Question

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 .

Answer #1

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

The polar curve r = 5sin3θ where 0 ≤ θ ≤ π. Find the area of one
loop of the curve. Find an equation for the line tangent which has
a positive slope to the curve at the pole.

4)
Consider the polar curve r=e2theta
a) Find the parametric equations x = f(θ), y =
g(θ) for this curve.
b) Find the slope of the line tangent to this curve when
θ=π.
6)
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the...

For r = f(θ) = sin(θ)−1
(A) Find the area contained within f(θ).
(B) Find the slope of the tangent line to f(θ) at θ = 0
,π,3π/2
.

Find the area inside the polar curve of r = 1 + 2 sin θ but
outside the smaller loop.

Find the slope of the curve r = 1 − sin θ, at θ = π/4 in the xy
plane at the point θ = π.

Sketch the curve.
r = 4 + 2 cos(θ) and find area enclosed by it.

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 =
.........?..................., .

The Cartesian coordinates of a point are given. (a) (−4, 4) (i)
Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤
θ < 2π. (r, θ) (ii) Find polar coordinates (r, θ) of the point,
where r < 0 and 0 ≤ θ < 2π. (r, θ) (b) (3, 3 3 ) (i) Find
polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ
< 2π. (r, θ) =...

2. Rotate the semicircle of radius 2 given by y = √(4 − x^2)
about the x-axis to generate a sphere of radius 2, and use this to
calculate the surface area of the sphere.
3. Consider the curve given by parametric equations x = 2
sin(t), y = 2 cos(t).
a. Find dy/dx
b. Find the arclength of the curve for 0 ≤ θ ≤ 2π.
4.
a. Sketch one loop of the curve r = sin(2θ) and find...

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