Question

If θ is in the interval [0, 2π) and cos(θ) = √ 2/2 , then θ must be π/4 .

State true or false.

Answer #1

Change from rectangular to spherical coordinates. (Let ρ ≥ 0, 0
≤ θ ≤ 2π, and 0 ≤ ϕ ≤ π.)
(a) (0, −3, 0):
(ρ, θ, ϕ) = (3, −π 2 , π 2) <---- (WRONG!!!!)
(b) (−1, 1, − 2 ):
(ρ, θ, ϕ) = (2, − π 4 , π 4) <------ (WRONG!!!!)

Find the dimension of the subspace U = span {1,sin^2(θ), cos 2θ}
of F[0, 2π]

a.
r=3 - 3cos(Θ), enter value for r on a table
when;
Θ=0, (π/3),(π/2),(2π/3),π,(4π/3),(3π/2),(5π/3) & 2π
b. plot points from a, sketch graph
c. use calculus to find slope at (π/2),(2π/3),(5π/3)
& 2π
d. find EXACT area inside the curve in 1st
quadrant

Let Θ ∼ Unif.([0, 2π]) and consider X = cos(Θ) and Y =
sin(Θ).
Can you find E[X], E[Y], and E[XY]?
clearly, x and y are not independent
I think E[X] = E[Y] = 0 but how do you find E[XY]?

Identify the surface with parametrization x = 3 cos θ sin φ, y =
3 sin θ sin φ, z = cos φ where 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π. Hint: Find
an equation of the form F(x, y, z) = 0 for this surface by
eliminating θ and φ from the equations above. (b) Calculate a
parametrization for the tangent plane to the surface at (θ, φ) =
(π/3, π/4).

1. Sketch the polar function r = (θ − π/4)(θ − 3π/4) on the
interval 0 ≤ θ ≤ 2π. Then find all lines tangent to this polar
function at the point (0, 0).
2. Find the area of the region enclosed by one loop of the curve
r = 5 sin(4θ).
3. Use the Monotone Sequence Theorem to determine that the
following sequence converges: an = 1/ 2n+3 .

Change from rectangular to spherical coordinates. (Let ρ ≥ 0, 0
≤ θ ≤ 2π, and 0 ≤ ϕ ≤ π.) (a) (0, −3, 0) (b) (−1, 1, − sqrt 2 )

Write the following numbers in the polar form
re^iθ,
0≤θ<2π
a) 7−7i
r =......8.98............. , θ
=.........?..................
Write each of the given numbers in the polar form
re^iθ,
−π<θ≤π
a) (2+2i) / (-sqrt(3)+i)
r =......sqrt(2)........, θ =
.........?..................., .

Evaluate, in spherical coordinates, the triple integral of
f(ρ,θ,ϕ)=cosϕ, over the region 0≤θ≤2π, π/6≤ϕ≤π/2, 3≤ρ≤8.
integral =

Consider the function on the interval (0, 2π).
f(x) =
sin(x)/
2 + (cos(x))2
(a) Find the open intervals on which the function is increasing
or decreasing. (Enter your answers using interval notation.)
increasing
decreasing
(b) Apply the First Derivative Test to identify the relative
extrema.
relative maximum
(x, y) =
relative minimum
(x, y) =

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