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Deck 11: Partial Derivatives
Full screen (f)
Question
Find the maximum value of the function 
subject to the constraint that 
.
A)
e.2
B)
f. 
c.1
G)4
D)
h. 
subject to the constraint that .A)
e.2B)
f. c.1G)4
D)
h. 
Question
Find the point at which the function 
has the maximum value subject to the constraint that 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h.None of these
has the maximum value subject to the constraint that .A)
e. b. f. c. g. d. h.None of these Question
Find the maximum value of the function 
subject to the constraint that 
.
A)
e. 
b.1
F)
c. 
g. 
d.2
H)
subject to the constraint that .A)
e. b.1F)
c. g. d.2H)
Question
Optimize 
subject to 
.
subject to . Question
Use the method of Lagrange multipliers to find points on the surface of 
where the function 
has
(a) a minimum
(b) a maximum.
where the function has(a) a minimum
(b) a maximum.
Question
Question
Question
Compute the minimum value of 
subject to the condition that 
.
subject to the condition that . Question
What is the shortest distance from the origin to the surface 
?
? Question
Find the point at which the function 
has the minimum value subject to the constraint that 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h.None of these
has the minimum value subject to the constraint that .A)
e. b. f. c. g. d. h.None of these Question
In using Lagrange multipliers to minimize the function 
subject to the constraint that 
, what is the value of the multiplier 
?
A)
e. 
b.1
F)3
C)
g. 
d.2
H)4
subject to the constraint that , what is the value of the multiplier ?A)
e. b.1F)3
C)
g. d.2H)4
Question
Question
Find the extreme values of 
on the circle 
.
on the circle . Question
Find the maximum value of the function 
subject to the constraint that 
.
A)
e.4
B)
f.5
C)
g.6
D)
h.7
subject to the constraint that .A)
e.4B)
f.5C)
g.6D)
h.7 Question
Find the point on the plane 
where 
is minimum.
where is minimum. Question
In using Lagrange multipliers to minimize the function 
subject to the constraint that 
, what is the value of the multiplier 
?
A)
e. 
b.1
F)3
C)
g. 
d.2
H)4
subject to the constraint that , what is the value of the multiplier ?A)
e. b.1F)3
C)
g. d.2H)4
Question
Find the minimum value of the function 
subject to the constraint that 
.
A)
e. 
b.1
F)3
C)
g. 
d.2
H)4
subject to the constraint that .A)
e. b.1F)3
C)
g. d.2H)4
Question
Find the maximum and minimum values of the function 
on the ellipse given by the equation 
.
on the ellipse given by the equation . Question
Find the minimum value of the function 
subject to the constraint that 
.
A)
e.4
B)
f.5
C)
g.6
D)
h.7
subject to the constraint that .A)
e.4B)
f.5C)
g.6D)
h.7 Question
Find the minimum value of the function 
subject to the constraint that 
.
A)
e. 
b.1
F)
c. 
g. 
d.2
H)
subject to the constraint that .A)
e. b.1F)
c. g. d.2H)
Question
The quality 
of a good produced by a company is given by 
, where 
is the quantity of capital and 
is the quantity of labor used. Capital costs are $20 per unit, labor costs are $10 per unit, and the company wants to keep costs for capital and labor combined to $150.(a) What combination of labor and capital should be used to produce maximum quantity? What is the maximum value?
(b) Draw the level curves of
and the graph of the budget constraint on the same set of axes.(c) Complete the value of 
. What does 
represent?
of a good produced by a company is given by , where is the quantity of capital and is the quantity of labor used. Capital costs are $20 per unit, labor costs are $10 per unit, and the company wants to keep costs for capital and labor combined to $150.(a) What combination of labor and capital should be used to produce maximum quantity? What is the maximum value?(b) Draw the level curves of
and the graph of the budget constraint on the same set of axes.(c) Complete the value of . What does represent? Question
Determine how many critical points the function 
has.
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
has.A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
Question
The function 
has one critical point. Determine its location and type.
A)
, saddle point
E)
, minimum point
B)
, minimum point
F)
, maximum point
C)
, maximum point
G)
, saddle point
D)
, saddle point
H)
, minimum point
has one critical point. Determine its location and type.A)
, saddle pointE)
, minimum pointB)
, minimum pointF)
, maximum pointC)
, maximum pointG)
, saddle pointD)
, saddle pointH)
, minimum point Question
A rancher with 300 ft of fence intends to enclose a rectangular corral, dividing it in half by a fence parallel to the short sides of the corral. What is the maximum area he can enclose? Compute the value of 
. What does 
represent?
. What does represent? Question
Find the maximum value of the function 
a.90
E)80
B)160
F)135
C)50
G)16
D)15
H)5
a.90E)80
B)160
F)135
C)50
G)16
D)15
H)5
Question
The function 
has one critical point. Determine its location and type.
A)
, saddle point
E)
, minimum point
B)
, minimum point
F)
, maximum point
C)
, maximum point
G)
, saddle point
D)
, saddle point
H)
, minimum point
has one critical point. Determine its location and type.A)
, saddle pointE)
, minimum pointB)
, minimum pointF)
, maximum pointC)
, maximum pointG)
, saddle pointD)
, saddle pointH)
, minimum point Question
The function 
has one critical point. Determine its location and type.
A)
, saddle point
E)
, minimum point
B)
, minimum point
F)
, maximum point
C)
, maximum point
G)
, saddle point
D)
, saddle point
H)
, minimum point
has one critical point. Determine its location and type.A)
, saddle pointE)
, minimum pointB)
, minimum pointF)
, maximum pointC)
, maximum pointG)
, saddle pointD)
, saddle pointH)
, minimum point Question
The level curves of a function 
and a curve with equation 
( 
constant) are given below. Estimate the point where 
has a maximum value and the point where 
has a minimum value, subject to the constraint that 
. Indicate your answer in the figure. 
and a curve with equation ( constant) are given below. Estimate the point where has a maximum value and the point where has a minimum value, subject to the constraint that . Indicate your answer in the figure. Question
Suppose that the quantity 
produced of a certain good depends on the number of units of labor 
and the quantity of capital 
according to the function 
. Suppose also that labor costs $100 per unit and that capital costs $200 per unit.(a) What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimum cost?
(b) Draw the level curves of
, the cost function, and the graph of the constraint on the same set of axes.(c) Complete the value of 
. What does 
represent?
produced of a certain good depends on the number of units of labor and the quantity of capital according to the function . Suppose also that labor costs $100 per unit and that capital costs $200 per unit.(a) What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimum cost?(b) Draw the level curves of
, the cost function, and the graph of the constraint on the same set of axes.(c) Complete the value of . What does represent? Question
What point on the surface 
, 
, 
, 
is closest to the origin?
, , , is closest to the origin? Question
Determine how many critical points the function 
has.
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
has.A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
Question
Question
Find the minimum value of the function 
a.8.5
E)10.5
B)18
F)9
C)19
G)9.5
D)10
H)50
a.8.5E)10.5
B)18
F)9
C)19
G)9.5
D)10
H)50
Question
A rectangular area of 3200 ft 
is to be fenced off. Two opposite sides will use fencing costing $1 per foot and the remaining sides will use fencing costing $2 per foot. Find the dimensions of the rectangle of least cost. Compute the value of 
. What does 
represent?
is to be fenced off. Two opposite sides will use fencing costing $1 per foot and the remaining sides will use fencing costing $2 per foot. Find the dimensions of the rectangle of least cost. Compute the value of . What does represent? Question
The function 
has one critical point. Determine its location and type.
A)
, saddle point
E)
, minimum point
B)
, minimum point
F)
, maximum point
C)
, maximum point
G)
, saddle point
D)
, saddle point
H)
, minimum point
has one critical point. Determine its location and type.A)
, saddle pointE)
, minimum pointB)
, minimum pointF)
, maximum pointC)
, maximum pointG)
, saddle pointD)
, saddle pointH)
, minimum point Question
A rancher intends to fence off a rectangular region along a river (which serves as a natural boundary requiring no fence). If the enclosed area is to be 1800 square yards, what is the least amount of fence needed? Compute the value of 
. What does 
represent?
. What does represent? Question
Determine how many critical points the function 
has.
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
has.A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
Question
Determine how many critical points the function 
has.
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
has.A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
Question
Find the maximum and minimum values of 
on the circle 
.
on the circle . Question
Determine how many critical points the function 
has.
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
has.A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
Question
Find the local maximum and minimum values and saddle points of the function 
.
. Question
Find the local maximum and minimum values and saddle points of the function 
.
. Question
Find the point at which the function 
has a local maximum.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
has a local maximum.A)
e. b. f. c. g. d. h. Question
For each of the following functions, find the critical point, if there is one, and determine if it is a local maximum, local minimum, saddle point, or otherwise.(a) 
(b) 
(c) 
(d) 
(b) (c) (d) Question
Use the level curves of 
shown below to estimate the critical points of 
. Indicate whether 
has a saddle point or a local maximum or minimum at each of those points. 
shown below to estimate the critical points of . Indicate whether has a saddle point or a local maximum or minimum at each of those points. Question
A cardboard box without a lid is to have volume 100 cubic inches, with total area of cardboard as small as possible. Find its height in inches.
A)2
E)
b. 
f. 
c.4
G)
d.5
H)
A)2
E)
b. f. c.4G)
d.5H)
Question
Find the local maximum and minimum values and saddle points of the function 
.
. Question
Find the local maximum and minimum values and saddle points of the function 
.
. Question
The function 
has a maximum. Find the values of 
and 
at which it occurs.
has a maximum. Find the values of and at which it occurs. Question
Use the level curves of 
shown below to estimate the critical points of 
. Indicate whether 
has a saddle point or a local maximum or minimum at each of those points. 
shown below to estimate the critical points of . Indicate whether has a saddle point or a local maximum or minimum at each of those points. Question
Find the local maximum and minimum values and saddle points of the function 
.
. Question
Compare the minimum value of 
and sketch a portion of the graph of 
near its lowest point. 
and sketch a portion of the graph of near its lowest point. Question
Find the absolute maximum and minimum values of 
over the region 
, where 
is a closed triangular region with vertices 
, 
, and 
.
over the region , where is a closed triangular region with vertices , , and . Question
Find three positive numbers 
, 
, and 
whose sum is 48 and product is maximum.
, , and whose sum is 48 and product is maximum. Question
Find the shortest distance from the origin to the surface 
.
A)
e.2
B)
f. 
c.1
G)4
D)
h. 
.A)
e.2B)
f. c.1G)4
D)
h. Question
Find the critical points (if any) for 
and determine if each is a local extreme value or a saddle point.
and determine if each is a local extreme value or a saddle point. Question
Find the local maximum and minimum values and saddle points of the function 
.
. Question
Find the absolute maximum and minimum value of 
on the square region 
with vertices 
, 
, 
, and 
.
on the square region with vertices , , , and . Question
Find the absolute maximum and minimum value of 
on the square region 
with vertices 
, 
, 
, and 
.
on the square region with vertices , , , and . Question
A company estimates that its annual profits for next year will be 
, where 
represents investment in research and 
represents investment in labor. All units are in tens of thousands of dollars. Determine the amount the company should spend on research and on labor to maximize its profit. What is maximum profit?
, where represents investment in research and represents investment in labor. All units are in tens of thousands of dollars. Determine the amount the company should spend on research and on labor to maximize its profit. What is maximum profit? Question
Find an equation of the tangent plane to the surface 
at the point 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
at the point .A)
e. b. f. c. g. d. h. Question
Let 
. Find the gradient vector 
at the point 
.
A)i
E)
b.j
F)
c.k
G)
d. 
h.0
. Find the gradient vector at the point .A)i
E)
b.jF)
c.kG)
d. h.0 Question
Find the directional derivative of the function 
at the point 
in the direction from 
toward the point 
..
A)
e.7
B)
f. 
c. 
g. 
d. 
h.8
at the point in the direction from toward the point ..A)
e.7B)
f. c. g. d. h.8 Question
Find the largest value of the directional derivative of the function 
at the point 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
at the point .A)
e. b. f. c. g. d. h. Question
Let 
. Find the gradient vector 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
. Find the gradient vector .A)
e. b. f. c. g. d. h. Question
Find a normal vector to the surface 
at the point 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
at the point .A)
e. b. f. c. g. d. h. Question
Let 
. Find the gradient vector 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
. Find the gradient vector .A)
e. b. f. c. g. d. h. Question
Find the directional derivative of the function 
at the point 
in the direction 
.
A)
e.1
B)0
F)
c. 
g. 
d.12
H)
at the point in the direction .A)
e.1B)0
F)
c. g. d.12H)
Question
Find the direction of maximum increase of the function 
at the point 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
at the point .A)
e. b. f. c. g. d. h. Question
Find the second directional derivative of 
at the point 
in the direction 
.
A)
e. 
b. 
f. 
c.2
G)
d.6
H)
at the point in the direction .A)
e. b. f. c.2G)
d.6H)
Question
Find three positive numbers 
, 
, and 
whose product is 343 and sum is minimum.
, , and whose product is 343 and sum is minimum. Question
Let 
. Find the gradient vector 
at the point 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
. Find the gradient vector at the point .A)
e. b. f. c. g. d. h. Question
Find an equation of the tangent plane to the hyperboloid 
at the point 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
at the point .A)
e. b. f. c. g. d. h. Question
Find the directional derivative of the function 
at the point 
in the direction 
.
A)
e.1
B)
f.2
C)
g.4
D)
h. 
at the point in the direction .A)
e.1B)
f.2C)
g.4D)
h. Question
Find the second directional derivative of 
at the point 
in the direction 
.
A)
e. 
b. 
f. 
c. 
g. 
d.14
H)
at the point in the direction .A)
e. b. f. c. g. d.14H)
Question
Find the direction 
in which the directional derivative of the function 
at the point 
is maximum.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
in which the directional derivative of the function at the point is maximum.A)
e. b. f. c. g. d. h. Question
Find the direction of maximum increase of the function 
at the point 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
at the point .A)
e. b. f. c. g. d. h. Question
Find an equation of the tangent plane to the surface 
at the point 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
at the point .A)
e. b. f. c. g. d. h. Question
Find the directional derivative of the function 
at the point 
in the direction of the vector 
.
A)
e. 
b. 
f. 
c. 
g. 
d.0
H)
at the point in the direction of the vector .A)
e. b. f. c. g. d.0H)
Question
Find the directional derivative of the function 
at the point 
in the direction 
.
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
at the point in the direction .A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
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Full screen (f)
Deck 11: Partial Derivatives
1
Find the maximum value of the function subject to the constraint that .
A) e.2
B) f. c.1
G)4
D) h.
subject to the constraint that .A)
e.2B)
f. c.1G)4
D)
h. C
2
Find the point at which the function has the maximum value subject to the constraint that .
A) e. b. f. c. g. d. h.None of these
has the maximum value subject to the constraint that .A)
e. b. f. c. g. d. h.None of theseB
3
Find the maximum value of the function subject to the constraint that .
A) e. b.1
F) c. g. d.2
H)
subject to the constraint that .A)
e. b.1F)
c. g. d.2H)
B
4
Optimize subject to .
subject to . Unlock Deck
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5
Use the method of Lagrange multipliers to find points on the surface of where the function has
(a) a minimum
(b) a maximum.
where the function has(a) a minimum
(b) a maximum.
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6
Find two positive numbers who sum is eighteen and whose product is a maximum, using the method of Lagrange multipliers
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7
Find the greatest product three numbers can have if the sum of their squares must be 48.
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8
Compute the minimum value of subject to the condition that .
subject to the condition that . Unlock Deck
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9
What is the shortest distance from the origin to the surface ?
? Unlock Deck
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10
Find the point at which the function has the minimum value subject to the constraint that .
A) e. b. f. c. g. d. h.None of these
has the minimum value subject to the constraint that .A)
e. b. f. c. g. d. h.None of these Unlock Deck
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11
In using Lagrange multipliers to minimize the function subject to the constraint that , what is the value of the multiplier ?
A) e. b.1
F)3
C) g. d.2
H)4
subject to the constraint that , what is the value of the multiplier ?A)
e. b.1F)3
C)
g. d.2H)4
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12
Solve completely, using Lagrange multipliers: Find the dimension of a box with volume 1000 which minimizes the total length of the 12 edges.
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13
Find the extreme values of on the circle .
on the circle . Unlock Deck
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14
Find the maximum value of the function subject to the constraint that .
A) e.4
B) f.5
C) g.6
D) h.7
subject to the constraint that .A)
e.4B)
f.5C)
g.6D)
h.7 Unlock Deck
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15
Find the point on the plane where is minimum.
where is minimum. Unlock Deck
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16
In using Lagrange multipliers to minimize the function subject to the constraint that , what is the value of the multiplier ?
A) e. b.1
F)3
C) g. d.2
H)4
subject to the constraint that , what is the value of the multiplier ?A)
e. b.1F)3
C)
g. d.2H)4
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17
Find the minimum value of the function subject to the constraint that .
A) e. b.1
F)3
C) g. d.2
H)4
subject to the constraint that .A)
e. b.1F)3
C)
g. d.2H)4
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18
Find the maximum and minimum values of the function on the ellipse given by the equation .
on the ellipse given by the equation . Unlock Deck
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19
Find the minimum value of the function subject to the constraint that .
A) e.4
B) f.5
C) g.6
D) h.7
subject to the constraint that .A)
e.4B)
f.5C)
g.6D)
h.7 Unlock Deck
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20
Find the minimum value of the function subject to the constraint that .
A) e. b.1
F) c. g. d.2
H)
subject to the constraint that .A)
e. b.1F)
c. g. d.2H)
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21
The quality of a good produced by a company is given by , where is the quantity of capital and is the quantity of labor used. Capital costs are $20 per unit, labor costs are $10 per unit, and the company wants to keep costs for capital and labor combined to $150.(a) What combination of labor and capital should be used to produce maximum quantity? What is the maximum value?
(b) Draw the level curves of and the graph of the budget constraint on the same set of axes.(c) Complete the value of . What does represent?
of a good produced by a company is given by , where is the quantity of capital and is the quantity of labor used. Capital costs are $20 per unit, labor costs are $10 per unit, and the company wants to keep costs for capital and labor combined to $150.(a) What combination of labor and capital should be used to produce maximum quantity? What is the maximum value?(b) Draw the level curves of
and the graph of the budget constraint on the same set of axes.(c) Complete the value of . What does represent? Unlock Deck
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22
Determine how many critical points the function has.
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
has.A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
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23
The function has one critical point. Determine its location and type.
A) , saddle point
E) , minimum point
B) , minimum point
F) , maximum point
C) , maximum point
G) , saddle point
D) , saddle point
H) , minimum point
has one critical point. Determine its location and type.A)
, saddle pointE)
, minimum pointB)
, minimum pointF)
, maximum pointC)
, maximum pointG)
, saddle pointD)
, saddle pointH)
, minimum point Unlock Deck
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24
A rancher with 300 ft of fence intends to enclose a rectangular corral, dividing it in half by a fence parallel to the short sides of the corral. What is the maximum area he can enclose? Compute the value of . What does represent?
. What does represent? Unlock Deck
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25
Find the maximum value of the function a.90
E)80
B)160
F)135
C)50
G)16
D)15
H)5
a.90E)80
B)160
F)135
C)50
G)16
D)15
H)5
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26
The function has one critical point. Determine its location and type.
A) , saddle point
E) , minimum point
B) , minimum point
F) , maximum point
C) , maximum point
G) , saddle point
D) , saddle point
H) , minimum point
has one critical point. Determine its location and type.A)
, saddle pointE)
, minimum pointB)
, minimum pointF)
, maximum pointC)
, maximum pointG)
, saddle pointD)
, saddle pointH)
, minimum point Unlock Deck
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27
The function has one critical point. Determine its location and type.
A) , saddle point
E) , minimum point
B) , minimum point
F) , maximum point
C) , maximum point
G) , saddle point
D) , saddle point
H) , minimum point
has one critical point. Determine its location and type.A)
, saddle pointE)
, minimum pointB)
, minimum pointF)
, maximum pointC)
, maximum pointG)
, saddle pointD)
, saddle pointH)
, minimum point Unlock Deck
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28
The level curves of a function and a curve with equation ( constant) are given below. Estimate the point where has a maximum value and the point where has a minimum value, subject to the constraint that . Indicate your answer in the figure.
and a curve with equation ( constant) are given below. Estimate the point where has a maximum value and the point where has a minimum value, subject to the constraint that . Indicate your answer in the figure. Unlock Deck
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29
Suppose that the quantity produced of a certain good depends on the number of units of labor and the quantity of capital according to the function . Suppose also that labor costs $100 per unit and that capital costs $200 per unit.(a) What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimum cost?
(b) Draw the level curves of , the cost function, and the graph of the constraint on the same set of axes.(c) Complete the value of . What does represent?
produced of a certain good depends on the number of units of labor and the quantity of capital according to the function . Suppose also that labor costs $100 per unit and that capital costs $200 per unit.(a) What combination of labor and capital should be used to produce 36,000 units of the goods at minimum cost? What is that minimum cost?(b) Draw the level curves of
, the cost function, and the graph of the constraint on the same set of axes.(c) Complete the value of . What does represent? Unlock Deck
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30
What point on the surface , , , is closest to the origin?
, , , is closest to the origin? Unlock Deck
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31
Determine how many critical points the function has.
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
has.A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
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32
A package in the shape of a cylindrical box can be mailed by the U.S. Postal Service if the sum of its height and girth (the circumference of the circular base) is at most 108 in. Find the dimension of the package with largest volume that can be mailed.
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33
Find the minimum value of the function a.8.5
E)10.5
B)18
F)9
C)19
G)9.5
D)10
H)50
a.8.5E)10.5
B)18
F)9
C)19
G)9.5
D)10
H)50
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34
A rectangular area of 3200 ft is to be fenced off. Two opposite sides will use fencing costing $1 per foot and the remaining sides will use fencing costing $2 per foot. Find the dimensions of the rectangle of least cost. Compute the value of . What does represent?
is to be fenced off. Two opposite sides will use fencing costing $1 per foot and the remaining sides will use fencing costing $2 per foot. Find the dimensions of the rectangle of least cost. Compute the value of . What does represent? Unlock Deck
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35
The function has one critical point. Determine its location and type.
A) , saddle point
E) , minimum point
B) , minimum point
F) , maximum point
C) , maximum point
G) , saddle point
D) , saddle point
H) , minimum point
has one critical point. Determine its location and type.A)
, saddle pointE)
, minimum pointB)
, minimum pointF)
, maximum pointC)
, maximum pointG)
, saddle pointD)
, saddle pointH)
, minimum point Unlock Deck
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36
A rancher intends to fence off a rectangular region along a river (which serves as a natural boundary requiring no fence). If the enclosed area is to be 1800 square yards, what is the least amount of fence needed? Compute the value of . What does represent?
. What does represent? Unlock Deck
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37
Determine how many critical points the function has.
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
has.A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
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38
Determine how many critical points the function has.
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
has.A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
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39
Find the maximum and minimum values of on the circle .
on the circle . Unlock Deck
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40
Determine how many critical points the function has.
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
has.A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
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41
Find the local maximum and minimum values and saddle points of the function .
. Unlock Deck
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42
Find the local maximum and minimum values and saddle points of the function .
. Unlock Deck
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43
Find the point at which the function has a local maximum.
A) e. b. f. c. g. d. h.
has a local maximum.A)
e. b. f. c. g. d. h. Unlock Deck
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44
For each of the following functions, find the critical point, if there is one, and determine if it is a local maximum, local minimum, saddle point, or otherwise.(a) (b) (c) (d)
(b) (c) (d) Unlock Deck
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45
Use the level curves of shown below to estimate the critical points of . Indicate whether has a saddle point or a local maximum or minimum at each of those points.
shown below to estimate the critical points of . Indicate whether has a saddle point or a local maximum or minimum at each of those points. Unlock Deck
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46
A cardboard box without a lid is to have volume 100 cubic inches, with total area of cardboard as small as possible. Find its height in inches.
A)2
E) b. f. c.4
G) d.5
H)
A)2
E)
b. f. c.4G)
d.5H)
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47
Find the local maximum and minimum values and saddle points of the function .
. Unlock Deck
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48
Find the local maximum and minimum values and saddle points of the function .
. Unlock Deck
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49
The function has a maximum. Find the values of and at which it occurs.
has a maximum. Find the values of and at which it occurs. Unlock Deck
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50
Use the level curves of shown below to estimate the critical points of . Indicate whether has a saddle point or a local maximum or minimum at each of those points.
shown below to estimate the critical points of . Indicate whether has a saddle point or a local maximum or minimum at each of those points. Unlock Deck
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51
Find the local maximum and minimum values and saddle points of the function .
. Unlock Deck
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52
Compare the minimum value of and sketch a portion of the graph of near its lowest point.
and sketch a portion of the graph of near its lowest point. Unlock Deck
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53
Find the absolute maximum and minimum values of over the region , where is a closed triangular region with vertices , , and .
over the region , where is a closed triangular region with vertices , , and . Unlock Deck
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54
Find three positive numbers , , and whose sum is 48 and product is maximum.
, , and whose sum is 48 and product is maximum. Unlock Deck
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55
Find the shortest distance from the origin to the surface .
A) e.2
B) f. c.1
G)4
D) h.
.A)
e.2B)
f. c.1G)4
D)
h. Unlock Deck
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56
Find the critical points (if any) for and determine if each is a local extreme value or a saddle point.
and determine if each is a local extreme value or a saddle point. Unlock Deck
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57
Find the local maximum and minimum values and saddle points of the function .
. Unlock Deck
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58
Find the absolute maximum and minimum value of on the square region with vertices , , , and .
on the square region with vertices , , , and . Unlock Deck
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59
Find the absolute maximum and minimum value of on the square region with vertices , , , and .
on the square region with vertices , , , and . Unlock Deck
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60
A company estimates that its annual profits for next year will be , where represents investment in research and represents investment in labor. All units are in tens of thousands of dollars. Determine the amount the company should spend on research and on labor to maximize its profit. What is maximum profit?
, where represents investment in research and represents investment in labor. All units are in tens of thousands of dollars. Determine the amount the company should spend on research and on labor to maximize its profit. What is maximum profit? Unlock Deck
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61
Find an equation of the tangent plane to the surface at the point .
A) e. b. f. c. g. d. h.
at the point .A)
e. b. f. c. g. d. h. Unlock Deck
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62
Let . Find the gradient vector at the point .
A)i
E) b.j
F) c.k
G) d. h.0
. Find the gradient vector at the point .A)i
E)
b.jF)
c.kG)
d. h.0 Unlock Deck
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63
Find the directional derivative of the function at the point in the direction from toward the point ..
A) e.7
B) f. c. g. d. h.8
at the point in the direction from toward the point ..A)
e.7B)
f. c. g. d. h.8 Unlock Deck
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64
Find the largest value of the directional derivative of the function at the point .
A) e. b. f. c. g. d. h.
at the point .A)
e. b. f. c. g. d. h. Unlock Deck
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65
Let . Find the gradient vector .
A) e. b. f. c. g. d. h.
. Find the gradient vector .A)
e. b. f. c. g. d. h. Unlock Deck
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66
Find a normal vector to the surface at the point .
A) e. b. f. c. g. d. h.
at the point .A)
e. b. f. c. g. d. h. Unlock Deck
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67
Let . Find the gradient vector .
A) e. b. f. c. g. d. h.
. Find the gradient vector .A)
e. b. f. c. g. d. h. Unlock Deck
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68
Find the directional derivative of the function at the point in the direction .
A) e.1
B)0
F) c. g. d.12
H)
at the point in the direction .A)
e.1B)0
F)
c. g. d.12H)
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69
Find the direction of maximum increase of the function at the point .
A) e. b. f. c. g. d. h.
at the point .A)
e. b. f. c. g. d. h. Unlock Deck
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70
Find the second directional derivative of at the point in the direction .
A) e. b. f. c.2
G) d.6
H)
at the point in the direction .A)
e. b. f. c.2G)
d.6H)
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71
Find three positive numbers , , and whose product is 343 and sum is minimum.
, , and whose product is 343 and sum is minimum. Unlock Deck
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72
Let . Find the gradient vector at the point .
A) e. b. f. c. g. d. h.
. Find the gradient vector at the point .A)
e. b. f. c. g. d. h. Unlock Deck
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73
Find an equation of the tangent plane to the hyperboloid at the point .
A) e. b. f. c. g. d. h.
at the point .A)
e. b. f. c. g. d. h. Unlock Deck
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74
Find the directional derivative of the function at the point in the direction .
A) e.1
B) f.2
C) g.4
D) h.
at the point in the direction .A)
e.1B)
f.2C)
g.4D)
h. Unlock Deck
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75
Find the second directional derivative of at the point in the direction .
A) e. b. f. c. g. d.14
H)
at the point in the direction .A)
e. b. f. c. g. d.14H)
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76
Find the direction in which the directional derivative of the function at the point is maximum.
A) e. b. f. c. g. d. h.
in which the directional derivative of the function at the point is maximum.A)
e. b. f. c. g. d. h. Unlock Deck
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77
Find the direction of maximum increase of the function at the point .
A) e. b. f. c. g. d. h.
at the point .A)
e. b. f. c. g. d. h. Unlock Deck
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78
Find an equation of the tangent plane to the surface at the point .
A) e. b. f. c. g. d. h.
at the point .A)
e. b. f. c. g. d. h. Unlock Deck
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79
Find the directional derivative of the function at the point in the direction of the vector .
A) e. b. f. c. g. d.0
H)
at the point in the direction of the vector .A)
e. b. f. c. g. d.0H)
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80
Find the directional derivative of the function at the point in the direction .
A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
at the point in the direction .A)0
E)4
B)1
F)5
C)2
G)6
D)3
H)7
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Unlock Deck
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