Question

The equation 4 = 2xy^3 - xyz is a level surface in
3-dimensional space. A person is standing on this surface, at the
point (1, 2, 6).

a. Write the function f for which the above surface is a level
surface, and find the gradient of this

function f. What meaning does the gradient have for the
person?

b. Find an equation for the tangent plane to this surface at
the point (1, 2, 6).

c. Find the equations of the normal line to this surface at
the point (1, 2, 6).

Answer #1

find the tangent plane to the surface x^2 + 2xy + z^3 = 4 at point
P (1,1,1)

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...

find an equation of the tangent plane to the surface
z=4x^5+8y^5+2xy at the point (-2,1,-124)

Find an equation of the tangent plane and find the equations for
the normal line to
the following surface at the given point.
3xyz = 18 at (1, 2, 3)

Let S be a surface in the 3-D space (but we don’t have an
equation
for S). Suppose that there are two curves
r _1(t) = < cos(t), sin(t), t >
and
r_ 2(s) = < (s + 1)^2, 2s, se^s >
that both lie on S. Find an equation of the tangent plane to the
surface S at the
point (1, 0, 0).

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.

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

Find equations of the tangent plane and normal line to the
surface
?=1?2+4?2−321x=1y2+4z2−321
at the point (7, 2, 9).
Tangent Plane: (make the coefficient of x equal to 1).

Consider the function F(x, y, z) =x2/2−
y3/3 + z6/6 − 1.
(a) Find the gradient vector ∇F.
(b) Find a scalar equation and a vector parametric form for the
tangent plane to the surface F(x, y, z) = 0 at the point (1, −1,
1).
(c) Let x = s + t, y = st and z = et^2 . Use the multivariable
chain rule to find ∂F/∂s . Write your answer in terms of s and
t.

Find equations of the tangent plane and normal line to the
surface x=3y^2+1z^2−40x at the point (-9, 3, 2).
Tangent Plane: (make the coefficient of x equal to 1).
=0.
Normal line: 〈−9,〈−9, , 〉〉
+t〈1,+t〈1, , 〉〉.

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