Gradients in three dimensions Consider the following functions f, points P, and unit vectors u . a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 60. f ( x , y , z ) = e x y x − 1 ; P ( 0 , 1 , − 1 ) ; 〈 − 2 3 ′ 2 3 ′ − 1 3 〉
Gradients in three dimensions Consider the following functions f, points P, and unit vectors u . a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 60. f ( x , y , z ) = e x y x − 1 ; P ( 0 , 1 , − 1 ) ; 〈 − 2 3 ′ 2 3 ′ − 1 3 〉
Gradients in three dimensionsConsider the following functions f, points P, and unit vectorsu.
a.Compute the gradient of f and evaluate it at P
b.Find the unit vector in the direction of maximum increase of f at P.
c.Find the rate of change of the function in the direction of maximum increase at P.
d.Find the directional derivative at P in the direction of the given vector.
60.
f
(
x
,
y
,
z
)
=
e
x
y
x
−
1
;
P
(
0
,
1
,
−
1
)
;
〈
−
2
3
′
2
3
′
−
1
3
〉
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Two cables tied together at C are loaded as shown. Given: Q = 130 lb.
8
30°
C
B
Q
3
4
Draw the free-body diagram needed to determine the range of values of P for which both cables remain taut.
Cable AB is 103 ft long and the tension in the cable is 3900 lb.
56 ft
A
50°
20°
B
x
C
Identify the angles 0.0, and 8, that define the direction of force.
1
By
N
2
Match each of the options above to the items below.
142.1°
57.1°
73.3°
3
8.
In the given figure, P = 51 lb .
65°
C
25°
35°
75 lb
P
Determine the corresponding magnitude of the resultant.
The corresponding magnitude of the resultant is|
lb.
Elementary Statistics: Picturing the World (7th Edition)
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