Question

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?

Answer #1

. 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 : 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.

Find the directional derivative of the function
f(x,y)=x^6+y^3/(x+y+6 ) at the point (2,-2) in the direction of the
vector < - 2 ,2>.
b) Also find the maximum rate of change of f at the given
point and the unit vector of the direction in which the maximum
occurs.

1. Let f(x, y) = 2x + xy^2 , x, y ∈ R.
(a) Find the directional derivative Duf of f at the point (1, 2)
in the direction of the vector →v = 3→i + 4→j .
(b) Find the maximum directional derivative of f and a unit
vector corresponding to the maximum directional derivative at the
point (1, 2).
(c) Find the minimum directional derivative and a unit vector in
the direction of maximal decrease at the point...

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)

Find the directional derivative of the function f (x, y) =
tan−1(xy) at the point (1, 3) in the direction of the unit vector
parallel to the vector v = 4i + j.

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.)

Find the gradient ∇f and the directional derivative at the point
P (1,−1,2) in the direction a = (2,−1,1) for the function f (x,y,z)
= x^3z − y(x^2) + z^2. In which direction is the directional
derivative at P decreasing most rapidly and what is its value?

) 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).

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?.

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