many of the inflection points of the function are also extrema?
A)3
B)2
C)1
D)None
,how many of the inflection points of the function are also extrema? A)3
B)2
C)1
D)None
None
3
Let be defined as in the open interval (-3,3).Determine the coordinate(s)of the relative maxima of in the open interval.Explain your reasoning.
be defined as in the open interval (-3,3).Determine the coordinate(s)of the relative maxima of in the open interval.Explain your reasoning.For 
,the relative maximum will occur at the point (-1,8).All values of the new function 
are made positive with the application of the absolute value to 
,and the relative minimum of 
becomes a relative maximum for 
.
,the relative maximum will occur at the point (-1,8).All values of the new function are made positive with the application of the absolute value to ,and the relative minimum of becomes a relative maximum for . 4
Let M represent the absolute maximum of f(x)in an interval. Let R represent a root of f(x)in the given interval.
Let m represent the absolute minimum of f(x)in the interval.
If ,then which of the following is true over the closed interval ?
A)M and R occur at a critical point and m occurs at an endpoint.
B)M and m occur at critical points.
C)M,m,and R occur at endpoints of the given interval.
D)M occurs at an endpoint,whereas m and R occur at a critical point.
Let m represent the absolute minimum of f(x)in the interval.
If
,then which of the following is true over the closed interval ?
A)M and R occur at a critical point and m occurs at an endpoint.
B)M and m occur at critical points.
C)M,m,and R occur at endpoints of the given interval.
D)M occurs at an endpoint,whereas m and R occur at a critical point.
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5
Find all values of x for which is concave down.Explain your reasoning.
is concave down.Explain your reasoning. فتح الحزمة
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6
The function is defined as on the open interval (-3,3)as illustrated in the graph shown.
Determine the coordinates of the relative extrema of in the open interval (-3,3).
is defined as on the open interval (-3,3)as illustrated in the graph shown.
Determine the coordinates of the relative extrema of
in the open interval (-3,3). فتح الحزمة
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7
The points of inflection for are at and .Which of the following is (are)true?
I.The points of inflection for are at and .
II.The points of inflection for are at and .
III.The points of inflection for are at and .
A)I only
B)II only
C)I and II only
D)I and III only
are at and .Which of the following is (are)true? I.The points of inflection for
are at and .II.The points of inflection for
are at and .III.The points of inflection for
are at and .
A)I only
B)II only
C)I and II only
D)I and III only
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8
The graph of f ′(x)is given below for .On which interval(s)is the function f(x)both increasing and concave up?
A)
B)
C)
D)
.On which interval(s)is the function f(x)both increasing and concave up? A)
B)
C)
D)
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9
A rectangle is drawn in the first quadrant so that it has two adjacent sides on the coordinate axes and one vertex on the curve .Find the x coordinate of the vertex for which the area of the rectangle is a maximum.
A)
B)
C)
D)
.Find the x coordinate of the vertex for which the area of the rectangle is a maximum. A)
B)
C)
D)
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10
Given the following conditions for ,which graph best illustrates ? : The domain of the function is the real numbers,but ;
; ; .
for all x where ,and does not exist at . for , for ,and does not exist at .
A)
B)
C)
D)
,which graph best illustrates ? : The domain of the function is the real numbers,but ;
; ; .
for all x where ,and does not exist at . for , for ,and does not exist at .
A)
B)
C)
D)
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11
A calculator may be used for the following question.Let be a function such that .In the interval ,the graph of has a point of inflection nearest .
A)0.37
B)1.233
C)2.141
D)2.268
be a function such that .In the interval ,the graph of has a point of inflection nearest . A)0.37
B)1.233
C)2.141
D)2.268
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12
If ,then on which interval(s)is the continuous function f(x)increasing?
A)
B)
C)
D)
,then on which interval(s)is the continuous function f(x)increasing? A)
B)
C)
D)
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13
For time ,a particle moves along the x-axis with position given by .What is the position of the particle when it is farthest to the left?
A)
B)-11
C)-16
D)
,a particle moves along the x-axis with position given by .What is the position of the particle when it is farthest to the left? A)
B)-11
C)-16
D)
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14
A farmer has 225 yards of fencing to form two identical rectangular pens and a third pen that is twice as long as the other two pens,as shown in the diagram below.All three pens have the same width,x.Which value of y produces the maximum total fenced area?
A)
B)15
C)
D)
A)
B)15
C)
D)
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15
Find the equation of the tangent line T to the graph of at the given point .
A)
B)
C)
D)
E)
at the given point . A)
B)
C)
D)
E)
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16
Find the equation of the tangent line T to the graph of at the given point .
A)
B)
C)
D)
E)
at the given point . A)
B)
C)
D)
E)
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17
For what values of x is not defined? Explain your reasoning.
not defined? Explain your reasoning. فتح الحزمة
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18
What value of c in the open interval (0,16)satisfies the Mean Value Theorem for ?
A)0
B)
C)
D)4
? A)0
B)
C)
D)4
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19
The position of an object moving along a straight line for is given by ,and the position of a second object moving along the same line is given by .If both objects begin at ,at what time is the distance between the objects a minimum?
A)2
B)
C)
D)0
is given by ,and the position of a second object moving along the same line is given by .If both objects begin at ,at what time is the distance between the objects a minimum? A)2
B)
C)
D)0
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20
Based on the graph of pictured above,how many points of inflection exist for the twice differentiable function on the interval ?
A)4
B)3
C)2
D)1
pictured above,how many points of inflection exist for the twice differentiable function on the interval ?
A)4
B)3
C)2
D)1
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21
Compare and for at with Give your answers to four decimal places.
A)
B)
C)
D)
E)
and for at with Give your answers to four decimal places. A)
B)
C)
D)
E)
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22
The measurement of the side of a square floor tile is 22 inches,with a possible error of inch.Use differentials to approximate the possible propagated error in computing the area of the square.
A) square inches
B) square inches
C) square inches
D) square inches
E) square inches
inch.Use differentials to approximate the possible propagated error in computing the area of the square. A)
square inchesB)
square inchesC)
square inchesD)
square inchesE)
square inches فتح الحزمة
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23
Find the differential of the function .
A)
B)
C)
D)
E)
of the function .A)
B)
C)
D)
E)
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24
The radius of a spherical balloon is measured to be 9 inches,with a possible error of 0.02 inch.Use differentials to approximate the maximum possible error in calculating the volume of the sphere.Round your answer to two decimal places.
A)±10.18 cubic inches
B)±5.65 cubic inches
C)±30.54 cubic inches
D)±20.36 cubic inches
E)±4.52 cubic inches
A)±10.18 cubic inches
B)±5.65 cubic inches
C)±30.54 cubic inches
D)±20.36 cubic inches
E)±4.52 cubic inches
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25
Use differentials to approximate the value of .Round your answer to four decimal places.
A)1.8650
B)1.8750
C)1.9050
D)1.8850
E)1.8950
.Round your answer to four decimal places. A)1.8650
B)1.8750
C)1.9050
D)1.8850
E)1.8950
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26
Find the length and width of a rectangle that has an area of 1,152 square feet and whose perimeter is a minimum.
A) feet; feet
B) feet; feet
C) feet
D) feet; feet
E) feet; feet
A)
feet; feetB)
feet; feetC)
feetD)
feet; feetE)
feet; feet فتح الحزمة
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27
The radius of a spherical balloon is measured to be 7 inches,with a possible error of 0.03 inch.Use differentials to approximate the maximum possible error in calculating the surface area of the sphere.Round your answer to two decimal places.
A)±6.6 square inches
B)±9.24 square inches
C)±27.71 square inches
D)±18.47 square inches
E)±5.28 square inches
A)±6.6 square inches
B)±9.24 square inches
C)±27.71 square inches
D)±18.47 square inches
E)±5.28 square inches
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28
Compare and for at with Give your answers to four decimal places.
A)
B)
C)
D)
E)
and for at with Give your answers to four decimal places. A)
B)
C)
D)
E)
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29
Find the point on the graph of the function that is closest to the point .
A)
B)
C)
D)
E)
that is closest to the point . A)
B)
C)
D)
E)
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30
The measurements of the base and altitude of a triangle are found to be 42 and 21 centimeters.The possible error in each measurement is 0.25 centimeter.Use differentials to estimate the propagated error in computing the area of the triangle.Round your answer to four decimal places.
A)±8.6625 square centimeters
B)±7.0875 square centimeters
C)±7.875 square centimeters
D)±9.0563 square centimeters
E)±9.45 square centimeters
A)±8.6625 square centimeters
B)±7.0875 square centimeters
C)±7.875 square centimeters
D)±9.0563 square centimeters
E)±9.45 square centimeters
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31
Use differentials to approximate the value of .Round your answer to four decimal places.
A)3.0470
B)3.0270
C)3.0170
D)3.0370
E)3.0570
.Round your answer to four decimal places. A)3.0470
B)3.0270
C)3.0170
D)3.0370
E)3.0570
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32
The range R of a projectile is where ν0 is the initial velocity in feet per second and θ is the angle of elevation.If ν0 = 2300 feet per second and θ is changed from 13° to 14° use differentials to approximate the change in the range.Round your answer to the nearest integer.
A)2593 ft
B)4568 ft
C)5623 ft
D)2811 ft
E)5186 ft
where ν0 is the initial velocity in feet per second and θ is the angle of elevation.If ν0 = 2300 feet per second and θ is changed from 13° to 14° use differentials to approximate the change in the range.Round your answer to the nearest integer. A)2593 ft
B)4568 ft
C)5623 ft
D)2811 ft
E)5186 ft
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33
The measurement of the radius of the end of a log is found to be 10 inches,with a possible error of inch.Use differentials to approximate the possible propagated error in computing the area of the end of the log.
A)±12π square inches
B)±16π square inches
C)±5π square inches
D)±20π square inches
E)±8π square inches
inch.Use differentials to approximate the possible propagated error in computing the area of the end of the log. A)±12π square inches
B)±16π square inches
C)±5π square inches
D)±20π square inches
E)±8π square inches
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34
Find the point on the graph of the function that is closest to the point .Round all numerical values in your answer to four decimal places.
A)
B)
C)
D)
E)
that is closest to the point .Round all numerical values in your answer to four decimal places. A)
B)
C)
D)
E)
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35
Find the differential of the function .
A)
B)
C)
D)
E)
of the function . A)
B)
C)
D)
E)
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36
The total stopping distance T of a vehicle is where T is in feet and x is the speed in miles per hour.Use differentials to approximate the percent change in total stopping distance as speed changes from x = 22 to x = 27 miles per hour.Round your answer to one decimal place.
A)51.8%
B)22.1%
C)42.7%
D)78.0%
E)122.7%
where T is in feet and x is the speed in miles per hour.Use differentials to approximate the percent change in total stopping distance as speed changes from x = 22 to x = 27 miles per hour.Round your answer to one decimal place. A)51.8%
B)22.1%
C)42.7%
D)78.0%
E)122.7%
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37
Find two positive numbers such that the sum of the first and twice the second is 96 and whose product is a maximum.
A)48 and 48
B)48 and 24
C)42 and 27
D)54 and 21
E)44 and 26
A)48 and 48
B)48 and 24
C)42 and 27
D)54 and 21
E)44 and 26
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38
Find the differential of the function .
A)
B)
C)
D)
E)
of the function . A)
B)
C)
D)
E)
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39
Find two positive numbers whose product is 186 and whose sum is a minimum.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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40
Find the length and width of a rectangle that has perimeter 40 meters and a maximum area.
A)10 m;10 m.
B)14 m;7 m.
C)1 m;19 m.
D)11 m;9 m.
E)5 m;15 m.
A)10 m;10 m.
B)14 m;7 m.
C)1 m;19 m.
D)11 m;9 m.
E)5 m;15 m.
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41
The graph of a function f is is shown below.Sketch the graph of the derivative .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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42
Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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43
Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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44
The graph of a function f is is shown below.Sketch the graph of the derivative .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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45
A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder.The total volume of the solid is 28 cubic centimeters.Find the radius,r,of the cylinder that produces the minimum surface area.Round your answer to two decimal places.
A) cm
B) cm
C) cm
D) cm
E) cm
A)
cmB)
cmC)
cmD)
cmE)
cm فتح الحزمة
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46
Determine the slant asymptote of the graph of .
A)
B)
C)
D)
E)no slant asymptotes
. A)
B)
C)
D)
E)no slant asymptotes
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47
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 38 feet.
A) feet; feet
B) feet; feet
C) feet; feet
D) feet; feet
E) feet; feet
A)
feet; feetB)
feet; feetC)
feet; feetD)
feet; feetE)
feet; feet فتح الحزمة
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48
The sum of the perimeters of an equilateral triangle and a square is 8.Find the dimensions of the triangle and the square that produce a minimum total area.
A)side of square: ;side of triangle:
B)side of square: ;side of triangle:
C)side of square: ;side of triangle:
D)side of square: ;side of triangle:
E)side of square: ;side of triangle:
A)side of square:
;side of triangle:B)side of square:
;side of triangle:C)side of square:
;side of triangle:D)side of square:
;side of triangle:E)side of square:
;side of triangle: فتح الحزمة
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49
The graph of a function f is is shown below.Sketch the graph of the derivative .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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50
Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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51
Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)none of the above.
. A)
B)
C)
D)
E)none of the above.
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52
A rectangular page is to contain 196 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)17,17
B)29,29
C)18,18
D)15,15
E)16,16
A)17,17
B)29,29
C)18,18
D)15,15
E)16,16
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53
Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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54
Determine the dimensions of a rectangular solid (with a square base)with maximum volume if its surface area is 121 square meters.
A)Dimensions:
B)Dimensions:
C)Dimensions:
D)Dimensions:
E)Dimensions:
A)Dimensions:
B)Dimensions:
C)Dimensions:
D)Dimensions:
E)Dimensions:
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55
Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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56
A sector with central angle is cut from a circle of radius 4 inches,and the edges of the sector are brought together to form a cone.Find the magnitude of such that the volume of the cone is a maximum.
A) radians
B) radians
C) radians
D) radians
E) radians
is cut from a circle of radius 4 inches,and the edges of the sector are brought together to form a cone.Find the magnitude of such that the volume of the cone is a maximum. A)
radiansB)
radiansC)
radiansD)
radiansE)
radians فتح الحزمة
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57
On a given day,the flow rate F (cars per hour)on a congested roadway is given by ,where v is the speed of the traffic in miles per hour.What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour.
A)19 miles per hour
B)17 miles per hour
C)18 miles per hour
D)10 miles per hour
E)20 miles per hour
,where v is the speed of the traffic in miles per hour.What speed will maximize the flow rate on the road? Round your answer to the nearest mile per hour. A)19 miles per hour
B)17 miles per hour
C)18 miles per hour
D)10 miles per hour
E)20 miles per hour
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58
A rectangle is bounded by the x- and y-axes and the graph of (see figure).What length and width should the rectangle have so that its area is a maximum?
A)
B)
C)
D)
E)
(see figure).What length and width should the rectangle have so that its area is a maximum? A)
B)
C)
D)
E)
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59
Assume that the amount of money deposited in a bank is proportional to the square of the interest rate the bank pays on this money.Furthermore,the bank can reinvest this money at .Find the interest rate the bank should pay to maximize profit.(Use the simple interest formula. )
A)
B)
C)
D)
E)
.Find the interest rate the bank should pay to maximize profit.(Use the simple interest formula. ) A)
B)
C)
D)
E)
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60
The graph of a function f is is shown below.
Sketch the graph of the derivative .
A)
B)
C)
D)
E)
Sketch the graph of the derivative
.
A)
B)
C)
D)
E)
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61
Find the points of inflection and discuss the concavity of the function on the interval .
A)no inflection points.concave up on
B)concave downward on ;concave upward on
;inflection point at
C)no inflection points.concave down on
D)concave upward on ;concave downward on
;inflection point at
E)none of the above
on the interval . A)no inflection points.concave up on
B)concave downward on
;concave upward on ;inflection point at
C)no inflection points.concave down on
D)concave upward on
;concave downward on ;inflection point atE)none of the above
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62
Sketch a graph of the function over the interval .
A)
B)
C)
D)
E)
over the interval . A)
B)
C)
D)
E)
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63
Find the point of inflection of the graph of the function on the interval .
A)
B)
C)
D)
E)
on the interval . A)
B)
C)
D)
E)
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64
Determine the open intervals on which the graph of is concave downward or concave upward.
A)concave downward on ;concave upward 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
;concave upward onB)concave downward on
;concave upward onC)concave upward on
;concave downward on
D)concave downward on
;concave upward onE)concave upward on
;concave downward on فتح الحزمة
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65
Sketch a graph of the function over the interval .
A)
B)
C)
D)
E)
over the interval . A)
B)
C)
D)
E)
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66
Find all points of inflection on the graph of the function .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
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67
The graph of f is shown below.For which values of x is zero?
A)
B)
C)
D)
E)
zero? A)
B)
C)
D)
E)
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68
Determine the open intervals on which the graph of the function is concave upward or concave downward.
A)concave upward: ;concave downward:
B)concave upward: ;concave downward:
C)concave upward: ;concave downward:
D)concave upward: ;concave downward:
E)concave upward: ;concave downward:
is concave upward or concave downward. A)concave upward:
;concave downward:B)concave upward:
;concave downward:C)concave upward:
;concave downward:D)concave upward:
;concave downward:E)concave upward:
;concave downward: فتح الحزمة
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69
Determine the open intervals on which the graph of is concave downward or concave upward.
A)concave downward on
B)concave upward on ;concave downward on
C)concave upward on ;concave downward on
D)concave downward on ;concave upward on
E)concave downward on ;concave upward on
is concave downward or concave upward. A)concave downward on
B)concave upward on
;concave downward onC)concave upward on
;concave downward onD)concave downward on
;concave upward onE)concave downward on
;concave upward on فتح الحزمة
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70
Analyze the graph of the function .Determine any intercepts,relative extrema,points of inflection and asymptotes.Also determine where the graph is increasing or decreasing and concave up or concave down.Then identify the graph from the choices below.
A)
B)
C)
D)
E)
.Determine any intercepts,relative extrema,points of inflection and asymptotes.Also determine where the graph is increasing or decreasing and concave up or concave down.Then identify the graph from the choices below. A)
B)
C)
D)
E)
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71
Determine the open intervals on which the graph of the function is concave upward or concave downward.
A)concave upward: ;concave downward:
B)concave upward: ;concave downward:
C)concave upward: ;concave downward:
D)concave upward: ;concave downward:
E)concave upward: ;concave downward:
is concave upward or concave downward. A)concave upward:
;concave downward:B)concave upward:
;concave downward:C)concave upward:
;concave downward:D)concave upward:
;concave downward:E)concave upward:
;concave downward: فتح الحزمة
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72
A valve on a storage tank is opened for 4 hours to release a chemical in a manufacturing process.The flow rate R (in liters per hour)at time t (in hours)is given by the linear model .Write the linear model in exponential form.Round to four decimal places.
A)
B)
C)
D)
E)
.Write the linear model in exponential form.Round to four decimal places. A)
B)
C)
D)
E)
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73
The graph of f is shown below.On what interval is an increasing function?
A)
B)
C)
D)
E)
an increasing function? A)
B)
C)
D)
E)
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74
Determine the open intervals on which the graph of is concave downward or concave upward.
A)concave upward on
B)concave downward on ;concave upward on
C)concave downward on ;concave upward on
D)concave downward on
E)concave upward on ;concave downward on
is concave downward or concave upward. A)concave upward on
B)concave downward on
;concave upward onC)concave downward on
;concave upward onD)concave downward on
E)concave upward on
;concave downward on فتح الحزمة
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75
Find all points of inflection,if any exist,of the graph of the function .Round your answers to two decimal places.
A)
B)
C)
D)
E)no points of inflection
.Round your answers to two decimal places. A)
B)
C)
D)
E)no points of inflection
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76
Analyze and sketch a graph of the function .
A)
B)
C)
D)
E)
. A)
B)
C)
D)
E)
فتح الحزمة
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77
Find the points of inflection and discuss the concavity of the function.
A)inflection point at ;concave upward on
;concave downward on
B)inflection point at ;concave downward on
;concave upward on
C)inflection point at ;concave downward on
;concave upward on
D)inflection point at ;concave downward on
;concave upward on
E)inflection point at ;concave upward on
;concave downward on
A)inflection point at
;concave upward on ;concave downward onB)inflection point at
;concave downward on ;concave upward onC)inflection point at
;concave downward on ;concave upward onD)inflection point at
;concave downward on ;concave upward onE)inflection point at
;concave upward on ;concave downward on فتح الحزمة
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78
The graph of f is shown below.For which value of x is zero?
A)
B)
C)
D)
E)
zero? A)
B)
C)
D)
E)
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79
Find the points of inflection and discuss the concavity of the function .
A)no inflection points;concave up on
B)no inflection points;concave down on
C)inflection point at ;concave up on
D)inflection point at ;concave up on
;concave down on
E)inflection point at ;concave down on
. A)no inflection points;concave up on
B)no inflection points;concave down on
C)inflection point at
;concave up onD)inflection point at
;concave up on ;concave down onE)inflection point at
;concave down on فتح الحزمة
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80
The graph of f is shown below.For which value of x is minimum?
A)
B)
C)
D)
E)
minimum? A)
B)
C)
D)
E)
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