Question

Let w = xy + yz + zx and x = rcosθ, y = rsinθ, z = rθ. Find ∂w/∂r and ∂w/∂θ when r = 1,θ = π/2.

Answer #1

Use the Chain Rule to find the indicated partial
derivatives.
w = xy + yz + zx, x = r
cos(θ), y = r
sin(θ), z = rθ;
∂w
∂r
,
∂w
∂θ
when r = 4, θ =
π
2
∂w
∂r
=
∂w
∂θ
=

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).

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

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.

Let x,y,zx,y,z be (non-zero) vectors and suppose
w=10x+10y−4zw=10x+10y−4z.
If z=2x+2yz=2x+2y, then w=w= x+x+ yy.
Using the calculation above, mark the statements below that must
be true.
A. Span(x, z) = Span(w, z)
B. Span(w, y, z) = Span(x, y)
C. Span(w, z) = Span(w, y)
D. Span(w, x) = Span(x, y, z)
E. Span(w, y) = Span(w, x, y)

Let W = {(x, y, z, w) ∈ R 4 | x − z = 0 and y + 2z = 0}
(a) Find a basis for W.
(b) Apply the Gram-Schmidt algorithm to find an orthogonal basis
for the subspace (2) U = Span{(1, 0, 1, 0),(1, 1, 0, 0),(0, 1, 0,
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.

Let w = (x 2 -z)/ y4 ,
x = t3+7,
y = cos(2t),
z = 4t.
Use the Chain Rule to express dw/ dt in terms of t. Then
evaluate dw/ dt at t = π/ 2

1. Let W be the set of all [x y z}^t in R^3 such that xyz = 0.
Is W a subspace of R^3?
2. Let C^0 (R) denote the space of all continuous real-valued
functions f(x) of x in R. Let W be the set of all continuous
functions f(x) such that f(1) = 0. Is W a subspace of C^0(R)?

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 11 minutes ago

asked 17 minutes ago

asked 23 minutes ago

asked 23 minutes ago

asked 32 minutes ago

asked 38 minutes ago

asked 43 minutes ago

asked 45 minutes ago

asked 49 minutes ago

asked 53 minutes ago

asked 55 minutes ago