Question

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.)

Answer #1

] Consider the function f : R 2 → R defined by f(x, y) = x ln(x
+ 2y). (a) Find the gradient of f(x, y) at the point P(e/3, e/3).
(b) Use the gradient to find the directional derivative of f at
P(e/3, e/3) in the direction of the vector ~u = h−4, 3i. (c) Find a
unit vector (based at P) pointing in the direction in which f
increases most rapidly at P.

s] Consider the function f : R 2 → R defined by f(x, y) = x ln(x
+ 2y). (a) Find the gradient of f(x, y) at the point P(e/3, e/3).
(b) Use the gradient to find the directional derivative of f at
P(e/3, e/3) in the direction of the vector ~u = h−4, 3i. (c) Find a
unit vector (based at P) pointing in the direction in which f
increases most rapidly at P.

the function f(x; y) = xye^x-y, at the point (2; 2) (1)find the
gradient. (2) find the directional derivative in the direction of
the vector 3i - j. (3)find the direction of which unit vector is
the rate of increase maximum? What is the maxi- mum rate of
increase? (4)find the direction of which unit vector(s) is the
directional derivative zero?

Consider the function f(x,y) = ( x2 +
z2)ln(y)
a)Find the gradient of f.
b) Find the rate of change of f at the point (2, 1, 1) in the
direction of ?⃗ = 〈−2, 4, −4〉

. For the function f(x, y) = xye^x−y , at the point (2, 2)
(a) find the gradient.
(b) find the directional derivative in the direction of the
vector 3i − j.
(c) in the direction of which unit vector is the rate of
increase maximum? What is the maximum rate of increase?
(d) in the direction of which unit vector(s) is the directional
derivative zero?

) Consider the function f(x,y)=−2x^2−y^2.
Find the the directional derivative of ff at the point (1,−3)(1,−3)
in the direction given by the angle θ=π/2.
Find the unit vector which describes the direction in which ff
is increasing most rapidly at (1,−3).

Consider the following. f(x, y, z) = xe5yz, P(1, 0, 2),
u=1/3,-2/3,2/3. (a) Find the gradient of f. ∇f(x, y, z) = (b)
Evaluate the gradient at the point P. ∇f(1, 0, 2) = (c) Find the
rate of change of f at P in the direction of the vector u. Duf(1,
0, 2) =

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)

6. Consider the function f defined by f (x, y) = ln(x
− y). (a) Determine the natural domain of f. (b) Sketch the level
curves of f for the values k = −2, 0, 2. (c) Find the gradient of f
at the point (2,1), that is ∇f(2,1). (d) In which unit vector
direction, at the point (2,1), is the directional derivative of f
the smallest and what is the directional derivative in that
direction?

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)

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