st_14_6 Lista 2 saeger P1

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920 | | | | CHAPTER 14 PARTIAL DERIVATIVES
7–10
(a) Find the gradient of .
(b) Evaluate the gradient at the point .
(c) Find the rate of change of at in the direction of the 
vector .
7. , ,
8. , ,
9. , ,
10. , ,
11–17 Find the directional derivative of the function at the given
point in the direction of the vector .
, ,
12. , ,
13. , ,
14. , ,
15. , ,
16. , ,
17. , ,
18. Use the figure to estimate .
Find the directional derivative of at in
the direction of .
20. Find the directional derivative of at
in the direction of .
21–26 Find the maximum rate of change of at the given point
and the direction in which it occurs.
21. ,
22. ,
,
24. ,
25. ,
26. , ��5, 1, 1�f �x, y, z� � tan�x � 2y � 3z�
�3, 6, �2�f �x, y, z� � sx 2 � y 2 � z 2 
�1, 1, �1�f �x, y, z� � �x � y�
z
�1, 0�f �x, y� � sin�xy�23.
�0, 0�f �p, q� � qe�p � pe�q
�2, 4�f �x, y� � y 2
x
f
Q�2, 4, 5�P�1, �1, 3�
f �x, y, z� � xy � yz � zx
Q�5, 4�
P�2, 8�f �x, y� � sxy 19.
y
x0
(2, 2)
±f(2, 2)
u
Du f �2, 2�
v � 2 j � k�1, 1, 2�t�x, y, z� � �x � 2y � 3z�3
2
v � ��1, �2, 2 ��3, 2, 6�f �x, y, z� � sxyz 
v � �5, 1, �2 ��0, 0, 0�f �x, y, z� � xe y � ye z � ze x
v � 5 i � 10 j�1, 2�t�r, s� � tan�1�rs�
v � i � 3 j�2, 1�t�p, q� � p4 � p2q3
v � ��1, 2 ��2, 1�f �x, y� � ln�x 2 � y 2�
v � �4, �3 ��3, 4�f �x, y� � 1 � 2xsy 11.
v
u � � 27 , 37 , 67 �P�1, 3, 1�f �x, y, z� � sx � yz 
u � � 23 , �23 , 13 �P�3, 0, 2�f �x, y, z� � xe 2 yz
u � 13 (2 i � s5 j)P�1, 2�f �x, y� � y 2
x
u � 12 (s3 i � j)P��6, 4�f �x, y� � sin�2x � 3y�
u
Pf
P
f
Level curves for barometric pressure (in millibars) are shown
for 6:00 AM on November 10, 1998. A deep low with pressure
972 mb is moving over northeast Iowa. The distance along the
red line from K (Kearney, Nebraska) to S (Sioux City, Iowa) is
300 km. Estimate the value of the directional derivative of the
pressure function at Kearney in the direction of Sioux City.
What are the units of the directional derivative?
2. The contour map shows the average maximum temperature for
November 2004 (in ). Estimate the value of the directional
derivative of this temperature function at Dubbo, New South
Wales, in the direction of Sydney. What are the units?
3. A table of values for the wind-chill index is given
in Exercise 3 on page 888. Use the table to estimate the value
of , where .
4–6 Find the directional derivative of at the given point in the
direction indicated by the angle .
4. , ,
5. , ,
6. , , � � 	
3�2, 0�f �x, y� � x sin�xy�
� � 2	
3�0, 4�f �x, y� � ye�x
� � 	
4�2, 1�f �x, y� � x 2y3 � y 4
�
f
u � �i � j�
s2 Du f ��20, 30�
W � f �T, v�
Sydney
Dubbo
30
27 24
24
21
18
0 100 200 300
(Distance in kilometres)
Copyright Commonwealth of Australia. Reproduced by permission.
�C
1012
1012
1008
1008
1004
1000
996
992
988
980
976
984
1016
1020
1024
972
K
S
From Meteorology Today, 8E by C. Donald Ahrens (2007 Thomson Brooks/Cole).
1.
EXERCISES14.6
SECTION 14.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR | | | | 921
35. Let be a function of two variables that has continuous 
partial derivatives and consider the points , ,
, and . The directional derivative of at in the
direction of the vector is 3 and the directional derivative at
in the direction of is 26. Find the directional derivative of
at in the direction of the vector .
36. For the given contour map draw the curves of steepest ascent
starting at and at .
37. Show that the operation of taking the gradient of a function has
the given property. Assume that and are differentiable func-
tions of and and that , are constants.
(a) (b)
(c) (d)
38. Sketch the gradient vector for the function whose
level curves are shown. Explain how you chose the direction
and length of this vector.
39–44 Find equations of (a) the tangent plane and (b) the normal
line to the given surface at the specified point.
39. ,
40. ,
41. ,
42. ,
,
44. , �0, 0, 1�yz � ln�x � z�
�1, 0, 0�z � 1 � xe y cos z43.
�1 � 	, 1, 1�x � z � 4 arctan�yz�
�2, 1, �1�x 2 � 2y 2 � z 2 � yz � 2
�4, 7, 3�y � x 2 � z 2
�3, 3, 5�2�x � 2�2 � �y � 1�2 � �z � 3�2 � 10
20
2
4
6
4 6 x
y
_1
0
1 3 5
_3
_5
(4, 6)
f� f �4, 6�
�un � nu n�1 �u��uv� � v �u � u �vv 2
��uv� � u �v � v �u��au � bv� � a �u � b �v
bayx
vu
60 50
40
30
20
Q
P
QP
AD
l
Af
AC
l
A
AB
l
AfD�6, 15�C�1, 7�
B�3, 3�A�1, 3�
f(a) Show that a differentiable function decreases most
rapidly at in the direction opposite to the gradient vector,
that is, in the direction of .
(b) Use the result of part (a) to find the direction in which the
function decreases fastest at the 
point .
28. Find the directions in which the directional derivative of
at the point has the value 1.
Find all points at which the direction of fastest change of the
function is .
30. Near a buoy, the depth of a lake at the point with coordinates
is , where , , and are
measured in meters. A fisherman in a small boat starts at the
point and moves toward the buoy, which is located at
. Is the water under the boat getting deeper or shallower
when he departs? Explain.
31. The temperature in a metal ball is inversely proportional to
the distance from the center of the ball, which we take to be the
origin. The temperature at the point is .
(a) Find the rate of change of at in the direction
toward the point .
(b) Show that at any point in the ball the direction of greatest
increase in temperature is given by a vector that points
toward the origin.
32. The temperature at a point is given by 
where is measured in and , , in meters.
(a) Find the rate of change of temperature at the point
in the direction toward the point .
(b) In which direction does the temperature increase fastest 
at ?
(c) Find the maximum rate of increase at .
Suppose that over a certain region of space the electrical poten-
tial is given by .
(a) Find the rate of change of the potential at in the
direction of the vector .
(b) In which direction does change most rapidly at ?
(c) What is the maximum rate of change at ?
34. Suppose you are climbing a hill whose shape is given by the
equation , where , , and are
measured in meters, and you are standing at a point with coor-
dinates . The positive -axis points east and the
positive -axis points north.
(a) If you walk due south, will you start to ascend or descend?
At what rate?
(b) If you walk northwest, will you start to ascend or descend?
At what rate?
(c) In which direction is the slope largest? What is the rate of
ascent in that direction? At what angle above the horizontal
does the path in that direction begin?
y
x�60, 40, 966�
zyxz � 1000 � 0.005x 2 � 0.01y 2
P
PV
v � i � j � k
P�3, 4, 5�
V�x, y, z� � 5x 2 � 3xy � xyzV
33.
P
P
�3, �3, 3�P�2, �1, 2�
zyx�CT
T�x, y, z� � 200e�x 2�3y 2�9z 2
�x, y, z�
�2, 1, 3�
�1, 2, 2�T
120��1, 2, 2�
T
�0, 0�
�80, 60�
zyxz � 200 � 0.02x 2 � 0.001y 3�x, y�
i � jf �x, y� � x 2 � y 2 � 2x � 4y
29.
�0, 2�f �x, y� � ye�xy
�2, �3�
f �x, y� � x 4y � x 2 y 3
�� f �x�
x
f27.
56. Show that every normal line to the sphere 
passes through the center of the sphere.
Show that the sum of the -, -, and -intercepts of any 
tangent plane to the surface is a 
constant.
58. Show that the pyramids cut off from the first octant by any
tangent planes to the surface at points in the first
octant must all have the same volume.
59. Find parametric equations for the tangent line to the curve of
intersection of the paraboloid and the ellipsoid
at the point .
60. (a) The plane intersects the cylinder 
in an ellipse. Find parametric equations for the tangent
line to this ellipse at the point .
; (b) Graph the cylinder, the plane, and the tangent line on the
same screen.
61. (a) Two surfaces are called orthogonal at a point of inter-
section if their normal lines are perpendicular at that
point. Show that surfaces with equations 
and are orthogonal at a point where
and if and only if
at 
(b) Use part (a) to show that the surfaces and
are orthogonal at every point of 
intersection. Can you see why this is true without using
calculus?62. (a) Show that the function is continuous and
the partial derivatives and exist at the origin but the
directional derivatives in all other directions do not exist.
; (b) Graph near the origin and comment on how the graph
confirms part (a).
Suppose that the directional derivatives of are known 
at a given point in two nonparallel directions given by unit 
vectors and . Is it possible to find at this point? If so,
how would you do it?
64. Show that if is differentiable at 
then
[Hint: Use Definition 14.4.7 directly.]
lim 
xl x 0
 
 f �x� � f �x0 � � � f �x0 � � �x � x0 �
� x � x0 � � 0
x0 � �x0, y0 �,z � f �x, y�
� fvu
f �x, y�63.
f
fyfx
f �x, y� � s3 xy 
x 2 � y 2 � z2 � r 2
z2 � x 2 � y 2
PFx Gx � FyGy � FzGz � 0
�G � 0�F � 0
PG�x, y, z� � 0
F�x, y, z� � 0
�1, 2, 1�
x 2 � y 2 � 5y � z � 3
��1, 1, 2�4x 2 � y 2 � z2 � 9
z � x 2 � y 2
xyz � 1
sx � sy � sz � sc 
zyx57.
x 2 � y 2 � z2 � r 2; 45–46 Use a computer to graph the surface, the tangent plane,
and the normal line on the same screen. Choose the domain care-
fully so that you avoid extraneous vertical planes. Choose the
viewpoint so that you get a good view of all three objects.
45. ,
46. ,
47. If , find the gradient vector and use it 
to find the tangent line to the level curve at the
point . Sketch the level curve, the tangent line, and the
gradient vector.
48. If , find the gradient vector 
and use it to find the tangent line to the level curve
at the point . Sketch the level curve, the 
tangent line, and the gradient vector.
49. Show that the equation of the tangent plane to the ellipsoid
at the point can be 
written as
50. Find the equation of the tangent plane to the hyperboloid
at and express it in a
form similar to the one in Exercise 49.
51. Show that the equation of the tangent plane to the elliptic
paraboloid at the point can
be written as
52. At what point on the paraboloid is the tangent
plane parallel to the plane ?
53. Are there any points on the hyperboloid 
where the tangent plane is parallel to the plane ?
54. Show that the ellipsoid and the sphere
are tangent to each
other at the point . (This means that they have a com-
mon tangent plane at the point.)
55. Show that every plane that is tangent to the cone
passes through the origin.x 2 � y 2 � z2
�1, 1, 2�
x 2 � y 2 � z2 � 8x � 6y � 8z � 24 � 0
3x 2 � 2y 2 � z2 � 9
z � x � y
x 2 � y 2 � z2 � 1
x � 2y � 3z � 1
y � x 2 � z2
2xx0
a 2
�
2yy0
b 2
�
z � z0
c
�x0, y0, z0 �z
c � x 2
a 2 � y 2
b 2
�x0, y0, z0 �x 2
a 2 � y 2
b 2 � z2
c 2 � 1
xx0
a 2
�
 yy0
b 2
�
zz0
c 2
� 1
�x0, y0, z0 �x 2
a 2 � y 2
b 2 � z2
c 2 � 1
�1, 2�t�x, y� � 1
�t�1, 2�t�x, y� � x 2 � y 2 � 4x
�3, 2�
f �x, y� � 6
� f �3, 2�f �x, y� � xy
�1, 2, 3�xyz � 6
�1, 1, 1�xy � yz � zx � 3
922 | | | | CHAPTER 14 PARTIAL DERIVATIVES
MAXIMUM AND MINIMUM VALUES
As we saw in Chapter 4, one of the main uses of ordinary derivatives is in finding maxi-
mum and minimum values. In this section we see how to use partial derivatives to locate
maxima and minima of functions of two variables. In particular, in Example 6 we will see
how to maximize the volume of a box without a lid if we have a fixed amount of cardboard
to work with.
14.7
A120 || | | APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES
61.
63.
65. 67.
69. 81.
87. No 89.
93.
95. (a)
(b) ,
(c) 0, 0 (e) No, since and are not continuous.
EXERCISES 14.4 N PAGE 899
1.
3.
5.
7. 9.
11. 13. 15.
19. ; 21.
23.
25.
27.
29.
31. 33. 35.
37. 150 39. 41. 2.3%
43.
EXERCISES 14.5 N PAGE 907
1.
3.
5.
7. ,
9. ,
11. ,
�z
�t
� e r
s cos � � tss 2 � t 2 sin ��
�z
�s
� e r
t cos � � sss 2 � t 2 sin ��
�z��t � 2st cos � cos � � s2 sin � sin �
�z��s � t 2 cos � cos � � 2st sin � sin �
�z��t � �2sxy 3 sin t � 3sx 2y 2 cos t
�z��s � 2xy 3 cos t � 3x 2y 2 sin t
e y�z�2t � �x�z� � �2xy�z2 �	
��x�t� � y sin t	�s1 � x 2 � y2
�2x � y� cos t � �2y � x�e t
�1 � �x, �2 � �y
1
17 � 0.059 �
16 cm35.4 cm2�z � 0.9225, dz � 0.9
dR � � 2 cos � d� � 2�� cos � d� � �� 2 sin � d�
dm � 5p4q3 dp � 3p5q2 dq
dz � 3x 2 ln�y 2� dx � �2x 3�y� dy
4T � H � 329; 129�F
3
7 x �
2
7 y �
6
7 z; 6.99142.846�
2
3 x �
7
3 y �
20
3
1 � �y19 x �
2
9 y �
2
32x �
1
4 y � 1
0
2 x
0
2y
_1
0z
1
400
200
0
y5
0 _5x
10
0
_10
z
z � y
x � y � 2z � 0
z � �8x � 2y
fyxfxy
fy�x, y� � x
5 � 4x 3y 2 � xy 4
�x 2 � y 2 �2
fx�x, y� � x
4y � 4x 2y 3 � y 5
�x 2 � y 2 �2
_0.2
0.2
0
_1
0
1
y
1
0
_1
x
z
�2
x � 1 � t, y � 2, z � 2 � 2t
R 2�R 12�12.2, �16.8, �23.25
4��y � 2z�3, 0�e r��2 sin � � � cos � � r� sin ��
24 sin�4x � 3y � 2z�, 12 sin�4x � 3y � 2z�
12xy, 72xy 13. 62 15.
17. , ,
19. ,
21. 23. 25.
27. 29.
31.
33.
35. 37.
39. (a) (b) (c)
41. 43.
45. (a) ,
51.
EXERCISES 14.6 N PAGE 920
1. 3. 5.
7. (a)
(b) (c)
9. (a) (b) (c)
11. 13. 15. 17.
19. 21. , 23.
25. 27. (b)
29. All points on the line 
31. (a)
33. (a) (b) (c) 35.
39. (a) (b)
41. (a) (b)
43. (a) (b)
45. 47. ,
53. No 59.
63. If and , then and are
known, so we solve linear equations for and .fyfx
c fx � dfyafx � bfyv � �c, d �u � �a, b �
x � �1 � 10t, y � 1 � 16t, z � 2 � 12t
y
x0
2x+3y=12
xy=6
(3, 2)
f (3, 2)
Î
1
_1
0
1
2
1 2x
2
z
y
2x � 3y � 12�2, 3�
x � 1 � y � �zx � y � z � 1
x � 2
4
�
y � 1
�5 �
z � 1
�1
4x � 5y � z � 4
x � 3 � y � 3 � z � 5x � y � z � 11
327
132s406�38, 6, 12 �32�s3
�40�(3s3)
y � x � 1
��12, 92�1, �3, 6, �2 �
1, �0, 1 ���1, 1 �4s22�5
9�(2s5)4�s30�8�s1023�10
�
22
3�1, 12, 0 ��e 2yz, 2xze 2yz, 2xye 2yz �
s3 � 32�2, 3 �
�f �x, y� � �2 cos�2x � 3y�, 3 cos�2x � 3y��
2 � s3�2�0.778��0.08 mb�km
4rs �2z��x 2 � �4r 2 � 4s 2 ��2z��x �y � 4rs �2z��y 2 � 2 �z��y
�z��� � ���z��x�r sin � � ��z��y�r cos �
�z��r � ��z��x� cos � � ��z��y� sin �
�1�(12s3) rad�s��0.27 L�s
0 m�s10 m2�s6 m3�s
� �0.33 m�s per minute2�C�s
1 � y 2z 2
1 � y � y 2z 2
, �
z
1 � y � y 2z 2
3yz � 2x
2z � 3xy
, 
3xz � 2y
2z � 3xy
sin�x � y� � e y
sin�x � y� � xe y
4�xy�3�2 � y
x � 2x 2sxy
36, 24, 3097, 
9
785, 178, 54
�w
�y
�
�w
�r
 
�r
�y
�
�w
�s
 
�s
�y
�
�w
�t
 
�t
�y
�
�w
�t
 
�t
�x
�w
�x
�
�w
�r
 
�r
�x
�
�w
�s
 
�s
�x
�u
�t
�
�u
�x
 
�x
�t
�
�u
�y
 
�y
�t
�u
�s
�
�u
�x
 
�x
�s
�
�u
�y
 
�y
�s
�u
�r
�
�u
�x
 
�x
�r
�
�u
�y
 
�y
�r
7, 2