Question

1. a) For the surface f(x, y, z) = xy + yz + xz = 3, find the equation of the tangent plane at (1, 1, 1).

b) For the surface f(x, y, z) = xy + yz + xz = 3, find the equation of the normal line to the surface at (1, 1, 1).

Answer #1

Consider F and C below.
F(x, y,
z) = yz i +
xz j + (xy +
18z) k
C is the line segment from (1, 0, −3) to (4,
4, 1)
(a) Find a function f such that F =
∇f.
f(x, y,
z) =
(b) Use part (a) to evaluate
C
∇f · dr
along the given curve C.

Consider F and C below.
F(x, y, z) = yz i + xz j + (xy + 12z) k
C is the line segment from (2, 0, −3) to (4, 6, 3)
(a) Find a function f such that F =
∇f.
f(x, y, z) =
(b) Use part (a) to evaluate
C
∇f · dr along the given curve C.

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.

Let F(x, y, z) = (yz, xz, xy) and the path c(t) = (cos3 t,sin3
t, 0) for 0 ≤ t ≤ 2π. Evaluate R c F · ds. Hint: Identify f such
that ∇f = F.

The tangent plane at (1,1,1) on the surface x2+y2+z2+xy+xz=5 is
given by
x+ y+ z=
(all values should be positive whole numbers with no common
factors)

Find the equation for the tangent plane to the
surface
xy + yz + zx = 11 at P(1, 2, 3)

Find the work done by the force ﬁeld F(x,y,z) = yz i + xz j + xy
k acting along the curve given by r(t) = t3 i + t2 j + tk from the
point (1,1,1) to the point (8,4,2).

(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 distance from (3,4,9) to each of the following
a) the xy-plane
b) the yz- plane
c) the xz-plane
d) the x-axis
e) the y-axis
f) the z-axis

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

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 6 minutes ago

asked 54 minutes ago

asked 55 minutes ago

asked 55 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago