Question

16.

a. Find the directional derivative of f (x, y) = xy at P0 = (1, 2) in the direction of v = 〈3, 4〉.

b. Find the equation of the tangent plane to the level surface xy2 + y3z4 = 2 at the point (1, 1, 1).

c. Determine all critical points of the function f(x,y)=y3 +3x2y−6x2 −6y2 +2.

Answer #1

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.

Find the directional derivative of the function at P in the
direction of v. f(x, y) = x3 − y3, P(8, 5), v = 2 2 (i + j)

Suppose that f(x,y)=xy. Find the directional derivative of
f(x,y) in the directional 〈−6,3〉 and at the point (x,y)=(1,−4).
Answer exactly or round to 2 decimal places.

a) evaluate the directional derivative of z=F(x,y) = sin(xy) in
the direction of u=(1,-1) at the point (0,pi/2)
b) Determine the slope of the tangent line
c) State the tangent vector

find the directional derivative of f(x, y, z) = xy arctan (z) at
the point (1, 1, pi/4) in the direction <1, -1, 2>.

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).

(a) Find an equation of the plane tangent to the surface xy ln x
− y^2 + z^2 + 5 = 0 at the point (1, −3, 2)
(b) Find the directional derivative of f(x, y, z) = xy ln x −
y^2 + z^2 + 5 at the point (1, −3, 2) in the direction of the
vector < 1, 0, −1 >. (Hint: Use the results of partial
derivatives from part(a))

Given the level surface S defined by f(x, y, z) = x −
y3 − 2z2 = 2 and the point P0(−4,
−2, 1).
Find the equation of the tangent plane to the surface S at the
point P0.
Find the derivative of f at P0in the direction of
r(t) =< 3, 6, −2 >
Find the direction and the value of the maximum rate of change
greatest increase of f at P0;
(d) Find the parametric equations of the...

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...

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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