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

958 chapter 14 Partial Derivatives
(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?
35. Let f be a function of two variables that has continuous
partial derivatives and consider the points As1, 3d, Bs3, 3d,
Cs1, 7d, and Ds6, 15d. The directional derivative of f at A in
the direction of the vector AB l is 3 and the directional derivative
at A in the direction of AC l is 26. Find the directional
derivative of f at A in the direction of the vector AD l .
36. Shown is a topographic map of Blue River Pine Provincial
Park in British Columbia. Draw curves of steepest descent
from point A (descending to Mud Lake) and from point B.
© 2016 Cengage Learning ®
1000 m
Blue River
North Thompson River
Blue River
Smoke Creek
2000 m
© Department of Natural Resources Canada. All rights reserved.
B
Mud Lake
Blue River Pine Provincial Park
Mud Creek
2200 m
2200 m
2200 m
37. Show that the operation of taking the gradient of a function
has
7et1406x36
the given property. Assume that u and v are differen tiable
functions of x and y and that a, b are constants.
05/04/10
(a) =sau 1 bvd − a =u 1 b =v
(b)
MastreID:
=suvd − u
01662
=v 1 v =u
(c) =S u vD −
v =u 2 u =v
v 2 (d) =u n − nu n21 =u
38. Sketch the gradient vector =f s4, 6d for the function f whose
level curves are shown. Explain how you chose the direction
and length of this vector.
y
6
4
2
0
_5
_3
2
_1
0
(4, 6)
1
A
3 5
4 6 x
;
39. The second directional derivative of f sx, yd is
D u 2 f sx, yd − D ufD u f sx, ydg
If f sx, yd − x 3 1 5x 2 y 1 y 3 and u − k 3 5 , 4 5 l, calculate
D u 2 f s2, 1d.
40. (a) If u − ka, bl is a unit vector and f has continuous
second partial derivatives, show that
D u 2 f − f xxa 2 1 2f xyab 1 f yyb 2
(b) Find the second directional derivative of f sx, yd − xe 2y in
the direction of v − k4, 6 l.
41–46 Find equations of (a) the tangent plane and (b) the normal
line to the given surface at the specified point.
41. 2sx 2 2d 2 1 sy 2 1d 2 1 sz 2 3d 2 − 10, s3, 3, 5d
42. x − y 2 1 z 2 1 1, s3, 1, 21d
43. xy 2 z 3 − 8, s2, 2, 1d
44. xy 1 yz 1 zx − 5, s1, 2, 1d
45. x 1 y 1 z − e xyz , s0, 0, 1d
46. x 4 1 y 4 1 z 4 − 3x 2 y 2 z 2 , s1, 1, 1d
47–48 Use a computer to graph the surface, the tangent plane,
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.
47. xy 1 yz 1 zx − 3, s1, 1, 1d
48. xyz − 6, s1, 2, 3d
49. If f sx, yd − xy, find the gradient vector =f s3, 2d and use it
to find the tangent line to the level curve f sx, yd − 6 at the
point s3, 2d. Sketch the level curve, the tangent line, and the
gradient vector.
50. If tsx, yd − x 2 1 y 2 2 4x, find the gradient vector =ts1, 2d
and use it to find the tangent line to the level curve
tsx, yd − 1 at the point s1, 2d. Sketch the level curve, the
tangent line, and the gradient vector.
51. Show that the equation of the tangent plane to the ellipsoid
x 2 ya 2 1 y 2 yb 2 1 z 2 yc 2 − 1 at the point sx 0, y 0, z 0d can be
written as
xx 0
a 2
1 yy0
b 2
1 zz0
c 2 − 1
52. Find the equation of the tangent plane to the hyperboloid
x 2 ya 2 1 y 2 yb 2 2 z 2 yc 2 − 1 at sx 0, y 0, z 0d and express it in a
form similar to the one in Exercise 51.
53. Show that the equation of the tangent plane to the elliptic
paraboloid zyc − x 2 ya 2 1 y 2 yb 2 at the point sx 0, y 0, z 0d can
be written as
2xx 0
a 2
1 2yy0
b 2
− z 1 z0
c
7et1406x38
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.