Question

Consider the function F(x, y, z) =x^{2}/2−
y^{3}/3 + z^{6}/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 = e^{t^2} . Use the multivariable
chain rule to find ∂F/∂s . Write your answer in terms of s and
t.

Answer #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...

for the surface
f(x/y/z)=x3+3x2y2+y3+4xy-z2=0
find any vector that is normal to the surface at the point
Q(1,1,3). use this to find the equation of the tangent plane to the
surface at q.

The paraboloid z = 5 − x − x2 − 2y2 intersects the plane x = 1
in a parabola. Find parametric equations in terms of t for the
tangent line to this parabola at the point (1, 4, −29). (Enter your
answer as a comma-separated list of equations. Let x, y, and z be
in terms of t.)

. Find the flux of the vector field F~ (x, y, z) =
<y,-x,z> over a surface with downward orientation, whose
parametric equation is given by r(s, t) = <2s, 2t, 5 − s 2 − t 2
> with s^2 + t^2 ≤ 1

The paraboloid
z = 5 − x −
x2 −
2y2
intersects the plane x = 4 in a parabola. Find
parametric equations in terms of t for the tangent line to
this parabola at the point
(4, 2, −23).
(Enter your answer as a comma-separated list of equations. Let
x, y, and z be in terms of
t.)

Consider the following. f(x, y, z) = xe5yz, P(1, 0, 2),
u=1/3,-2/3,2/3. (a) Find the gradient of f. ∇f(x, y, z) = (b)
Evaluate the gradient at the point P. ∇f(1, 0, 2) = (c) Find the
rate of change of f at P in the direction of the vector u. Duf(1,
0, 2) =

(1) The paraboloid z = 9 − x − x2 − 7y2 intersects the plane x =
1 in a parabola. Find parametric equations in terms of t for the
tangent line to this parabola at the point (1, 2, −21). (Enter your
answer as a comma-separated list of equations. Let x, y, and z be
in terms of t.)
(2)Find the first partial derivatives of the function.
(Sn = x1 +
2x2 + ... + nxn; i
= 1,...

Find equations of the following.
x2 − 3y2 + z2 + yz =
52, (7, 2, −5)
(a) the tangent plane
(b) parametric equations of the normal line to the given surface at
the specified point. (Enter your answer as a comma-separated list
of equations. Let x, y, and z be in
terms of t.)

Find the gradient of the scalar function T (x, y, z) = (x +
3y)z2.
find divergence and the curl too

Let f(x, y) =sqrt(1−xy) and consider the surface S defined by
z=f(x, y).
find a vector normal to S at (1,-3)

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