Question

(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 of the vector 〈1, 2〉.

Answer #1

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)

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...

using the change of variable x =u/v, y=v evaluate "double
integral(x^2+2y^2)dxdy: R is the region in the first quadrant
bounded by the graphs of xy=1, xy=2, y=x, y=2x

Calculate double integral D f(x, y) dA as an iterated integral,
where f(x, y) = −4x 2y 3 + 4y and D is the region bounded by y = −x
− 3 and y = 3 − x 2 .

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:

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.

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?

. 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?

Set up the triple integral, including limits, of the function
over the region.
f(x, y, z) = sin z, x ≥ 0, y ≥ 0, and below the plane 2x + 2y +
z = 2

Find the maximum and minimum of the function f (x, y) = 6x − 2y
subject to the constraint . 3x^2 + y ^2 = 4

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