Assumindo que o limite existe,

SECTION 14.2 LIMITS AND CONTINUITY
14.2
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877
EXERCISES
1. Suppose that lim 共x, y兲 l 共3, 1兲 f 共x, y兲苷 6. What can you say
24.
about the value of f 共3, 1兲? What if f is continuous?
lim
共x, y兲 l 共0, 0兲
xy3
x ⫹ y6
2
2. Explain why each function is continuous or discontinuous.
(a) The outdoor temperature as a function of longitude,
latitude, and time
(b) Elevation (height above sea level) as a function of longitude, latitude, and time
(c) The cost of a taxi ride as a function of distance traveled
and time
25–26 Find h共x, y兲苷 t共 f 共x, y兲兲 and the set on which h is
continuous.
25. t共t兲苷 t 2 ⫹ st ,
26. t共t兲苷 t ⫹ ln t,
f 共x, y兲苷 2 x ⫹ 3y ⫺ 6
f 共x, y兲苷
1 ⫺ xy
1 ⫹ x2y2
3– 4 Use a table of numerical values of f 共x, y兲 for 共x, y兲 near the
origin to make a conjecture about the value of the limit of f 共x, y兲
as 共x, y兲 l 共0, 0兲. Then explain why your guess is correct.
x y ⫹x y ⫺5
2 ⫺ xy
2
3. f 共x, y兲苷
3
3
2
2x y
x 2 ⫹ 2y 2
4. f 共x, y兲苷
; 27–28 Graph the function and observe where it is discontinuous.
Then use the formula to explain what you have observed.
27. f 共x, y兲苷 e 1兾共x⫺y兲
1
1 ⫺ x2 ⫺ y2
28. f 共x, y兲苷
5–22 Find the limit, if it exists, or show that the limit does
29–38 Determine the set of points at which the function is
not exist.
continuous.
5.
7.
9.
11.
13.
lim
共5x ⫺ x y 兲
6.
lim
4 ⫺ xy
x 2 ⫹ 3y 2
8.
lim
y4
x 4 ⫹ 3y 4
10.
lim
x y cos y
3x 2 ⫹ y 2
12.
lim
xy
sx 2 ⫹ y 2
14.
共x, y兲 l 共1, 2兲
共x, y兲 l 共2, 1兲
共x, y兲 l 共0, 0兲
共x, y兲 l 共0, 0兲
共x, y兲 l 共0, 0兲
3
2
15.
2
x ye
x 4 ⫹ 4y 2
lim
x ⫹y
sx 2 ⫹ y 2 ⫹ 1 ⫺ 1
19.
20.
21.
22.
共x, y兲 l 共0, 0兲
⫺xy
16.
lim
x 2 ⫹ 2y 2 ⫹ 3z 2
x 2 ⫹ y 2 ⫹ z2
lim
共x, y, z兲 l 共0, 0, 0兲
lim
共x, y, z兲 l 共0, 0, 0兲
ln
lim
x 2 ⫹ sin 2 y
2x 2 ⫹ y 2
lim
6x 3 y
2x 4 ⫹ y 4
lim
x4 ⫺ y4
x2 ⫹ y2
共x, y兲 l 共1, 0兲
共x, y兲 l 共0, 0兲
共x, y兲 l 共0, 0兲
共x, y兲 l 共0, 0兲
冉
1 ⫹ y2
x 2 ⫹ xy
lim
共x, y兲 l 共0, 0兲
冊
2
x sin y
x 2 ⫹ 2y 2
29. F共x, y兲苷
18.
lim
共x, y兲 l 共0, 0兲
xy
x2 ⫹ y8
x y ⫹ yz 2 ⫹ xz 2
x2 ⫹ y2 ⫹ z4
yz
x 2 ⫹ 4y 2 ⫹ 9z 2
; 23–24 Use a computer graph of the function to explain why the
23.
lim
共x, y兲 l 共0, 0兲
30. F共x, y兲苷
x⫺y
1 ⫹ x2 ⫹ y2
32. F共x, y兲苷 e x y ⫹ sx ⫹ y 2
33. G共x, y兲苷 ln共x 2 ⫹ y 2 ⫺ 4 兲
34. G共x, y兲苷 tan⫺1(共x ⫹ y兲⫺2)
2
sy
x2 ⫺ y2 ⫹ z2
35. f 共x, y, z兲苷
36. f 共x, y, z兲苷 sx ⫹ y ⫹ z
再
再
x2y3
37. f 共x, y兲苷 2 x 2 ⫹ y 2
1
38. f 共x, y兲苷
if 共x, y兲苷共0, 0兲
if 共x, y兲苷共0, 0兲
xy
x2 ⫹ xy ⫹ y2
0
if 共x, y兲苷共0, 0兲
if 共x, y兲苷共0, 0兲
Assumindo que o limite existe,
39– 41 Use polar coordinates to find the limit. [If 共r, ␪ 兲 are
polar coordinates of the point 共x, y兲 with r 艌 0, note that r l 0 ⫹
as 共x, y兲 l 共0, 0兲.]
39.
40.
lim
x3 ⫹ y3
x2 ⫹ y2
lim
共x 2 ⫹ y 2 兲 ln共x 2 ⫹ y 2 兲
lim
e⫺x ⫺y ⫺ 1
x2 ⫹ y2
共x, y兲 l 共0, 0兲
共x, y兲 l 共0, 0兲
limit does not exist.
2x 2 ⫹ 3x y ⫹ 4y 2
3x 2 ⫹ 5y 2
sin共x y兲
e x ⫺ y2
31. F共x, y兲苷 arctan( x ⫹ sy )
4
sin共␲ z兾2兲
e
共x, y, z兲 l 共0, 0, 0兲
lim
2
lim
共x, y, z兲 l 共3, 0, 1兲
cos共x ⫹ y兲
e
2
lim
共x, y兲 l 共0, 0兲
⫺xy
lim
共x, y兲 l 共1, ⫺1兲
y
2
17.
2
2
41.
共x, y兲 l 共0, 0兲
2
14.3
SECTION 14.3 PARTIAL DERIVATIVES
5– 8 Determine the signs of the partial derivatives for the function
||||
889
10. A contour map is given for a function f . Use it to estimate
fx 共2, 1兲 and fy 共2, 1兲.
f whose graph is shown.
y
z
3
_4
0
6
_2
1
x
8
10
12
14
16
4
2
2
y
1
5. (a) fx 共1, 2兲
(b) fy 共1, 2兲
6. (a) fx 共1, 2兲
(b) fy 共1, 2兲
7. (a) fxx 共1, 2兲
(b) fyy 共1, 2兲
8. (a) fxy 共1, 2兲
(b) fxy 共1, 2兲
3
x
18
11. If f 共x, y兲苷 16 4x 2 y 2, find fx 共1, 2兲 and fy 共1, 2兲 and inter-
pret these numbers as slopes. Illustrate with either hand-drawn
sketches or computer plots.
12. If f 共x, y兲苷 s4 x 2 4y 2 , find fx 共1, 0兲 and fy 共1, 0兲 and
interpret these numbers as slopes. Illustrate with either handdrawn sketches or computer plots.
9. The following surfaces, labeled a, b, and c, are graphs of a
function f and its partial derivatives fx and fy . Identify each
surface and give reasons for your choices.
; 13–14 Find fx and fy and graph f , fx , and fy with domains and
viewpoints that enable you to see the relationships between them.
13. f 共x, y兲苷 x 2 y 2 x 2 y
14. f 共x, y兲苷 xex
y 2
2
15–38 Find the first partial derivatives of the function.
8
4
15. f 共x, y兲苷 y 5 3xy
16. f 共x, y兲苷 x 4 y 3 8x 2 y
z 0
17. f 共x, t兲苷 et cos x
18. f 共x, t兲苷 sx ln t
19. z 苷共2x 3y兲
20. z 苷 tan xy
_4
_8
_3 _2 _1
10
a
0
y
1
2
3
_2
0
x
2
21. f 共x, y兲苷
xy
xy
22. f 共x, y兲苷 x y
23. w 苷 sin cos 24. w 苷 e v兾共u v 2 兲
25. f 共r, s兲苷 r ln共r 2 s 2 兲
26. f 共x, t兲苷 arctan ( x st )
27. u 苷 te w兾t
28. f 共x, y兲苷
29. f 共x, y, z兲苷 xz 5x 2 y 3z 4
30. f 共x, y, z兲苷 x sin共 y z兲
31. w 苷 ln共x 2y 3z兲
32. w 苷 ze xyz
33. u 苷 xy sin1共 yz兲
34. u 苷 x y兾z
35. f 共x, y, z, t兲苷 x yz 2 tan共 yt兲
36. f 共x, y, z, t兲苷
4
z 0
b
_4
_3 _2 _1
0
y
1
2
3
2
0
_2
x
37. u 苷 sx 12 x 22 x n2
38. u 苷 sin共x 1 2x 2 nx n 兲
8
4
39– 42 Find the indicated partial derivatives.
z 0
39. f 共x, y兲苷 ln ( x sx 2 y 2 );
_4
_8
_3 _2 _1
c
0
y
1
2
3
2
0
_2
x
40. f 共x, y兲苷 arctan共 y兾x兲;
fx 共3, 4兲
fx 共2, 3兲
y
; fy 共2, 1, 1兲
41. f 共x, y, z兲苷
xyz
y
x
y
cos共t 2 兲 dt
xy2
t 2z
890
||||
CHAPTER 14 PARTIAL DERIVATIVES
42. f 共x, y, z兲苷 ssin 2 x sin 2 y sin 2 z ;
fz 共0, 0, 兾4兲
69. Use the table of values of f 共x, y兲 to estimate the values of
fx 共3, 2兲, fx 共3, 2.2兲, and fx y 共3, 2兲.
43– 44 Use the definition of partial derivatives as limits (4) to find
44. f 共x, y兲苷
43. f 共x, y兲苷 xy 2 x 3y
y
1.8
2.0
2.2
2.5
12. 5
10. 2
9.3
3.0
18. 1
17. 5
15. 9
3.5
20. 0
22. 4
26. 1
x
fx 共x, y兲 and fy 共x, y兲.
x
x y2
45– 48 Use implicit differentiation to find z兾x and z兾y.
45. x 2 y 2 z 2 苷 3x yz
46. yz 苷 ln共x z兲
47. x z 苷 arctan共 yz兲
48. sin共x yz兲苷 x 2y 3z
49–50 Find z兾x and z兾y.
70. Level curves are shown for a function f . Determine whether
the following partial derivatives are positive or negative at the
point P.
(a) fx
(b) fy
(c) fxx
(d) fxy
(e) fyy
y
49. (a) z 苷 f 共x兲 t共 y兲
(b) z 苷 f 共x y兲
50. (a) z 苷 f 共x兲 t共 y兲
(b) z 苷 f 共x y兲
10 8
(c) z 苷 f 共x兾y兲
6
4
2
P
51–56 Find all the second partial derivatives.
x
51. f 共x, y兲苷 x 3 y 5 2x 4 y
52. f 共x, y兲苷 sin 2 共mx ny兲
53. w 苷 su 2 v 2
54. v 苷
55. z 苷 arctan
xy
1 xy
xy
xy
56. v 苷 e xe
71. Verify that the function u 苷 e
2 2
k t
sin kx is a solution of the
heat conduction equation u t 苷 2u xx .
72. Determine whether each of the following functions is a solution
y
57–60 Verify that the conclusion of Clairaut’s Theorem holds, that
is, u x y 苷 u yx .
57. u 苷 x sin共x 2y兲
58. u 苷 x 4 y 2 2xy 5
59. u 苷 ln sx 2 y 2
60. u 苷 x ye
y
of Laplace’s equation u xx u yy 苷 0 .
(a) u 苷 x 2 y 2
(b) u 苷 x 2 y 2
3
2
(c) u 苷 x 3xy
(d) u 苷 ln sx 2 y 2
(e) u 苷 sin x cosh y cos x sinh y
(f) u 苷 ex cos y ey cos x
73. Verify that the function u 苷 1兾sx 2 y 2 z 2 is a solution of
the three-dimensional Laplace equation u xx u yy u zz 苷 0 .
74. Show that each of the following functions is a solution of the
wave equation u t t 苷 a 2u xx .
(a) u 苷 sin共k x兲 sin共ak t兲
(b) u 苷 t兾共a 2t 2 x 2 兲
(c) u 苷共x at兲6 共x at兲6
(d) u 苷 sin共x at兲 ln共x at兲
61–68 Find the indicated partial derivative.
61. f 共x, y兲苷 3x y 4 x 3 y 2;
2 ct
62. f 共x, t兲苷 x e
;
fttt ,
fxxy ,
fyyy
ftxx
75. If f and t are twice differentiable functions of a single vari-
63. f 共x, y, z兲苷 cos共4x 3y 2z兲;
64. f 共r, s, t兲苷 r ln共rs 2 t 3 兲;
frss ,
67. w 苷
x
;
y 2z
68. u 苷 x a y bz c;
able, show that the function
u共x, t兲苷 f 共x at兲 t共x at兲
is a solution of the wave equation given in Exercise 74.
76. If u 苷 e a1 x1a2 x2an x n, where a 12 a 22 a n2 苷 1,
show that
z
u v w
3
66. z 苷 us v w ;
fyzz
frst
3u
r 2 65. u 苷 e r sin ;
fxy z ,
3w
,
z y x
6u
x y 2 z 3
3w
x 2 y
2u
2u
2u
苷u
2 2 x1
x 2
x n2
77. Verify that the function z 苷 ln共e x e y 兲 is a solution of the
differential equations
z
z
苷1
x
y
SECTION 14.3 PARTIAL DERIVATIVES
and
2z 2 z
x 2 y 2
冉冊
2z
x y
are fx 共x, y兲苷 x 4y and fy 共x, y兲苷 3x y. Should you
believe it?
苷0
2
2
; 88. The paraboloid z 苷 6 x x 2y intersects the plane
x 苷 1 in a parabola. Find parametric equations for the tangent
line to this parabola at the point 共1, 2, 4兲. Use a computer to
graph the paraboloid, the parabola, and the tangent line on the
same screen.
satisfies the equation
P
P
K
苷共兲P
L
K
89. The ellipsoid 4x 2 2y 2 z 2 苷 16 intersects the plane y 苷 2
79. Show that the Cobb-Douglas production function satisfies
P共L, K0 兲苷 C1共K0 兲L by solving the differential equation
in an ellipse. Find parametric equations for the tangent line to
this ellipse at the point 共1, 2, 2兲.
P
dP
苷
dL
L
90. In a study of frost penetration it was found that the temperature
(See Equation 5.)
T at time t (measured in days) at a depth x (measured in feet)
can be modeled by the function
80. The temperature at a point 共x, y兲 on a flat metal plate is given
by T共x, y兲苷 60兾共1 x 2 y 2 兲, where T is measured in C
and x, y in meters. Find the rate of change of temperature with
respect to distance at the point 共2, 1兲 in (a) the x-direction and
(b) the y-direction.
T共x, t兲苷 T0 T1 e x sin共 t x兲
81. The total resistance R produced by three conductors with resis-
tances R1 , R2 , R3 connected in a parallel electrical circuit is
given by the formula
;
1
1
1
1
苷
R
R1
R2
R3
Find R兾R1.
where 苷 2兾365 and is a positive constant.
(a) Find T兾x. What is its physical significance?
(b) Find T兾t. What is its physical significance?
(c) Show that T satisfies the heat equation Tt 苷 kTxx for a certain constant k.
(d) If 苷 0.2, T0 苷 0, and T1 苷 10, use a computer to
graph T共x, t兲.
(e) What is the physical significance of the term x in the
expression sin共 t x兲?
91. Use Clairaut’s Theorem to show that if the third-order partial
82. The gas law for a fixed mass m of an ideal gas at absolute tem-
derivatives of f are continuous, then
perature T, pressure P, and volume V is PV 苷 mRT, where R is
the gas constant. Show that
fx yy 苷 fyx y 苷 fyyx
P V T
苷 1
V T P
92. (a) How many nth-order partial derivatives does a function of
two variables have?
(b) If these partial derivatives are all continuous, how many of
them can be distinct?
(c) Answer the question in part (a) for a function of three
variables.
83. For the ideal gas of Exercise 82, show that
T
891
87. You are told that there is a function f whose partial derivatives
2
78. Show that the Cobb-Douglas production function P 苷 bLK L
||||
P V
苷 mR
T T
93. If f 共x, y兲苷 x共x 2 y 2 兲3兾2e sin共x y兲, find fx 共1, 0兲.
84. The wind-chill index is modeled by the function
2
[Hint: Instead of finding fx 共x, y兲 first, note that it’s easier to
use Equation 1 or Equation 2.]
W 苷 13.12 0.6215T 11.37v 0.16 0.3965T v 0.16
where T is the temperature 共C兲 and v is the wind speed
共km兾h兲. When T 苷 15C and v 苷 30 km兾h, by how much
would you expect the apparent temperature W to drop if the
actual temperature decreases by 1C ? What if the wind speed
increases by 1 km兾h ?
3
x 3 y 3 , find fx 共0, 0兲.
94. If f 共x, y兲苷 s
95. Let
85. The kinetic energy of a body with mass m and velocity v is
K 苷 12 mv 2. Show that
K 2K
苷K
m v 2
;
86. If a, b, c are the sides of a triangle and A, B, C are the opposite
angles, find A兾a, A兾b, A兾c by implicit differentiation of
the Law of Cosines.
再
x 3y xy 3
x2 y2
f 共x, y兲苷
0
CAS
(a)
(b)
(c)
(d)
(e)
if 共x, y兲苷共0, 0兲
if 共x, y兲苷共0, 0兲
Use a computer to graph f .
Find fx 共x, y兲 and fy 共x, y兲 when 共x, y兲苷共0, 0兲.
Find fx 共0, 0兲 and fy 共0, 0兲 using Equations 2 and 3.
Show that fxy 共0, 0兲苷 1 and fyx 共0, 0兲苷 1.
Does the result of part (d) contradict Clairaut’s Theorem?
Use graphs of fxy and fyx to illustrate your answer.
SECTION 14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS
1–6 Find an equation of the tangent plane to the given surface at
1. z 苷 4x 2 ⫺ y 2 ⫹ 2y,
共⫺1, 2, 4兲
2. z 苷 3共x ⫺ 1兲 ⫹ 2共 y ⫹ 3兲2 ⫹ 7,
2
3. z 苷 sxy ,
f 共x, y兲苷 ln共x ⫺ 3y兲 at 共7, 2兲 and use it to approximate
f 共6.9, 2.06兲. Illustrate by graphing f and the tangent plane.
共1, 4, 0兲
5. z 苷 y cos共x ⫺ y兲,
6. z 苷 e x ⫺y ,
2
; 20. Find the linear approximation of the function
共2, ⫺2, 12兲
共1, 1, 1兲
4. z 苷 y ln x,
2
21. Find the linear approximation of the function
f 共x, y, z兲苷 sx 2 ⫹ y 2 ⫹ z 2 at 共3, 2, 6兲 and use it to
approximate the number s共3.02兲 2 ⫹ 共1.97兲 2 ⫹ 共5.99兲 2 .
共2, 2, 2兲
共1, ⫺1, 1兲
22. The wave heights h in the open sea depend on the speed v
; 7– 8 Graph the surface and the tangent plane at the given point.
(Choose the domain and viewpoint so that you get a good view of
both the surface and the tangent plane.) Then zoom in until the
surface and the tangent plane become indistinguishable.
7. z 苷 x 2 ⫹ xy ⫹ 3y 2,
8. z 苷 arctan共xy 2 兲,
of the wind and the length of time t that the wind has been
blowing at that speed. Values of the function h 苷 f 共v, t兲 are
recorded in feet in the following table.
Duration (hours)
共1, 1, ␲兾4兲
point. (Use your computer algebra system both to compute the
partial derivatives and to graph the surface and its tangent plane.)
Then zoom in until the surface and the tangent plane become
indistinguishable.
xy sin共x ⫺ y兲
,
1 ⫹ x2 ⫹ y2
t
v
共1, 1, 5兲
9–10 Draw the graph of f and its tangent plane at the given
9. f 共x, y兲苷
19. Find the linear approximation of the function
f 共x, y兲苷 s20 ⫺ x 2 ⫺ 7y 2 at 共2, 1兲 and use it to
approximate f 共1.95, 1.08兲.
the specified point.
10. f 共x, y兲苷 e⫺xy兾10 (sx ⫹ sy ⫹ sxy ),
5
10
15
20
30
40
50
20
5
7
8
8
9
9
9
30
9
13
16
17
18
19
19
40
14
21
25
28
31
33
33
50
19
29
36
40
45
48
50
60
24
37
47
54
62
67
69
Use the table to find a linear approximation to the wave
height function when v is near 40 knots and t is near
20 hours. Then estimate the wave heights when the wind has
been blowing for 24 hours at 43 knots.
共1, 1, 0兲
共1, 1, 3e⫺0.1兲
23. Use the table in Example 3 to find a linear approximation to
the heat index function when the temperature is near 94⬚F
and the relative humidity is near 80%. Then estimate the heat
index when the temperature is 95⬚F and the relative humidity
is 78%.
11–16 Explain why the function is differentiable at the given
point. Then find the linearization L共x, y兲 of the function at
that point.
11. f 共x, y兲苷 x sy ,
12. f 共x, y兲苷 x y ,
3
13. f 共x, y兲苷
4
共1, 4兲
24. The wind-chill index W is the perceived temperature when the
actual temperature is T and the wind speed is v, so we can
write W 苷 f 共T, v兲. The following table of values is an excerpt
共1, 1兲
x
, 共2, 1兲
x⫹y
14. f 共x, y兲苷 sx ⫹ e 4y ,
⫺xy
15. f 共x, y兲苷 e
cos y,
from Table 1 in Section 14.1.
Wind speed (km/h)
共3, 0兲
Actual temperature (°C)
CAS
899
EXERCISES
Wind speed (knots)
14.4
||||
共␲, 0兲
16. f 共x, y兲苷 sin共2 x ⫹ 3y兲,
共⫺3, 2兲
17–18 Verify the linear approximation at 共0, 0兲.
17.
2x ⫹ 3
⬇ 3 ⫹ 2x ⫺ 12y
4y ⫹ 1
v
20
30
40
50
60
70
⫺10
⫺18
⫺20
⫺21
⫺22
⫺23
⫺23
⫺15
⫺24
⫺26
⫺27
⫺29
⫺30
⫺30
⫺20
⫺30
⫺33
⫺34
⫺35
⫺36
⫺37
⫺25
⫺37
⫺39
⫺41
⫺42
⫺43
⫺44
T
18. sy ⫹ cos 2 x ⬇ 1 ⫹ 2 y
1
Use the table to find a linear approximation to the wind-chill
900
||||
CHAPTER 14 PARTIAL DERIVATIVES
index function when T is near ⫺15⬚C and v is near 50 km兾h.
Then estimate the wind-chill index when the temperature is
⫺17⬚C and the wind speed is 55 km兾h.
39. If R is the total resistance of three resistors, connected in par-
allel, with resistances R1 , R2 , R3 , then
1
1
1
1
苷
⫹
⫹
R
R1
R2
R3
25–30 Find the differential of the function.
25. z 苷 x 3 ln共 y 2 兲
26. v 苷 y cos xy
27. m 苷 p 5q 3
28. T 苷
29. R 苷 ␣␤ 2 cos ␥
30. w 苷 xye xz
v
1 ⫹ u vw
31. If z 苷 5x 2 ⫹ y 2 and 共x, y兲 changes from 共1, 2兲 to 共1.05, 2.1兲,
compare the values of ⌬z and dz.
32. If z 苷 x 2 ⫺ xy ⫹ 3y 2 and 共x, y兲 changes from 共3, ⫺1兲 to
共2.96, ⫺0.95兲, compare the values of ⌬z and dz.
33. The length and width of a rectangle are measured as 30 cm and
24 cm, respectively, with an error in measurement of at most
0.1 cm in each. Use differentials to estimate the maximum
error in the calculated area of the rectangle.
34. The dimensions of a closed rectangular box are measured as
80 cm, 60 cm, and 50 cm, respectively, with a possible error
of 0.2 cm in each dimension. Use differentials to estimate the
maximum error in calculating the surface area of the box.
35. Use differentials to estimate the amount of tin in a closed tin
can with diameter 8 cm and height 12 cm if the tin is 0.04 cm
thick.
36. Use differentials to estimate the amount of metal in a closed
cylindrical can that is 10 cm high and 4 cm in diameter if the
metal in the top and bottom is 0.1 cm thick and the metal in the
sides is 0.05 cm thick.
If the resistances are measured in ohms as R1 苷 25 ⍀,
R2 苷 40 ⍀, and R3 苷 50 ⍀, with a possible error of 0.5% in
each case, estimate the maximum error in the calculated value
of R.
40. Four positive numbers, each less than 50, are rounded to the
first decimal place and then multiplied together. Use differentials to estimate the maximum possible error in the computed
product that might result from the rounding.
41. A model for the surface area of a human body is given by
S 苷 0.1091w 0.425 h 0.725, where w is the weight (in pounds), h is
the height (in inches), and S is measured in square feet. If the
errors in measurement of w and h are at most 2%, use differentials to estimate the maximum percentage error in the
calculated surface area.
42. Suppose you need to know an equation of the tangent plane to
a surface S at the point P共2, 1, 3兲. You don’t have an equation
for S but you know that the curves
r1共t兲苷具2 ⫹ 3t, 1 ⫺ t 2, 3 ⫺ 4t ⫹ t 2 典
r2共u兲苷具1 ⫹ u 2, 2u 3 ⫺ 1, 2u ⫹ 1 典
both lie on S. Find an equation of the tangent plane at P.
43– 44 Show that the function is differentiable by finding values
of ␧1 and ␧2 that satisfy Definition 7.
43. f 共x, y兲苷 x 2 ⫹ y 2
45. Prove that if f is a function of two variables that is differen-
tiable at 共a, b兲, then f is continuous at 共a, b兲.
Hint: Show that
lim
共⌬x, ⌬y兲 l 共0, 0兲
37. A boundary stripe 3 in. wide is painted around a rectangle
whose dimensions are 100 ft by 200 ft. Use differentials to
approximate the number of square feet of paint in the stripe.
46. (a) The function
f 共x, y兲苷
38. The pressure, volume, and temperature of a mole of an ideal
gas are related by the equation PV 苷 8.31T , where P is measured in kilopascals, V in liters, and T in kelvins. Use differentials to find the approximate change in the pressure if the
volume increases from 12 L to 12.3 L and the temperature
decreases from 310 K to 305 K.
44. f 共x, y兲苷 xy ⫺ 5y 2
f 共a ⫹ ⌬x, b ⫹ ⌬y兲苷 f 共a, b兲
再
xy
x2 ⫹ y2
0
if 共x, y兲苷共0, 0兲
if 共x, y兲苷共0, 0兲
was graphed in Figure 4. Show that fx 共0, 0兲 and fy 共0, 0兲
both exist but f is not differentiable at 共0, 0兲. [Hint: Use
the result of Exercise 45.]
(b) Explain why fx and fy are not continuous at 共0, 0兲.
SECTION 14.5 THE CHAIN RULE
EXAMPLE 9 Find
||||
907
z
z
and
if x 3 y 3 z 3 6 xyz 苷 1.
x
y
SOLUTION Let F共x, y, z兲苷 x 3 y 3 z 3 6xyz 1. Then, from Equations 7, we have
Fx
x 2 2yz
z
3x 2 6yz
苷
苷 2
苷 2
x
Fz
3z 6xy
z 2xy
The solution to Example 9 should be
compared to the one in Example 4 in
Section 14.3.
N
14.5
z
3y 2 6xz
Fy
y 2 2xz
苷 2
苷
苷 2
y
Fz
3z 6xy
z 2xy
M
EXERCISES
1–6 Use the Chain Rule to find dz兾dt or dw兾dt.
1. z 苷 x y xy,
2
x 苷 sin t,
2
2. z 苷 cos共x 4y兲,
5. w 苷 xe y兾z,
14. Let W共s, t兲苷 F共u共s, t兲, v共s, t兲兲, where F, u, and v are
differentiable, and
u共1, 0兲苷 2
v共1, 0兲苷 3
us共1, 0兲苷 2
vs共1, 0兲苷 5
u t 共1, 0兲苷 6
vt 共1, 0兲苷 4
Fu共2, 3兲苷 1
Fv共2, 3兲苷 10
x 苷 5t 4, y 苷 1兾t
3. z 苷 s1 x 2 y 2 ,
4. z 苷 tan1共 y兾x兲,
y苷e
t
x 苷 ln t,
x 苷 e t,
x 苷 t 2,
y 苷 cos t
y 苷 1 et
y 苷 1 t,
6. w 苷 ln sx 2 y 2 z 2 ,
z 苷 1 2t
x 苷 sin t, y 苷 cos t, z 苷 tan t
Find Ws 共1, 0兲 and Wt 共1, 0兲.
15. Suppose f is a differentiable function of x and y, and
t共u, v兲苷 f 共e u sin v, e u cos v兲. Use the table of values
7–12 Use the Chain Rule to find z兾s and z兾t.
7. z 苷 x y ,
2
3
x 苷 s cos t,
8. z 苷 arcsin共x y兲,
9. z 苷 sin cos ,
10. z 苷 e x2y,
y 苷 s sin t
x苷s t ,
2
2
y 苷 1 2st
苷 st 2, 苷 s 2 t
x 苷 s兾t,
y 苷 t兾s
苷 ss 2 t 2
r 苷 st,
12. z 苷 tan共u兾v兲,
u 苷 2s 3t, v 苷 3s 2t
13. If z 苷 f 共x, y兲, where f is differentiable, and
x 苷 t共t兲
y 苷 h共t兲
t共3兲苷 2
h共3兲苷 7
t共3兲苷 5
h共3兲苷 4
fx 共2, 7兲苷 6
fy 共2, 7兲苷 8
find dz兾dt when t 苷 3.
f
t
fx
fy
共0, 0兲
3
6
4
8
共1, 2兲
6
3
2
5
16. Suppose f is a differentiable function of x and y, and
11. z 苷 e cos ,
r
to calculate tu共0, 0兲 and tv共0, 0兲.
t共r, s兲苷 f 共2r s, s 2 4r兲. Use the table of values in
Exercise 15 to calculate tr 共1, 2兲 and ts 共1, 2兲.
17–20 Use a tree diagram to write out the Chain Rule for the given
case. Assume all functions are differentiable.
17. u 苷 f 共x, y兲,
where x 苷 x共r, s, t兲, y 苷 y共r, s, t兲
18. R 苷 f 共x, y, z, t兲, where x 苷 x共u, v, w兲, y 苷 y共u, v, w兲,
z 苷 z共u, v, w兲, t 苷 t共u, v, w兲
19. w 苷 f 共r, s, t兲,
where r 苷 r共x, y兲, s 苷 s共x, y兲, t 苷 t共x, y兲
20. t 苷 f 共u, v, w兲, where u 苷 u共 p, q, r, s兲, v 苷 v共 p, q, r, s兲,
w 苷 w共 p, q, r, s兲
908
||||
CHAPTER 14 PARTIAL DERIVATIVES
21–26 Use the Chain Rule to find the indicated partial derivatives.
21. z 苷 x xy ,
2
x 苷 uv w ,
3
2
z z z
,
,
u v w
u u u
,
,
x y t
C 苷 1449.2 4.6T 0.055T 2 0.00029T 3 0.016D
where C is the speed of sound (in meters per second), T is the
temperature (in degrees Celsius), and D is the depth below the
ocean surface (in meters). A scuba diver began a leisurely dive
into the ocean water; the diver’s depth and the surrounding
water temperature over time are recorded in the following
graphs. Estimate the rate of change (with respect to time) of
the speed of sound through the ocean water experienced by the
diver 20 minutes into the dive. What are the units?
r 苷 y x cos t, s 苷 x y sin t ;
when x 苷 1, y 苷 2, t 苷 0
23. R 苷 ln共u 2 v 2 w 2 兲,
u 苷 x 2y, v 苷 2x y,
w 苷 2xy;
when x 苷 y 苷 1
2
24. M 苷 xe yz ,
M M
,
u v
w
when u 苷 2, v 苷 1, w 苷 0
22. u 苷 sr 2 s 2 ,
R R
,
x y
37. The speed of sound traveling through ocean water with salinity
35 parts per thousand has been modeled by the equation
y 苷 u ve ;
3
x 苷 2u v,
y 苷 u v,
z 苷 u v;
when u 苷 3, v 苷 1
x 苷 pr cos ,
25. u 苷 x yz,
2
u u u
,
,
p r y 苷 pr sin ,
z 苷 p r;
Y Y Y
,
,
r s t
u 苷 r s,
v 苷 s t,
w 苷 t r;
when r 苷 1, s 苷 0, t 苷 1
27. sxy 苷 1 x y
28. y x y 苷 1 ye
29. cos共x y兲苷 xe
5
y
20
14
15
12
10
10
5
8
20
30
40
t
(min)
10
20
30
40 t
(min)
38. The radius of a right circular cone is increasing at a rate of
1.8 in兾s while its height is decreasing at a rate of 2.5 in兾s. At
what rate is the volume of the cone changing when the radius is
120 in. and the height is 140 in.?
39. The length 艎, width w, and height h of a box change with
27–30 Use Equation 6 to find dy兾dx.
2
T
16
10
when p 苷 2, r 苷 3, 苷 0
26. Y 苷 w tan1共u v兲,
D
2
3
x2
30. sin x cos y 苷 sin x cos y
31–34 Use Equations 7 to find z兾x and z兾y.
31. x 2 y 2 z 2 苷 3x yz
32. x yz 苷 cos共x y z兲
33. x z 苷 arctan共 yz兲
34. yz 苷 ln共x z兲
35. The temperature at a point 共x, y兲 is T共x, y兲, measured in degrees
Celsius. A bug crawls so that its position after t seconds is
given by x 苷 s1 t , y 苷 2 13 t, where x and y are measured
in centimeters. The temperature function satisfies Tx 共2, 3兲苷 4
and Ty 共2, 3兲苷 3. How fast is the temperature rising on the
bug’s path after 3 seconds?
36. Wheat production W in a given year depends on the average
temperature T and the annual rainfall R. Scientists estimate
that the average temperature is rising at a rate of 0.15°C兾year
and rainfall is decreasing at a rate of 0.1 cm兾year. They also
estimate that, at current production levels, W兾T 苷 2
and W兾R 苷 8.
(a) What is the significance of the signs of these partial
derivatives?
(b) Estimate the current rate of change of wheat production,
dW兾dt.
time. At a certain instant the dimensions are 艎苷 1 m and
w 苷 h 苷 2 m, and 艎 and w are increasing at a rate of 2 m兾s
while h is decreasing at a rate of 3 m兾s. At that instant find the
rates at which the following quantities are changing.
(a) The volume
(b) The surface area
(c) The length of a diagonal
40. The voltage V in a simple electrical circuit is slowly decreasing
as the battery wears out. The resistance R is slowly increasing
as the resistor heats up. Use Ohm’s Law, V 苷 IR, to find how
the current I is changing at the moment when R 苷 400 ,
I 苷 0.08 A, dV兾dt 苷 0.01 V兾s, and dR兾dt 苷 0.03 兾s.
41. The pressure of 1 mole of an ideal gas is increasing at a rate
of 0.05 kPa兾s and the temperature is increasing at a rate of
0.15 K兾s. Use the equation in Example 2 to find the rate of
change of the volume when the pressure is 20 kPa and the
temperature is 320 K.
42. Car A is traveling north on Highway 16 and car B is traveling
west on Highway 83. Each car is approaching the intersection
of these highways. At a certain moment, car A is 0.3 km from
the intersection and traveling at 90 km兾h while car B is 0.4 km
from the intersection and traveling at 80 km兾h. How fast is the
distance between the cars changing at that moment?
43. One side of a triangle is increasing at a rate of 3 cm兾s and a
second side is decreasing at a rate of 2 cm兾s. If the area of the
APPENDIX I ANSWERS TO ODD-NUMBERED EXERCISES
EXERCISES 14.2
N
5. 1
7.
2
7
9. Does not exist
13. 0
15. Does not exist
21. Does not exist
3. 5
2
11. Does not exist
17. 2
19. 1
10
0
_2
x
25. h共x, y兲苷共2 x 3y 6兲 2 s2x 3y 6 ;
ⱍ
兵共x, y兲 2x 3y 6其
ⱍ
ⱍ
33. 兵共x, y兲 x 2 y 2 4其
ⱍ
ⱍ
_2
0
_2
0
2
y
fx
_10
_2
41. 1
39. 0
2
z 0
35. 兵共x, y, z兲 y 0, y 苷 sx 2 z 2 其
37. 兵共x, y兲共x, y兲苷共0, 0兲其
0
10
29. 兵共x, y兲 y 苷 e x兾2 其
ⱍ
31. 兵共x, y兲 y 0其
f
z
23. The graph shows that the function approaches different numbers along different lines.
27. Along the line y 苷 x
A119
13. fx 苷 2x 2xy, fy 苷 2y x 2
PAGE 877
1. Nothing; if f is continuous, f 共3, 1兲苷 6
||||
x
0
2
2
y
43.
z
2
1
z 0
fy
0
_1
_2
_2
y
0
f is continuous on ⺢
EXERCISES 14.3
N
0x
2
2
_2
x
2
PAGE 888
1. (a) The rate of change of temperature as longitude varies, with
latitude and time fixed; the rate of change as only latitude varies;
the rate of change as only time varies.
(b) Positive, negative, positive
3. (a) fT 共15, 30兲 ⬇ 1.3; for a temperature of 15C and wind
speed of 30 km兾h, the wind-chill index rises by 1.3C for each
degree the temperature increases. fv 共15, 30兲 ⬇ 0.15; for a
temperature of 15C and wind speed of 30 km兾h, the wind-chill
index decreases by 0.15C for each km兾h the wind speed
increases.
(b) Positive, negative (c) 0
(b) Negative
5. (a) Positive
7. (a) Positive
(b) Negative
9. c 苷 f, b 苷 fx, a 苷 fy
11. fx 共1, 2兲苷 8 苷 slope of C1 , fy共1, 2兲苷 4 苷 slope of C2
z
16
16
(1, 2, 8)
(1, 2, 8)
C¡
0
x
0
4
2
y
(1, 2)
C™
4
2
x
y
(1, 2)
2
_2
0
2
y
fx 共x, y兲苷 3y, fy 共x, y兲苷 5y 4 3x
fx 共x, t兲苷 e t sin x, ft 共x, t兲苷 et cos x
z兾x 苷 20共2x 3y兲 9, z兾y 苷 30共2x 3y兲 9
fx 共x, y兲苷 2y兾共x y兲2, fy共x, y兲苷 2x兾共x y兲2
w兾苷 cos cos , w兾苷 sin sin 2r 2
2rs
ln共r 2 s 2 兲, fs共r, s兲苷 2
25. fr共r, s兲苷 2
r s2
r s2
27. u兾t 苷 e w兾t (1 w兾t), u兾w 苷 e w兾t
29. fx 苷 z 10xy 3z 4, fy 苷 15x 2 y 2z 4, fz 苷 x 20x 2 y 3z 3
31. w兾x 苷 1兾共x 2y 3z兲, w兾y 苷 2兾共x 2y 3z兲,
w兾z 苷 3兾共x 2y 3z兲
33. u兾x 苷 y sin1 共 yz兲, u兾y 苷 x sin1 共 yz兲 xyz兾s1 y 2 z 2,
u兾z 苷 xy 2兾s1 y 2 z 2
35. fx 苷 yz 2 tan共 yt兲, fy 苷 xyz 2 t sec 2共 yt兲 xz 2 tan共 yt兲,
fz 苷 2xyz tan共 yt兲, ft 苷 xy 2z 2 sec 2共 yt兲
37. u兾xi 苷 xi兾sx 12 x 22 x n2
15.
17.
19.
21.
23.
39.
1
5
41.
1
4
43. fx 共x, y兲苷 y 2 3x 2 y , fy 共x, y兲苷 2xy x 3
z
3yz 2x z
3xz 2y
,
苷
苷
x
2z 3xy y
2z 3xy
z
1 y 2z 2
z
z
47.
,
苷
苷
x
1 y y 2z 2 y
1 y y 2z 2
(b) f 共x y兲, f 共x y兲
49. (a) f 共x兲, t共 y兲
51. fxx 苷 6xy 5 24x 2 y, fxy 苷 15x 2 y 4 8x 3 苷 fyx , fyy 苷 20x 3 y 3
53. wuu 苷 v 2兾共u 2 v 2 兲3兾2, wuv 苷 uv兾共u 2 v 2 兲3兾2 苷 wvu,
wvv 苷 u 2兾共u 2 v 2 兲3兾2
55. zxx 苷 2x兾共1 x 2 兲 2, zxy 苷 0 苷 zyx , zyy 苷 2y兾共1 y 2 兲 2
45.
z
0
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 .

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