James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

984 chapter 14 Partial Derivatives
42. If cossxyzd − 1 1 x 2 y 2 1 z 2 , find −z −z
and
−x −y .
43. Find the gradient of the function f sx, y, zd − x 2 e yz 2 .
44. (a) When is the directional derivative of f a maximum?
(b) When is it a minimum?
(c) When is it 0?
(d) When is it half of its maximum value?
45–46 Find the directional derivative of f at the given point in
the indicated direction.
45. f sx, yd − x 2 e 2y , s22, 0d,
in the direction toward the point s2, 23d
46. f sx, y, zd − x 2 y 1 xs1 1 z , s1, 2, 3d,
in the direction of v − 2i 1 j 2 2k
47. Find the maximum rate of change of f sx, yd − x 2 y 1 sy
at the point s2, 1d. In which direction does it occur?
48. Find the direction in which f sx, y, zd − ze x y increases most
rapidly at the point s0, 1, 2d. What is the maximum rate of
increase?
49. The contour map shows wind speed in knots during Hurricane
Andrew on August 24, 1992. Use it to estimate the value
of the directional derivative of the wind speed at Homestead,
Florida, in the direction of the eye of the hurricane.
;
;
52. f sx, yd − x 3 2 6xy 1 8y 3
53. f sx, yd − 3xy 2 x 2 y 2 xy 2
54. f sx, yd − sx 2 1 yde yy2
55–56 Find the absolute maximum and minimum values of f on
the set D.
55. f sx, yd − 4xy 2 2 x 2 y 2 2 xy 3 ; D is the closed triangular
region in the xy-plane with vertices s0, 0d, s0, 6d, and s6, 0d
56. f sx, yd − e 2x 2 2y 2 sx 2 1 2y 2 d; D is the disk x 2 1 y 2 < 4
57. Use a graph or level curves or both to estimate the local
maximum and minimum values and saddle points of
f sx, yd − x 3 2 3x 1 y 4 2 2y 2 . Then use calculus to find
these values precisely.
58. Use a graphing calculator or computer (or Newton’s method
or a computer algebra system) to find the critical points of
f sx, yd − 12 1 10y 2 2x 2 2 8xy 2 y 4 correct to three
decimal places. Then classify the critical points and find
the highest point on the graph.
59–62 Use Lagrange multipliers to find the maximum and
minimum values of f subject to the given constraint(s).
59. f sx, yd − x 2 y; x 2 1 y 2 − 1
60. f sx, yd − 1 x 1 1 y ; 1 x 2 1 1 y 2 − 1
Based © 2016 on Cengage NOAA / AOML Learning Hurricane ® Research Division
70
60 70 80
65
55
65
75
60
55
50
Key West
45
40
35
30
7et14rx49
05/12/10
MasterID: 01648
0 10 20 30 40
(Distance in miles)
Homestead
50. Find parametric equations of the tangent line at the point
s22, 2, 4d to the curve of intersection of the surface
z − 2x 2 2 y 2 and the plane z − 4.
51–54 Find the local maximum and minimum values and saddle
points of the function. If you have three-dimensional graphing
software, graph the function with a domain and viewpoint that
reveal all the important aspects of the function.
51. f sx, yd − x 2 2 xy 1 y 2 1 9x 2 6y 1 10
61. f sx, y, zd − xyz; x 2 1 y 2 1 z 2 − 3
62. f sx, y, zd − x 2 1 2y 2 1 3z 2 ;
x 1 y 1 z − 1, x 2 y 1 2z − 2
63. Find the points on the surface xy 2 z 3 − 2 that are closest to
the origin.
64. A package in the shape of a rectangular box can be mailed by
the US Postal Service if the sum of its length and girth (the
perimeter of a cross-section perpendicular to the length) is at
most 108 in. Find the dimensions of the package with largest
volume that can be mailed.
65. A pentagon is formed by placing an isosceles triangle on a
rectangle, as shown in the figure. If the pentagon has fixed
perimeter P, find the lengths of the sides of the pentagon that
maximize the area of the pentagon.
¨
=
=
7et14rx65
05/12/10
MasteraID: 01649
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.