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. 56. f ( x , y , z ) = 4 − x 2 + 3 y 2 + z 2 2 ; P ( 0 , 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. 56. f ( x , y , z ) = 4 − x 2 + 3 y 2 + z 2 2 ; P ( 0 , 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.
56.
f
(
x
,
y
,
z
)
=
4
−
x
2
+
3
y
2
+
z
2
2
;
P
(
0
,
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.
Suppose f(x,y)=x/y, P=(−2,−3) and v=4i−3j
A. Find the gradient of f.
B. Find the gradient of f at the point P.
C. Find the directional derivative of f at P in the direction of v.
D. Find the maximum rate of change of f at P.
E. Find the (unit) direction vector w in which the maximum rate of change occurs at P.
I would need help with a, b, and c as mention below.
(a) Find the gradient of f.(b) Evaluate the gradient at the point P.(c) Find the rate of change of f at P in the direction of the vector u.
Given the problem:
Find the gradient of f at P0
Find the unit vector u in the direction of A
The derivative of f(x,y) at the point P0 in the direction of A
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