920
||||
14.6
CHAPTER 14 PARTIAL DERIVATIVES
EXERCISES
1. 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?
7–10
(a) Find the gradient of f .
(b) Evaluate the gradient at the point P.
(c) Find the rate of change of f at P in the direction of the
vector u.
8. f 共x, y兲 苷 y 2兾x,
1012
1016
1008
1004
1000
996
992
988
984
980
S 976
972
1012
1020
1024
(
P共⫺6, 4兲, u 苷 12 s3 i ⫺ j)
7. f 共x, y兲 苷 sin共2x ⫹ 3y兲,
u苷
P共1, 2兲,
9. f 共x, y, z兲 苷 xe 2 yz,
1
3
u 苷 具 23 , ⫺ 23 , 13 典
P共3, 0, 2兲,
10. f 共x, y, z兲 苷 sx ⫹ yz ,
(2 i ⫹ s5 j)
u 苷 具 7, 7, 7 典
2 3 6
P共1, 3, 1兲,
11–17 Find the directional derivative of the function at the given
point in the direction of the vector v.
11. f 共x, y兲 苷 1 ⫹ 2x sy ,
K
共3, 4兲, v 苷 具4, ⫺3典
12. f 共x, y兲 苷 ln共x ⫹ y 兲,
2
13. t共 p, q兲 苷 p ⫺ p q ,
4
1008
14. t共r, s兲 苷 tan 共rs兲,
2. The contour map shows the average maximum temperature for
November 2004 (in ⬚C ). 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?
共2, 1兲,
2 3
⫺1
From Meteorology Today, 8E by C. Donald Ahrens (2007 Thomson Brooks/Cole).
共2, 1兲, v 苷 具⫺1, 2 典
2
v 苷 i ⫹ 3j
共1, 2兲, v 苷 5 i ⫹ 10 j
15. f 共x, y, z兲 苷 xe ⫹ ye z ⫹ ze x,
共0, 0, 0兲,
16. f 共x, y, z兲 苷 sxyz ,
v 苷 具⫺1, ⫺2, 2 典
y
共3, 2, 6兲,
17. t共x, y, z兲 苷 共x ⫹ 2y ⫹ 3z兲
v 苷 具5, 1, ⫺2典
共1, 1, 2兲, v 苷 2 j ⫺ k
3兾2
,
18. Use the figure to estimate Du f 共2, 2兲.
0
100 200 300
(Distance in kilometres)
y
(2, 2)
24
u
±f (2, 2)
Dubbo
30
0
27
24
Sydney
21
18
Copyright Commonwealth of Australia. Reproduced by permission.
3. A table of values for the wind-chill index W 苷 f 共T, v兲 is given
in Exercise 3 on page 888. Use the table to estimate the value
of Du f 共⫺20, 30兲, where u 苷 共i ⫹ j兲兾s2 .
19. Find the directional derivative of f 共x, y兲 苷 sxy at P共2, 8兲 in
the direction of Q共5, 4兲.
20. Find the directional derivative of f 共x, y, z兲 苷 xy ⫹ yz ⫹ zx at
P共1, ⫺1, 3兲 in the direction of Q共2, 4, 5兲.
21–26 Find the maximum rate of change of f at the given point
and the direction in which it occurs.
21. f 共x, y兲 苷 y 2兾x,
⫺p
共2, 4兲
⫹ pe ⫺q, 共0, 0兲
4 –6 Find the directional derivative of f at the given point in the
22. f 共 p, q兲 苷 qe
direction indicated by the angle ␪.
23. f 共x, y兲 苷 sin共xy兲,
4. f 共x, y兲 苷 x y ⫺ y ,
2 3
⫺x
5. f 共x, y兲 苷 ye ,
4
共2, 1兲,
共0, 4兲,
6. f 共x, y兲 苷 x sin共xy兲,
␪ 苷 ␲兾4
␪ 苷 2␲兾3
共2, 0兲,
␪ 苷 ␲兾3
x
共1, 0兲
24. f 共x, y, z兲 苷 共x ⫹ y兲兾z,
共1, 1, ⫺1兲
25. f 共x, y, z兲 苷 sx 2 ⫹ y 2 ⫹ z 2 ,
共3, 6, ⫺2兲
26. f 共x, y, z兲 苷 tan共x ⫹ 2y ⫹ 3z兲,
共⫺5, 1, 1兲
SECTION 14.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR
27. (a) Show that a differentiable function f decreases most
rapidly at x in the direction opposite to the gradient vector,
that is, in the direction of ⫺ⵜ f 共x兲.
(b) Use the result of part (a) to find the direction in which the
function f 共x, y兲 苷 x 4 y ⫺ x 2 y 3 decreases fastest at the
point 共2, ⫺3兲.
28. Find the directions in which the directional derivative of
f 共x, y兲 苷 ye⫺xy at the point 共0, 2兲 has the value 1.
||||
921
35. Let f be a function of two variables that has continuous
partial derivatives and consider the points A共1, 3兲, B共3, 3兲,
C共1, 7兲, and D共6, 15兲. The
directional derivative of f at A in the
l
AB is 3 and the directional derivative at
direction of the vectorl
A in the direction of AC is 26. Find the directional derivative of
l
f at A in the direction of the vector AD .
36. For the given contour map draw the curves of steepest ascent
starting at P and at Q.
Q
29. Find all points at which the direction of fastest change of the
function f 共x, y兲 苷 x 2 ⫹ y 2 ⫺ 2 x ⫺ 4y is i ⫹ j.
30. Near a buoy, the depth of a lake at the point with coordinates
60
共x, y兲 is z 苷 200 ⫹ 0.02x 2 ⫺ 0.001y 3, where x, y, and z are
measured in meters. A fisherman in a small boat starts at the
point 共80, 60兲 and moves toward the buoy, which is located at
共0, 0兲. Is the water under the boat getting deeper or shallower
when he departs? Explain.
31. The temperature T 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 共1, 2, 2兲 is 120⬚.
(a) Find the rate of change of T at 共1, 2, 2兲 in the direction
toward the point 共2, 1, 3兲.
(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 共x, y, z兲 is given by
T共x, y, z兲 苷 200e⫺x
2
tions of x and y and that a, b are constants.
(a) ⵜ共au ⫹ b v兲 苷 a ⵜu ⫹ b ⵜv (b) ⵜ共u v兲 苷 u ⵜv ⫹ v ⵜu
(c) ⵜ
冉冊
u
v
苷
v ⵜu ⫺ u ⵜv
(d) ⵜu n 苷 nu n⫺1 ⵜu
v2
38. Sketch the gradient vector ⵜ f 共4, 6兲 for the function f whose
level curves are shown. Explain how you chose the direction
and length of this vector.
y
_5
6
(4, 6)
_3
_1
4
0
1
3
5
2
33. Suppose that over a certain region of space the electrical poten-
equation z 苷 1000 ⫺ 0.005x 2 ⫺ 0.01y 2, where x, y, and z are
measured in meters, and you are standing at a point with coordinates 共60, 40, 966兲. The positive x-axis points east and the
positive y-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?
30
37. Show that the operation of taking the gradient of a function has
the given property. Assume that u and v are differentiable func-
where T is measured in ⬚C and x, y, z in meters.
(a) Find the rate of change of temperature at the point
P共2, ⫺1, 2兲 in the direction toward the point 共3, ⫺3, 3兲.
(b) In which direction does the temperature increase fastest
at P ?
(c) Find the maximum rate of increase at P.
34. Suppose you are climbing a hill whose shape is given by the
40
P
⫺3y 2⫺9z 2
tial V is given by V共x, y, z兲 苷 5x 2 ⫺ 3xy ⫹ xyz.
(a) Find the rate of change of the potential at P共3, 4, 5兲 in the
direction of the vector v 苷 i ⫹ j ⫺ k.
(b) In which direction does V change most rapidly at P ?
(c) What is the maximum rate of change at P ?
20
50
0
2
4
6
x
39– 44 Find equations of (a) the tangent plane and (b) the normal
line to the given surface at the specified point.
39. 2共x ⫺ 2兲 2 ⫹ 共 y ⫺ 1兲 2 ⫹ 共z ⫺ 3兲 2 苷 10,
40. y 苷 x 2 ⫺ z 2,
共4, 7, 3兲
41. x 2 ⫺ 2y 2 ⫹ z 2 ⫹ yz 苷 2,
42. x ⫺ z 苷 4 arctan共 yz兲,
43. z ⫹ 1 苷 xe y cos z,
44. yz 苷 ln共x ⫹ z兲,
共2, 1, ⫺1兲
共1 ⫹ ␲, 1, 1兲
共1, 0, 0兲
共0, 0, 1兲
共3, 3, 5兲
922
||||
CHAPTER 14 PARTIAL DERIVATIVES
56. Show that every normal line to the sphere x 2 ⫹ y 2 ⫹ z 2 苷 r 2
; 45– 46 Use a computer to graph the surface, the tangent plane,
passes through the center of the sphere.
and the normal line on the same screen. Choose the domain carefully so that you avoid extraneous vertical planes. Choose the
viewpoint so that you get a good view of all three objects.
45. x y ⫹ yz ⫹ zx 苷 3,
46. x yz 苷 6,
57. Show that the sum of the x-, y-, and z -intercepts of any
tangent plane to the surface sx ⫹ sy ⫹ sz 苷 sc is a
constant.
共1, 1, 1兲
58. Show that the pyramids cut off from the first octant by any
共1, 2, 3兲
tangent planes to the surface xyz 苷 1 at points in the first
octant must all have the same volume.
47. If f 共x, y兲 苷 xy, find the gradient vector ⵜ f 共3, 2兲 and use it
59. Find parametric equations for the tangent line to the curve of
to find the tangent line to the level curve f 共x, y兲 苷 6 at the
point 共3, 2兲. Sketch the level curve, the tangent line, and the
gradient vector.
intersection of the paraboloid z 苷 x 2 ⫹ y 2 and the ellipsoid
4x 2 ⫹ y 2 ⫹ z 2 苷 9 at the point 共⫺1, 1, 2兲.
60. (a) The plane y ⫹ z 苷 3 intersects the cylinder x 2 ⫹ y 2 苷 5
48. If t共x, y兲 苷 x 2 ⫹ y 2 ⫺ 4x, find the gradient vector ⵜt共1, 2兲
and use it to find the tangent line to the level curve
t共x, y兲 苷 1 at the point 共1, 2兲. Sketch the level curve, the
tangent line, and the gradient vector.
;
49. Show that the equation of the tangent plane to the ellipsoid
61. (a) Two surfaces are called orthogonal at a point of inter-
x 2兾a 2 ⫹ y 2兾b 2 ⫹ z 2兾c 2 苷 1 at the point 共x 0 , y0 , z0 兲 can be
written as
yy0
zz0
xx 0
⫹ 2 ⫹ 2 苷1
a2
b
c
section if their normal lines are perpendicular at that
point. Show that surfaces with equations F共x, y, z兲 苷 0
and G共x, y, z兲 苷 0 are orthogonal at a point P where
ⵜF 苷 0 and ⵜG 苷 0 if and only if
Fx Gx ⫹ Fy Gy ⫹ Fz Gz 苷 0 at P
50. Find the equation of the tangent plane to the hyperboloid
(b) Use part (a) to show that the surfaces z 2 苷 x 2 ⫹ y 2 and
x 2 ⫹ y 2 ⫹ z 2 苷 r 2 are orthogonal at every point of
intersection. Can you see why this is true without using
calculus?
x 兾a ⫹ y 兾b ⫺ z 兾c 苷 1 at 共x 0 , y0 , z0 兲 and express it in a
form similar to the one in Exercise 49.
2
2
2
2
2
in an ellipse. Find parametric equations for the tangent
line to this ellipse at the point 共1, 2, 1兲.
(b) Graph the cylinder, the plane, and the tangent line on the
same screen.
2
51. Show that the equation of the tangent plane to the elliptic
paraboloid z兾c 苷 x 2兾a 2 ⫹ y 2兾b 2 at the point 共x 0 , y0 , z0 兲 can
be written as
3
x y is continuous and
62. (a) Show that the function f 共x, y兲 苷 s
2
the partial derivatives fx and fy exist at the origin but the
directional derivatives in all other directions do not exist.
(b) Graph f near the origin and comment on how the graph
confirms part (a).
plane parallel to the plane x ⫹ 2y ⫹ 3z 苷 1?
63. Suppose that the directional derivatives of f 共x, y兲 are known
2xx 0
2yy0
z ⫹ z0
⫹
苷
a2
b2
c
;
52. At what point on the paraboloid y 苷 x ⫹ z is the tangent
2
53. Are there any points on the hyperboloid x 2 ⫺ y 2 ⫺ z 2 苷 1
where the tangent plane is parallel to the plane z 苷 x ⫹ y?
54. Show that the ellipsoid 3x 2 ⫹ 2y 2 ⫹ z 2 苷 9 and the sphere
x 2 ⫹ y 2 ⫹ z 2 ⫺ 8x ⫺ 6y ⫺ 8z ⫹ 24 苷 0 are tangent to each
other at the point 共1, 1, 2兲. (This means that they have a common tangent plane at the point.)
at a given point in two nonparallel directions given by unit
vectors u and v. Is it possible to find ⵜ f at this point? If so,
how would you do it?
64. Show that if z 苷 f 共x, y兲 is differentiable at x 0 苷 具x 0 , y0 典,
then
lim
x l x0
55. Show that every plane that is tangent to the cone
x 2 ⫹ y 2 苷 z 2 passes through the origin.
14.7
f 共x兲 ⫺ f 共x 0 兲 ⫺ ⵜ f 共x 0 兲 ⴢ 共x ⫺ x 0 兲
苷0
x ⫺ x0
ⱍ
ⱍ
[Hint: Use Definition 14.4.7 directly.]
MAXIMUM AND MINIMUM VALUES
As we saw in Chapter 4, one of the main uses of ordinary derivatives is in finding maximum 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.
A120
61.
63.
65.
69.
87.
93.
||||
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES
15. 7, 2
u x
u y u
u x
u y
u
,
,
苷
苷
17.
r
x r
y r s
x s
y s
u
u x
u y
苷
t
x t
y t
13. 62
12xy, 72xy
24 sin共4x 3y 2z兲, 12 sin共4x 3y 2z兲
e r共2 sin cos r sin 兲
67. 4兾共 y 2z兲 3 , 0
2
2
81. R 兾R 1
⬇12.2, ⬇16.8, ⬇23.25
89. x 苷 1 t, y 苷 2, z 苷 2 2t
No
2
w r
w s
w
苷
x
r x
s x
w r
w s
w
w
苷
y
r y
s y
t
9 9
21. 85, 178, 54
23. 7 , 7
19.
95. (a)
0.2
z 0
_0.2
y 0
1
0
1
_1
x
x 4y 4x 2y 3 y 5
x 5 4x 3y 2 xy 4
, fy共x, y兲 苷
2
2 2
共x y 兲
共x 2 y 2 兲2
(e) No, since fxy and fyx are not continuous.
(b) fx 共x, y兲 苷
EXERCISES 14.4
N
PAGE 899
1. z 苷 8x 2y
3. x y 2z 苷 0
5. z 苷 y
7.
9.
400
EXERCISES 14.6
1
_1
5
11. 2x 4 y 1
43.
_5
0
2
2
x
y
1
19. x y 37.
0
y
13. 9 x 9 y 1
2
15. 1 y
2
3
21. x y 7 z; 6.9914
; 2.846
4T H 329; 129F
dz 苷 3x 2 ln共 y 2 兲 dx 共2x 3兾y兲 dy
dm 苷 5p 4q 3 dp 3p 5q 2 dq
dR 苷 2 cos d 2 cos d 2 sin d 33. 5.4 cm 2
35. 16 cm 3
z 苷 0.9225, dz 苷 0.9
1
39. 17 ⬇ 0.059 41. 2.3%
150
1 苷 x, 2 苷 y
2
3
23.
25.
27.
29.
31.
0
7
3
20
3
3
7
PAGE 920
(b) 具2, 3 典
(c) s3 2
(b) 具1, 12, 0典
(c) 223
9. (a) 具e 2yz, 2xze 2yz, 2xye 2yz 典
11. 23兾10
13. 8兾s10
15. 4兾s30
17. 9兾 (2s5 )
19. 2兾5
21. 4s2, 具1, 1 典
23. 1, 具0, 1典
25. 1, 具3, 6, 2 典
27. (b) 具12, 92典
29. All points on the line y 苷 x 1
31. (a) 40兾(3 s3 )
327
(b) 具38, 6, 12 典
(c) 2 s406
33. (a) 32兾s3
35. 13
39. (a) x y z 苷 11
(b) x 3 苷 y 3 苷 z 5
y1
z1
x2
41. (a) 4x 5y z 苷 4
苷
苷
(b)
4
5
1
43. (a) x y z 苷 1
(b) x 1 苷 y 苷 z
45.
47. 具2, 3典 , 2x 3y 苷 12
3
0
x 0 _10
N
1. ⬇ 0.08 mb兾km
3. ⬇ 0.778
5. 2 s3兾2
7. (a) f 共x, y兲 苷 具2 cos共2x 3y兲, 3 cos共2x 3y兲典
z 0
z 200
10
sin共x y兲 e y
4共xy兲 3兾2 y
29.
x 2x 2sxy
sin共x y兲 xe y
3yz 2x 3xz 2y
,
31.
2z 3xy 2z 3xy
2 2
z
1y z
,
33.
1 y y 2z 2
1 y y 2z 2
35. 2C兾s
37. ⬇ 0.33 m兾s per minute
(b) 10 m 2兾s (c) 0 m兾s
39. (a) 6 m3兾s
41. ⬇ 0.27 L兾s
43. 1兾 (12 s3 ) rad兾s
45. (a) z兾r 苷 共z兾x兲 cos 共z兾y兲 sin ,
z兾 苷 共z兾x兲r sin 共z兾y兲r cos 51. 4rs 2z兾x 2 共4r 2 4s 2 兲2z兾x y 4rs 2z兾y 2 2 z兾y
27.
_1
(c) 0, 0
w t
,
t x
t
y
25. 36, 24, 30
2
7
6
y
xy=6
EXERCISES 14.5
N
PAGE 907
2
1. 共2x y兲 cos t 共2y x兲e t
3. 关共x兾t兲 y sin t兴兾s1 x 2 y 2
5. e y兾z 关2t 共x兾z兲 共2xy兾z 2 兲兴
7. z兾s 苷 2xy 3 cos t 3x 2 y 2 sin t,
z兾t 苷 2sxy 3 sin t 3sx 2 y 2 cos t
9. z兾s 苷 t 2 cos cos 2st sin sin ,
z兾t 苷 2st cos cos s 2 sin sin 冉
冊
z
s
11.
苷 e r t cos sin ,
s
ss 2 t 2
t
z
sin 苷 e r s cos t
ss 2 t 2
冉
冊
Î
f (3, 2)
2x+3y=12
z 1
(3, 2)
0
0
x
_1
1
x
2
1
y
2
53. No
59. x 苷 1 10t, y 苷 1 16t, z 苷 2 12t
63. If u 苷 具a, b典 and v 苷 具c, d 典 , then afx bfy and c fx d fy are
known, so we solve linear equations for fx and fy .