Question

(1 point) If the gradient of f is ∇f=z3j⃗ −2yi⃗ −xzk⃗ and the point P=(10,−7,−4) lies on the level surface f(x,y,z)=0, find an equation for the tangent plane to the surface at the point P.

z=

Answer #1

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.

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.

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

Find the gradient ∇f and the directional derivative at the point
P (1,−1,2) in the direction a = (2,−1,1) for the function f (x,y,z)
= x^3z − y(x^2) + z^2. In which direction is the directional
derivative at P decreasing most rapidly and what is its value?

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) =

. The point P = (0, 2, 1) is on the surface 2x + y + 3z = 5e xyz
.
(a) Find a normal vector to the surface at P.
(b) Find an equation for the plane tangent to the surface at
P.

1). Consider the following function and point.
f(x) = x3 + x + 3; (−2,
−7)
(a) Find an equation of the tangent line to the graph of the
function at the given point.
y =
2) Consider the following function and point. See Example
10.
f(x) = (5x + 1)2; (0, 1)
(a) Find an equation of the tangent line to the graph of the
function at the given point.
y =

Let f(x, y) = sqrt( x^2 − y − 4) ln(xy).
• Plot the domain of f(x, y) on the xy-plane.
• Find the equation for the tangent plane to the surface at the
point (4, 1/4 , 0).
Give full explanation of your work

Find the equation for the tangent plane to the surface
z=(xy)/(y+x) at the point P(1,1,1/2).

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.

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