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 z ; 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 z ; P ( 0 , 2 , − 1 ) ; 〈 0 , 1 2 , − 1 2 〉
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
z
;
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.
The graph of f(x) is given in the figure below. draw tangent lines to the graph at x=-3,x=-2,x=1,and x=4. estimate f'(-3),f'(-2),f'(1),and f'(4). Round your answers to one decimal place.
Elementary Statistics: Picturing the World (7th Edition)
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