Question

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.

Answer #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)

(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))

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).

Calculus III. Please show all work and mark the
answer(s)!
1) Use Lagrange multipliers to find the maximum and minimum
values of the function f(x, y) = x^2 + y^2 subject to the
constraint xy = 1.
2) Use Lagrange Multipliers to find the point on the curve 2x +
3y = 6 that is closest to the origin. Hint: let f(x, y) be the
distance squared from the origin to the point (x, y), then find the
minimum of...

8).
a) Find an equation of the tangent plane to the surface z = x at
(−4, 2, −1).
b) Explain why f(x, y) = x2ey is differentiable at (1, 0). Then
find the linearization L(x, y) of the function at that point.

Problem 1. Find an equation of the tangent plane to the given
surface at the specified point. i) z = 2x 2 + y 2 − 5y, (1, 2, −4).
ii) z = e x−y , (2, 2, 1). iii) z = x sin(x + y), (−1, 1, 0)

Find an equation of the tangent plane to the given surface at
the specified point.
z = 2(x − 1)2 + 4(y + 3)2 +
9, (2, −2, 15)

Find an equation of the tangent plane to the given surface at
the specified point.
z = 2(x − 1)2 + 4(y + 3)2 +
1, (3, −1, 25)
Answer as z=

Find the equation for the tangent plane to the
surface
xy + yz + zx = 11 at P(1, 2, 3)

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