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Deck 4: Applications of the Derivative
Full screen (f)
Question
Find the linearization of the given function centered at 
, and use it to estimate 
.
A)
B)
, and use it to estimate . A)
B)
Question
A ball produced on an assembly line is supposed to have a volume of 
. Use linearization to estimate the maximum allowable error in the radius 
if the volume of the sphere must have an error of less than 
.
. Use linearization to estimate the maximum allowable error in the radius if the volume of the sphere must have an error of less than . Question
By Rolle's Theorem the equation 
has at most two roots.
has at most two roots. Question
The radius 
of a sphere is measured at 4 m. Use linear approximation to estimate the maximum error in the surface area of the sphere if 
is accurate to within 
m.
of a sphere is measured at 4 m. Use linear approximation to estimate the maximum error in the surface area of the sphere if is accurate to within m. Question
Find the maximum value 
and the minimum value 
of the function in the given interval
A)
B)
and the minimum value of the function in the given interval A)
B)
Question
Find the maximum value 
and the minimum value 
of the function in the given interval
A)
B)
and the minimum value of the function in the given interval A)
B)
Question
Find the maximum value 
and the minimum value 
of the function in the given interval or in its natural domain
A)
B)
C)
and the minimum value of the function in the given interval or in its natural domain A)
B)
C)
Question
Estimate 
using linear approximation.
using linear approximation. Question
The minimum and maximum of which of the following functions does not occur at a critical point in the open interval:
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Let 
. By Rolle's Theorem we can prove that
A)
has 11 real roots.
B)
has at most 3 real roots.
C)
has no real roots.
D)
has at most 2 real roots.
E)
has 10 real roots.
. By Rolle's Theorem we can prove thatA)
has 11 real roots.B)
has at most 3 real roots.C)
has no real roots.D)
has at most 2 real roots.E)
has 10 real roots. Question
Find the maximum value 
and the minimum value 
of the function in the given interval.
A)
B)
and the minimum value of the function in the given interval. A)
B)
Question
Each of the following functions restricted to the given interval attains a minimum and maximum value. Which attain their maximum or minimum value at a point in the interior of the interval?
A)
on 
B)
on 
C)
on 
D)
on 
A)
on B)
on C)
on D)
on Question
Estimate the length of the hypotenuse in a right triangle with sides 3 and 4.01, using the linear approximation for 
.
. Question
Find the linearization of the given function centered at 
and use it to estimate 
.
A)
B)
and use it to estimate . A)
B)
Question
Estimate the roots of the equation 
to three decimal places using the linear approximation for 
.
to three decimal places using the linear approximation for . Question
Estimate the roots of the equation 
using the linear approximation for 
at 
.
using the linear approximation for at . Question
Estimate the roots of the equation 
using the linear approximation for 
.
using the linear approximation for . Question
Estimate the value of 
using the linear approximation and find the error using a calculator.
using the linear approximation and find the error using a calculator. Question
Suppose that a function 
satisfies the following equation for small values of 
: 
.
Also,
and 
.
A) Find the linearization of
at 
.
B) Replace
by its linearization and find a quadratic equation for 
.
C) Estimate the roots of the quadratic equation in
to 4 decimal digits using linearization for 
.
satisfies the following equation for small values of : .Also,
and . A) Find the linearization of
at . B) Replace
by its linearization and find a quadratic equation for . C) Estimate the roots of the quadratic equation in
to 4 decimal digits using linearization for . Question
Find all the critical points of the function
A)
B)
A)
B)
Question
The following function has a local extremum at a point in the interval 
:
A)
B)
C)
D)
E) None of the above.
:A)
B)
C)
D)
E) None of the above.
Question
Determine the greater between 
and 
without using a calculator, by considering the function 
for 
and without using a calculator, by considering the function for Question
The function 
has
A) local minimum at
, inflection at 
.
B) local minimum at
, local maximum at 
, inflection at 
.
C) local minimum at
, local maximum at 
, inflection at 
.
D) local minimum at
, local maximum at 
, inflection at 
.
E) None of the above.
hasA) local minimum at
, inflection at .B) local minimum at
, local maximum at , inflection at .C) local minimum at
, local maximum at , inflection at .D) local minimum at
, local maximum at , inflection at .E) None of the above.
Question
The increase/decrease intervals of the function 
are
A) increasing on
, 
and decreasing on 
.
B) increasing on
and decreasing on 
.
C) increasing on
and decreasing on 
.
D) increasing on
and decreasing on 
.
E) None of the above.
areA) increasing on
, and decreasing on .B) increasing on
and decreasing on .C) increasing on
and decreasing on .D) increasing on
and decreasing on .E) None of the above.
Question
Given 
,
Which of the following statements is correct?
A)
is increasing on 
and decreasing on 
.
B)
is increasing on 
.
C)
is decreasing on 
.
D)
is decreasing on 
and increasing on 
.
E)
is decreasing on 
and increasing on 
.
,Which of the following statements is correct?
A)
is increasing on and decreasing on .B)
is increasing on .C)
is decreasing on .D)
is decreasing on and increasing on .E)
is decreasing on and increasing on . Question
Given the function 
is
A) a local maximum point.
B) a local minimum point.
C) is not a point of local extremum nor an inflection point.
D) an inflection point.
E) not in the domain of the function.
isA) a local maximum point.
B) a local minimum point.
C) is not a point of local extremum nor an inflection point.
D) an inflection point.
E) not in the domain of the function.
Question
Find the maximum value 
and the minimum value 
of the function in the given interval.
A)
B)
and the minimum value of the function in the given interval. A)
B)
Question
Given 
such that 
and 
for all 
, is 
for all 
?
(Hint: Use the MVT.)
A) Yes
B) No
such that and for all , is for all ?(Hint: Use the MVT.)
A) Yes
B) No
Question
Find a point 
satisfying the conclusion of the Mean Value Theorem for the function 
in the interval 
.
satisfying the conclusion of the Mean Value Theorem for the function in the interval . Question
Find all the critical points of the following functions (if they exist)
A)
B)
C)
A)
B)
C)
Question
Let 
.
A) Show that there is no
such that 
.
B) Explain why the above does not contradict the MVT.
. A) Show that there is no
such that . B) Explain why the above does not contradict the MVT.
Question
The following function has a local minimum at a point on the interval 
:
A)
B)
C)
D)
: A)
B)
C)
D)
Question
Which of the following functions has a local maximum at a point in the interval 
:
A)
B)
C)
D)
: A)
B)
C)
D)
Question
Given 
such that 
and 
for all 
.
Use the Mean Value Theorem to find which of the following statements is correct
A)
B)
C)
D)
E) None of the above.
such that and for all .Use the Mean Value Theorem to find which of the following statements is correct
A)
B)
C)
D)
E) None of the above.
Question
Let 
for 
Based on the MVT, which of the following statements is correct?
A)
B)
C)
D)
E)
for Based on the MVT, which of the following statements is correct?A)
B)
C)
D)
E)
Question
True or False: Let 
. By Rolle's Theorem, 
has at most 3 real roots.
. By Rolle's Theorem, has at most 3 real roots. Question
Suppose that 
is differentiable for 
, 
is the tangent line to the graph of 
at 
and 
. The following can be concluded:
A)
is concave up on 
.
B)
is not concave down on 
.
C)
is concave down on 
.
D)
is not concave up on 
.
E) None of the above.
is differentiable for , is the tangent line to the graph of at and . The following can be concluded:A)
is concave up on .B)
is not concave down on .C)
is concave down on .D)
is not concave up on .E) None of the above.
Question
Find the intervals on which the function is concave up and concave down and indicate the points of inflection
A)
B)
A)
B)
Question
Let 
be the function on 
given by 
and the graph: 
The MVT can be applied on the following interval:
A)
B)
C)
D)
E) A and C are correct.
be the function on given by and the graph: The MVT can be applied on the following interval:A)
B)
C)
D)
E) A and C are correct.
Question
The function 
is
A) increasing on
and decreasing on 
.
B) increasing on
and decreasing on 
.
C) increasing on
and decreasing on 
.
D) increasing in all its domain.
E) decreasing in all its domain.
isA) increasing on
and decreasing on .B) increasing on
and decreasing on .C) increasing on
and decreasing on .D) increasing in all its domain.
E) decreasing in all its domain.
Question
The following table describes the signs of the first and second derivatives of a function 
: 
0
1
2
+
+
+
+
+
0
-
-
0
+
-
-
-
-
Which of the following is a possible graph of
A)
B)
C)
D)
E) None of the above.
: 01
2
++
+
+
+
0
-
-0
+
-
-
-
-
Which of the following is a possible graph of
A)
B)
C)
D)
E) None of the above.
Question
Sketch the graph of the following function. Indicate the asymptotes, local extrema, and points of inflection. 
.
. Question
Sketch the graph of the function 
and indicate the transition points and asymptotes.
and indicate the transition points and asymptotes. Question
The following is the graph of 
Where do the points of inflection of 
occur, and on which intervals is 
concave up? 
Where do the points of inflection of occur, and on which intervals is concave up? Question
Sketch the graph of the function. Indicate the asymptotes, local extrema, and points of inflection. 
Question
Sketch the graph of a function such that 
for 
for 
for 
for 
for for for for Question
Find the sides of the isosceles triangle with maximum area, inscribed in a circle of radius 
.
. Question
Find the dimensions and the perimeter of the rectangle with maximum perimeter, inscribed in the ellipse 
. 
. Question
Find the intervals on which the function is concave up and concave down and indicate the points of inflection
A)
B)
A)
B)
Question
Match the functions with their graphs
(a)
(b) 
(c) 
(d) 
(A) 
(B) 
(C) 
(D) 
(a)
(b) (c) (d) (A) (B) (C) (D) Question
A manufacturer sells instruments for 
per unit.
The cost of producing
instruments is 
, and no more than 
instruments can be produced in a week.
Find the amount of instruments that should be produced in a week to obtain maximum profit.
per unit.The cost of producing
instruments is , and no more than instruments can be produced in a week.Find the amount of instruments that should be produced in a week to obtain maximum profit.
Question
The following table describes the signs of the first and second derivatives of a function 
: 
0
1
2
0 
0 
0 
0 
If 
, sketch a possible graph of 
: 01
2
0 0 0 0 If , sketch a possible graph of Question
Find the angles of the isosceles triangle of minimum area in which a circle of radius 
is inscribed.
What are the sides of this triangle?
(You don't have to prove that the area is a minimum.)
is inscribed.What are the sides of this triangle?
(You don't have to prove that the area is a minimum.)
Question
For the function 
, find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph using this information.
, find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph using this information. Question
The function 
has inflection points at 
and 
. Find 
and 
.
has inflection points at and . Find and . Question
The tangent line to 
at a point 
intersects the 
and 
axes at points 
and 
, respectively. Find 
such that the area of the triangle 
is maximum. 
at a point intersects the and axes at points and , respectively. Find such that the area of the triangle is maximum. Question
What can you conclude about the graph of 
from the following table?
-1
-6
+
0
-
-
-
+
0
-
0
-
A)
is concave down on 
and 
B)
is an inflection point and 
C)
is an inflection point and 
D)
is concave up on 
and 
is increasing on 
from the following table?-1
-6
+0
-
-
-
+0
-
0
-
A)
is concave down on and B)
is an inflection point and C)
is an inflection point and D)
is concave up on and is increasing on Question
Sketch the graph of the function 
.
Indicate the asymptotes, local extrema, and points of inflection if they exist.
.Indicate the asymptotes, local extrema, and points of inflection if they exist.
Question
Which of the following functions does not have horizontal asymptotes?
A)
B)
C)
D)
E) None of the above.
A)
B)
C)
D)
E) None of the above.
Question
The following is the graph of 
Where do the points of inflection of 
occur, and on which intervals is 
concave up? 
Where do the points of inflection of occur, and on which intervals is concave up? Question
Use Newton's Method with 
to find an approximate root for 
and use it to approximate 
to within an error of at most 
.
to find an approximate root for and use it to approximate to within an error of at most . Question
A factory produces 2000 products each month. The expenses on each product are 
and the income from each product is 
. For each additional product beyond the first 2000 products, the income for each product reduces by 
(for example, for 2001 products the income from each product is 
).
A) Find the profit in a month for which
additional product are produced.
B) How many additional products should be produced to obtain maximum profit in a month?
and the income from each product is . For each additional product beyond the first 2000 products, the income for each product reduces by (for example, for 2001 products the income from each product is ). A) Find the profit in a month for which
additional product are produced. B) How many additional products should be produced to obtain maximum profit in a month?
Question
Use the given graph of 
and Newton's Method to approximate the positive root to within an error of at most 
. 
and Newton's Method to approximate the positive root to within an error of at most . Question
Apply Newton's Method to 
with an initial guess of 
to calculate 
, 
, and 
.
with an initial guess of to calculate , , and . Question
Show that the equation 
has a root 
in the interval 
and use Newton's Method to approximate it to four decimal places.
has a root in the interval and use Newton's Method to approximate it to four decimal places. Question
Find the point on the graph of 
that is closest to the point 
.
that is closest to the point . Question
Show that the equation 
has a root in the interval 
and use Newton's Method to approximate it with an error of at most 
.
has a root in the interval and use Newton's Method to approximate it with an error of at most . Question
A producer can sell 
instruments per week for 
dollars each, where 
. His cost for producing 
instruments is 
dollars.
Find the amount of instruments that should be produced in a week to obtain maximum profit.
instruments per week for dollars each, where . His cost for producing instruments is dollars.Find the amount of instruments that should be produced in a week to obtain maximum profit.
Question
Show that the equation 
has a solution in the interval 
and use Newton's Method to approximate it within an error of at most 
.
has a solution in the interval and use Newton's Method to approximate it within an error of at most . Question
Use Newton's Method to approximate the only positive root for 
to within an error of at most 
to within an error of at most Question
Show that the equation 
has a solution in the interval 
and use Newton's Method to approximate it to within an error of at most 
.
has a solution in the interval and use Newton's Method to approximate it to within an error of at most . Question
Show that the equation 
has a solution 
in the interval 
and approximate it by Newton's Method to seven decimal places.
has a solution in the interval and approximate it by Newton's Method to seven decimal places. Question
Question
A cylinder of radius 
and height 
has surface area 
and volume 
. Find the dimensions of a cylinder with volume 
and minimal surface area.
and height has surface area and volume . Find the dimensions of a cylinder with volume and minimal surface area. Question
Find the dimensions and the area of the rectangle with maximum area inscribed in the ellipse 
. 
. Question
A right circular cylinder is to be constructed with a volume of 400 
. The material used to build the top and base of the cylinder costs 
. The material for the remainder of the cylinder costs 
. Find the radius value which will minimize the material cost of constructing the cylinder.
. The material used to build the top and base of the cylinder costs . The material for the remainder of the cylinder costs . Find the radius value which will minimize the material cost of constructing the cylinder. Question
Apply Newton's Method to 
with an initial guess of 
to calculate 
, 
, and 
.
with an initial guess of to calculate , , and .
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Full screen (f)
Deck 4: Applications of the Derivative
1
Find the linearization of the given function centered at , and use it to estimate .
A)
B)
, and use it to estimate . A)
B)
A) 
B) 
B) 2
A ball produced on an assembly line is supposed to have a volume of . Use linearization to estimate the maximum allowable error in the radius if the volume of the sphere must have an error of less than .
. Use linearization to estimate the maximum allowable error in the radius if the volume of the sphere must have an error of less than .Radius error must be less than 
inch.
inch. 3
By Rolle's Theorem the equation has at most two roots.
has at most two roots.True
4
The radius of a sphere is measured at 4 m. Use linear approximation to estimate the maximum error in the surface area of the sphere if is accurate to within m.
of a sphere is measured at 4 m. Use linear approximation to estimate the maximum error in the surface area of the sphere if is accurate to within m. Unlock Deck
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5
Find the maximum value and the minimum value of the function in the given interval
A)
B)
and the minimum value of the function in the given interval A)
B)
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6
Find the maximum value and the minimum value of the function in the given interval
A)
B)
and the minimum value of the function in the given interval A)
B)
Unlock Deck
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7
Find the maximum value and the minimum value of the function in the given interval or in its natural domain
A)
B)
C)
and the minimum value of the function in the given interval or in its natural domain A)
B)
C)
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8
Estimate using linear approximation.
using linear approximation. Unlock Deck
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9
The minimum and maximum of which of the following functions does not occur at a critical point in the open interval:
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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10
Let . By Rolle's Theorem we can prove that
A) has 11 real roots.
B) has at most 3 real roots.
C) has no real roots.
D) has at most 2 real roots.
E) has 10 real roots.
. By Rolle's Theorem we can prove thatA)
has 11 real roots.B)
has at most 3 real roots.C)
has no real roots.D)
has at most 2 real roots.E)
has 10 real roots. Unlock Deck
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11
Find the maximum value and the minimum value of the function in the given interval.
A)
B)
and the minimum value of the function in the given interval. A)
B)
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12
Each of the following functions restricted to the given interval attains a minimum and maximum value. Which attain their maximum or minimum value at a point in the interior of the interval?
A) on
B) on
C) on
D) on
A)
on B)
on C)
on D)
on Unlock Deck
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13
Estimate the length of the hypotenuse in a right triangle with sides 3 and 4.01, using the linear approximation for .
. Unlock Deck
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14
Find the linearization of the given function centered at and use it to estimate .
A)
B)
and use it to estimate . A)
B)
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15
Estimate the roots of the equation to three decimal places using the linear approximation for .
to three decimal places using the linear approximation for . Unlock Deck
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16
Estimate the roots of the equation using the linear approximation for at .
using the linear approximation for at . Unlock Deck
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17
Estimate the roots of the equation using the linear approximation for .
using the linear approximation for . Unlock Deck
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18
Estimate the value of using the linear approximation and find the error using a calculator.
using the linear approximation and find the error using a calculator. Unlock Deck
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19
Suppose that a function satisfies the following equation for small values of : .
Also, and .
A) Find the linearization of at .
B) Replace by its linearization and find a quadratic equation for .
C) Estimate the roots of the quadratic equation in to 4 decimal digits using linearization for .
satisfies the following equation for small values of : .Also,
and . A) Find the linearization of
at . B) Replace
by its linearization and find a quadratic equation for . C) Estimate the roots of the quadratic equation in
to 4 decimal digits using linearization for . Unlock Deck
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20
Find all the critical points of the function
A)
B)
A)
B)
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21
The following function has a local extremum at a point in the interval :
A)
B)
C)
D)
E) None of the above.
:A)
B)
C)
D)
E) None of the above.
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22
Determine the greater between and without using a calculator, by considering the function for
and without using a calculator, by considering the function for Unlock Deck
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23
The function has
A) local minimum at , inflection at .
B) local minimum at , local maximum at , inflection at .
C) local minimum at , local maximum at , inflection at .
D) local minimum at , local maximum at , inflection at .
E) None of the above.
hasA) local minimum at
, inflection at .B) local minimum at
, local maximum at , inflection at .C) local minimum at
, local maximum at , inflection at .D) local minimum at
, local maximum at , inflection at .E) None of the above.
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24
The increase/decrease intervals of the function are
A) increasing on , and decreasing on .
B) increasing on and decreasing on .
C) increasing on and decreasing on .
D) increasing on and decreasing on .
E) None of the above.
areA) increasing on
, and decreasing on .B) increasing on
and decreasing on .C) increasing on
and decreasing on .D) increasing on
and decreasing on .E) None of the above.
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25
Given ,
Which of the following statements is correct?
A) is increasing on and decreasing on .
B) is increasing on .
C) is decreasing on .
D) is decreasing on and increasing on .
E) is decreasing on and increasing on .
,Which of the following statements is correct?
A)
is increasing on and decreasing on .B)
is increasing on .C)
is decreasing on .D)
is decreasing on and increasing on .E)
is decreasing on and increasing on . Unlock Deck
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26
Given the function is
A) a local maximum point.
B) a local minimum point.
C) is not a point of local extremum nor an inflection point.
D) an inflection point.
E) not in the domain of the function.
isA) a local maximum point.
B) a local minimum point.
C) is not a point of local extremum nor an inflection point.
D) an inflection point.
E) not in the domain of the function.
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27
Find the maximum value and the minimum value of the function in the given interval.
A)
B)
and the minimum value of the function in the given interval. A)
B)
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28
Given such that and for all , is for all ?
(Hint: Use the MVT.)
A) Yes
B) No
such that and for all , is for all ?(Hint: Use the MVT.)
A) Yes
B) No
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29
Find a point satisfying the conclusion of the Mean Value Theorem for the function in the interval .
satisfying the conclusion of the Mean Value Theorem for the function in the interval . Unlock Deck
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30
Find all the critical points of the following functions (if they exist)
A)
B)
C)
A)
B)
C)
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31
Let .
A) Show that there is no such that .
B) Explain why the above does not contradict the MVT.
. A) Show that there is no
such that . B) Explain why the above does not contradict the MVT.
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32
The following function has a local minimum at a point on the interval :
A)
B)
C)
D)
: A)
B)
C)
D)
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33
Which of the following functions has a local maximum at a point in the interval :
A)
B)
C)
D)
: A)
B)
C)
D)
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34
Given such that and for all .
Use the Mean Value Theorem to find which of the following statements is correct
A)
B)
C)
D)
E) None of the above.
such that and for all .Use the Mean Value Theorem to find which of the following statements is correct
A)
B)
C)
D)
E) None of the above.
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35
Let for Based on the MVT, which of the following statements is correct?
A)
B)
C)
D)
E)
for Based on the MVT, which of the following statements is correct?A)
B)
C)
D)
E)
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36
True or False: Let . By Rolle's Theorem, has at most 3 real roots.
. By Rolle's Theorem, has at most 3 real roots. Unlock Deck
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37
Suppose that is differentiable for , is the tangent line to the graph of at and . The following can be concluded:
A) is concave up on .
B) is not concave down on .
C) is concave down on .
D) is not concave up on .
E) None of the above.
is differentiable for , is the tangent line to the graph of at and . The following can be concluded:A)
is concave up on .B)
is not concave down on .C)
is concave down on .D)
is not concave up on .E) None of the above.
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38
Find the intervals on which the function is concave up and concave down and indicate the points of inflection
A)
B)
A)
B)
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39
Let be the function on given by and the graph: The MVT can be applied on the following interval:
A)
B)
C)
D)
E) A and C are correct.
be the function on given by and the graph: The MVT can be applied on the following interval:A)
B)
C)
D)
E) A and C are correct.
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40
The function is
A) increasing on and decreasing on .
B) increasing on and decreasing on .
C) increasing on and decreasing on .
D) increasing in all its domain.
E) decreasing in all its domain.
isA) increasing on
and decreasing on .B) increasing on
and decreasing on .C) increasing on
and decreasing on .D) increasing in all its domain.
E) decreasing in all its domain.
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41
The following table describes the signs of the first and second derivatives of a function : 0
1
2 +
+
+
+
+
0
- -
0
+
-
-
-
-
Which of the following is a possible graph of
A)
B)
C)
D)
E) None of the above.
: 01
2
++
+
+
+
0
-
-0
+
-
-
-
-
Which of the following is a possible graph of
A)
B)
C)
D)
E) None of the above.
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42
Sketch the graph of the following function. Indicate the asymptotes, local extrema, and points of inflection. .
. Unlock Deck
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43
Sketch the graph of the function and indicate the transition points and asymptotes.
and indicate the transition points and asymptotes. Unlock Deck
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44
The following is the graph of Where do the points of inflection of occur, and on which intervals is concave up?
Where do the points of inflection of occur, and on which intervals is concave up? Unlock Deck
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45
Sketch the graph of the function. Indicate the asymptotes, local extrema, and points of inflection.
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46
Sketch the graph of a function such that for for for for
for for for for Unlock Deck
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47
Find the sides of the isosceles triangle with maximum area, inscribed in a circle of radius .
. Unlock Deck
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48
Find the dimensions and the perimeter of the rectangle with maximum perimeter, inscribed in the ellipse .
. Unlock Deck
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49
Find the intervals on which the function is concave up and concave down and indicate the points of inflection
A)
B)
A)
B)
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50
Match the functions with their graphs
(a) (b) (c) (d) (A) (B) (C) (D)
(a)
(b) (c) (d) (A) (B) (C) (D) Unlock Deck
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51
A manufacturer sells instruments for per unit.
The cost of producing instruments is , and no more than instruments can be produced in a week.
Find the amount of instruments that should be produced in a week to obtain maximum profit.
per unit.The cost of producing
instruments is , and no more than instruments can be produced in a week.Find the amount of instruments that should be produced in a week to obtain maximum profit.
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Unlock for access to all 77 flashcards in this deck.
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k this deck
52
The following table describes the signs of the first and second derivatives of a function : 0
1
2 0 0 0 0 If , sketch a possible graph of
: 01
2
0 0 0 0 If , sketch a possible graph of Unlock Deck
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53
Find the angles of the isosceles triangle of minimum area in which a circle of radius is inscribed.
What are the sides of this triangle?
(You don't have to prove that the area is a minimum.)
is inscribed.What are the sides of this triangle?
(You don't have to prove that the area is a minimum.)
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54
For the function , find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph using this information.
, find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph using this information. Unlock Deck
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55
The function has inflection points at and . Find and .
has inflection points at and . Find and . Unlock Deck
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56
The tangent line to at a point intersects the and axes at points and , respectively. Find such that the area of the triangle is maximum.
at a point intersects the and axes at points and , respectively. Find such that the area of the triangle is maximum. Unlock Deck
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57
What can you conclude about the graph of from the following table?
-1
-6 +
0
-
-
- +
0
-
0
-
A) is concave down on and
B) is an inflection point and
C) is an inflection point and
D) is concave up on and is increasing on
from the following table?-1
-6
+0
-
-
-
+0
-
0
-
A)
is concave down on and B)
is an inflection point and C)
is an inflection point and D)
is concave up on and is increasing on Unlock Deck
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58
Sketch the graph of the function .
Indicate the asymptotes, local extrema, and points of inflection if they exist.
.Indicate the asymptotes, local extrema, and points of inflection if they exist.
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k this deck
59
Which of the following functions does not have horizontal asymptotes?
A)
B)
C)
D)
E) None of the above.
A)
B)
C)
D)
E) None of the above.
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60
The following is the graph of Where do the points of inflection of occur, and on which intervals is concave up?
Where do the points of inflection of occur, and on which intervals is concave up? Unlock Deck
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61
Use Newton's Method with to find an approximate root for and use it to approximate to within an error of at most .
to find an approximate root for and use it to approximate to within an error of at most . Unlock Deck
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62
A factory produces 2000 products each month. The expenses on each product are and the income from each product is . For each additional product beyond the first 2000 products, the income for each product reduces by (for example, for 2001 products the income from each product is ).
A) Find the profit in a month for which additional product are produced.
B) How many additional products should be produced to obtain maximum profit in a month?
and the income from each product is . For each additional product beyond the first 2000 products, the income for each product reduces by (for example, for 2001 products the income from each product is ). A) Find the profit in a month for which
additional product are produced. B) How many additional products should be produced to obtain maximum profit in a month?
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63
Use the given graph of and Newton's Method to approximate the positive root to within an error of at most .
and Newton's Method to approximate the positive root to within an error of at most . Unlock Deck
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64
Apply Newton's Method to with an initial guess of to calculate , , and .
with an initial guess of to calculate , , and . Unlock Deck
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65
Show that the equation has a root in the interval and use Newton's Method to approximate it to four decimal places.
has a root in the interval and use Newton's Method to approximate it to four decimal places. Unlock Deck
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66
Find the point on the graph of that is closest to the point .
that is closest to the point . Unlock Deck
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67
Show that the equation has a root in the interval and use Newton's Method to approximate it with an error of at most .
has a root in the interval and use Newton's Method to approximate it with an error of at most . Unlock Deck
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68
A producer can sell instruments per week for dollars each, where . His cost for producing instruments is dollars.
Find the amount of instruments that should be produced in a week to obtain maximum profit.
instruments per week for dollars each, where . His cost for producing instruments is dollars.Find the amount of instruments that should be produced in a week to obtain maximum profit.
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k this deck
69
Show that the equation has a solution in the interval and use Newton's Method to approximate it within an error of at most .
has a solution in the interval and use Newton's Method to approximate it within an error of at most . Unlock Deck
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70
Use Newton's Method to approximate the only positive root for to within an error of at most
to within an error of at most Unlock Deck
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71
Show that the equation has a solution in the interval and use Newton's Method to approximate it to within an error of at most .
has a solution in the interval and use Newton's Method to approximate it to within an error of at most . Unlock Deck
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72
Show that the equation has a solution in the interval and approximate it by Newton's Method to seven decimal places.
has a solution in the interval and approximate it by Newton's Method to seven decimal places. Unlock Deck
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73
A farmer wants to fence in a rectangular field for livestock. The boundary for one
side of the field is a long, straight river. No fencing is needed on this side. For the
remaining three sides, he has 400 meters of fencing available. What are the dimensions of the largest rectangular area that can be formed with this amount of fencing?
side of the field is a long, straight river. No fencing is needed on this side. For the
remaining three sides, he has 400 meters of fencing available. What are the dimensions of the largest rectangular area that can be formed with this amount of fencing?
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74
A cylinder of radius and height has surface area and volume . Find the dimensions of a cylinder with volume and minimal surface area.
and height has surface area and volume . Find the dimensions of a cylinder with volume and minimal surface area. Unlock Deck
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75
Find the dimensions and the area of the rectangle with maximum area inscribed in the ellipse .
. Unlock Deck
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76
A right circular cylinder is to be constructed with a volume of 400 . The material used to build the top and base of the cylinder costs . The material for the remainder of the cylinder costs . Find the radius value which will minimize the material cost of constructing the cylinder.
. The material used to build the top and base of the cylinder costs . The material for the remainder of the cylinder costs . Find the radius value which will minimize the material cost of constructing the cylinder. Unlock Deck
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77
Apply Newton's Method to with an initial guess of to calculate , , and .
with an initial guess of to calculate , , and . Unlock Deck
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Unlock Deck
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