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. 58. f ( x , y , z ) = x y + y z + x z + 4 ; P ( 2 , − 2 , 1 ) ; 〈 0 , − 1 2 , − 1 2 〉
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. 58. f ( x , y , z ) = x y + y z + x z + 4 ; P ( 2 , − 2 , 1 ) ; 〈 0 , − 1 2 , − 1 2 〉
Solution Summary: The author explains how the gradient of f(x,y,z) is computed as follows.
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.
58.
f
(
x
,
y
,
z
)
=
x
y
+
y
z
+
x
z
+
4
;
P
(
2
,
−
2
,
1
)
;
〈
0
,
−
1
2
,
−
1
2
〉
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.
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