Question

Given the level surface S defined by f(x, y, z) = x −
y^{3} − 2z^{2} = 2 and the point P_{0}(−4,
−2, 1).

Find the equation of the tangent plane to the surface S at the
point P_{0}.

Find the derivative of f at P_{0}in 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 P_{0};

(d) Find the parametric equations of the tangent line at P0 to the curve of intersection of S and 2x − 3y − z = -3.

Answer #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.

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)

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.

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.

We are given a level surface F ( x , y , z ) = 0 where F ( x , y
, z ) = x^3 - y^2 + z^4 - 20 . Find the equation of the tangent
plane to the surface at the point P ( 2 , 2 , 2 ) . Write the final
answer in the form a x + b y + c z + d = 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.

Compute the surface integral of F(x, y, z) = (y,z,x) over the
surface S, where S is the portion of the cone x = sqrt(y^2+z^2)
(orientation is in the negative x direction) between the planes x =
0, x = 5, and above the xy-plane.
PLEASE EXPLAIN

Consider the surface defined by z = f(x,y) = x+y^2+1.
a）Sketch axes that cover the region -2<=x<=2 and
-2<=y<=2.On the axes , draw and clearly label the contours
for the eights z=0 ,z=1,and z=2.
b)evaluate the gradients of f(x,y) at the point (x,y) = (0.-1),
and draw the gradient vector on the contour diagrqam .
c)compute the directional derivative at(x,y) = (0,-1) in the
direction V =<2,1>.

(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))

Given S is the surface of the paraboloid z= 4-x^2-y^2
and C is the curve of intersection of the paraboloid with the plane
z=0. Verify stokes theorem for the field F=2zi+xj+y^2k.( you might
verify it by checking both sides of the theorem)

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