Question

In regards to the function f(x,y) = xye^y+2x. Find Duf(1,0) in the direction of the vector <3,4>. And determine the greatest slope along the surface that occurs at the point (1,0).

Answer #1

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...

. 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?

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?

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.

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)=1/2x ln x^4, (-1,0)
a) find an equation of the tangent line to the graph of the
function at the indicated point.
b) Use a graphing utility to graph the function and its tangent
line at the point.

(9)
(a)Find the double integral of the function f (x, y) = x + 2y
over the region in the plane bounded by the lines x = 0, y = x, and
y = 3 − 2x.
(b)Find the maximum and minimum values of 2x − 6y + 5 subject to
the constraint x^2 + 3(y^2) = 1.
(c)Consider the function f(x,y) = x^2 + xy. Find the directional
derivative of f at the point (−1, 3) in the direction...

) 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).

For the function f(x, y)=ln(1+xy)
a.Find the value of the directional derivative of f at the point
(-1, -2) in the direction <3,4>.
b.Find the unit vector that gives the direction of steepest
increase of f at the point (2,3).

let
f(x,y) = xe^(xy)
Find the directional derivative of f at point (2,0) in the
direction of vector <-6,8>. Find the maximum rate of change
of f at point (2,0) and the direction in which it occurs.

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