Question

(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))

Answer #1

Find an equation of the tangent plane to the surface z = x^2 +
xy + 3y^2 at the point (1, 1, 5)

Find the equation for the tangent plane to the surface
z=(xy)/(y+x) at the point P(1,1,1/2).

Find an equation of the tangent plane to the surface x y 2 + 3 x
− z 2 = 4 at the point ( 2 , 1 , − 2 ) An equation of the tangent
plane is

1)Find an equation of the tangent plane to the surface given by
the equation xy + e^2xz +3yz = −5, at the point, (0, −1, 2)
2)Find the local maximum and minimum values and saddle points
for the following function: f(x, y) = x − y+ 1 xy .
3)Use Lagrange multipliers to find the maximum and minimum
values of the function, f(x, y) = x^2 − y^2 subject to, x^2 + y 4 =
16.

Let f(x, y) = x tan(xy^2) + ln(2y). Find the equation of the
tangent plane at (π, 1⁄2).

Find an equation of the tangent plane to the surface given by
the equation xy + e 2xz+3yz = −5, at the point, (0, −1, 2)

Find the equation of the tangent plane to the surface given by z =
ln (2tan x - tan y) at (pi/4, pi/4, 0).

Let f(x, y) = sqrt( x^2 − y − 4) ln(xy).
• Plot the domain of f(x, y) on the xy-plane.
• Find the equation for the tangent plane to the surface at the
point (4, 1/4 , 0).
Give full explanation of your work

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.

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...

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