Question

For f(x,y,z) = sqrt(35-x^2-4y^2-2z) 1. Find the gradient of f(x,y,z) 2. Evaluate delta f(x,y,z) 3. Find the unit vectors U+ and U- , that give the direction of steepest ascent and the steepest descent respectively.

Answer #1

Consider the function F(x,y) = e^(((-x^2)/2)-((y^2)/2)) and the
point P(-3,3).
a. Find the unit vectors that give the direction of the steepest
ascent and the steepest descent at P.
b. Find a vector that points in a direction of no change at P.

Consider the function
F(x,y)=e^((-x^2/4)-(y^2/4)) and the point P(−1,1).
a. Find the unit vectors that give the
direction of steepest ascent and steepest descent at P.
b. Find a vector that points in a direction of
no change in the function at P.

For f(x, y) = x2 + 4xy - y2 at the point P(2, 1), a) find the
unit vector u in the direction of steepest ascent; b) find the unit
vector u in the direction of steepest descent; and c) find a unit
vector u that points in the direction of no change in the function.
show all work please
a)
b)
c)

Consider the function f(x, y) = sin(2x − 2y) (a) Solve and find
the gradient of the function.
(b) Find the directional derivative of the function at the point
P(π/2,π/6) in the direction of the vector
v = <sqrt(3), −1>
(c) Compute the unit vector in the direction of the steepest
ascent at A (π/2,π/2)

f(x, y, z) =
xe4yz, P(1, 0, 3),
u = <2/3, -1/3, 2/3>
(a) Find the gradient of f.
∇f(x, y, z) =
< , , >
(b) Evaluate the gradient at the point P.
∇f(1, 0, 3) = < , ,
>
(c) Find the rate of change of f at P in the
direction of the vector u.
Duf(1, 0, 3) =

In the following functions: a) Find the gradient of f. , b)
Evaluate
the gradient at point P. and
c) Find the rate of change of f in P, in the direction of
vector.
1- f(x. y) = 5xy^2 - 4x^3y, P( I , 2), u = ( 5/13, 12/13 )
2- f(x, y, z) = xe^2yz , P(3, 0, 2), u = (2/3, -2/3, 1/3)

Let f(x, y) = x^2 ln(x^3 + y).
(a) Find the gradient of f.
(b) Find the direction in which the function decreases most
rapidly at the point P(2, 1). (Give the direction as a unit
vector.)
(c) Find the directions of zero change of f at the point P(2,
1). (Give both directions as a unit vector.)

What is the gradient of the function f(x,y,z) = 5x^2 + 4y^3 +
4z^4 + xyz ?

f(x,y,z) = xey+z.
(a) Find the gradient of f, ∇f.
(b) Find the directional derivative of f at the point (2, 1, 2)
in the direction of ? = 3? + 4?.

find the directional derivative of f(x,y) = x^2y^3 +2x^4y at the
point (3,-1) in the direction theta= 5pi/6
the gradient of f is f(x,y)=
the gradient of f (3,-1)=
the directional derivative is:

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