Question

Consider the surface S in R3 defined implicitly by x**2 y = 4ze**(x+y) − 35 .

(a) Find the equations of the implicit partial derivatives ∂z ∂x and ∂z ∂y in terms of x, y, z. (b) Find equations of the tangent plane and the norma line to the surface S at the point (3, −3, 2)

Answer #1

M := {(x, y, z) ∈ R3 : x 2 y − 4ze^x+y = −35} is a surface. Find
the equation of the tangent plane to M at p = (3, −3, 2).

An implicitly defined function of x, y and z is given along with
a point P that lies on the surface: sin(xy) + cos(yz) = 0, at P =
(2, π/12, 4)
Use the gradient ∇F to:
(a) find the equation of the normal line to the surface at
P.
(b) find the equation of the plane tangent to the surface at
P.

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...

Suppose z is implicitly implicitly defined by the equation:
F(x, y, z) = 4x^ −1 − 3x 3 yz + e^ z/ (x − 2) = c where c is a
constant.
Compute the first and second order partial derivatives of z with
respect to x and y

(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 the equation of the tangent plane of the surface implicitly
defined by xy^2z^3=8 at the point (2,2,1).

Consider the surface x^7z^2+sin(y^7z^2)+10=0
Use implicit differentiation to find the following partial
derivatives.
∂z/∂x=
∂z/∂y=

Let D be the solid region defined by D = {(x, y, z) ∈ R3; y^2 +
z^2 + x^2 <= 1},
and V be the vector field in R3 defined by: V(x, y, z) = (y^2z +
2z^2y)i + (x^3 − 5^z)j + (z^3 + z) k.
1. Find I = (Triple integral) (3z^2 + 1)dxdydz.
2. Calculate double integral V · ndS, where n is pointing
outward the border surface of V .

1. Consider x=h(y,z) as a parametrized surface in the natural
way. Write the equation of the tangent plane to
the surface at the point (5,3,−4) given that ∂h/∂y(3,−4)=1 and
∂h/∂z(3,−4)=0.
2. Find the equation of the tangent plane to the surface
z=0y^2−9x^2 at the point (3,−1,−81). z=?

Compute equations of tangent plane and normal line to the
surface z = x cos (x+y) at point (π/2, π/3, -√3π/4).

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