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. 55. f ( x , y , z ) = x 2 + 2 y 2 + 4 z 2 + 10 ; P ( 1 , 0 , 4 ) ; 〈 1 2 , 0 , 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. 55. f ( x , y , z ) = x 2 + 2 y 2 + 4 z 2 + 10 ; P ( 1 , 0 , 4 ) ; 〈 1 2 , 0 , 1 2 〉
Gradients in three dimensionsConsider 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.
55.
f
(
x
,
y
,
z
)
=
x
2
+
2
y
2
+
4
z
2
+
10
;
P
(
1
,
0
,
4
)
;
〈
1
2
,
0
,
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.
Consider the function f and the point P.
f(x, y) = xey + yex, P(0, 0)
) Find the gradient of f and evaluate the gradient at the point P.
∇f(0, 0)
Find the directional derivative of f at P in the direction of the vector v =
(5, 1).
Duf(0, 0)
Find the maximum rate of change of f at P and the direction in which it occurs.
maximum rate of change
direction vector
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