Question

1) Show that the formulas below represent the equation of a circle.

x = h + r cos θ

y = k + r sin θ

2) Use the equations in the preceding problem to find a set of parametric equations for a circle whose radius is r = 4 and whose center is (-1,-2).

3) Plot each of the following points on the polar plane. A(2, π/4), B(1, 3π/2), C(4, π)

Answer #1

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...

If a circle C with radius 1 rolls along the outside of the
circle x2 + y2 = 49, a fixed point P on C traces out a curve called
an epicycloid, with parametric equations x = 8 cos(t) − cos(8t), y
= 8 sin(t) − sin(8t). Use one of the formulas below to find the
area it encloses. A = C x dy = −C y dx = 1/2 C x dy − y
dx

Find the center (h,k) and the radius r of the circle 4 x^2 + 7 x
+4 y^2 - 6 y - 9 = 0 .
h=? k=? r=?

a) Find the parametric equations for the circle centered at
(1,0) of radius 2 generated clockwise starting from
(1+21/2 , 21/2). <---( one plus square
root 2, square root 2)
b) When given x(t) = tcost, y(t) = sint, 0 <_ t. Find dy/dx
as a function of t.
c) When given the parametric equations x(t) =
eatsin2*(pi)*t, y(t) = eatcos2*(pi)*t where a
is a real number. Find the arc length as a function of a for 0
<_ t...

1. Graph the curve given in parametric form by x = e t sin(t)
and y = e t cos(t) on the interval 0 ≤ t ≤ π2.
2. Find the length of the curve in the previous problem.
3. In the polar curve defined by r = 1 − sin(θ) find the points
where the tangent line is vertical.

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 .

With the parametric equation x=cos(t)+tsin(t), y=sin(t)-tcos(t)
, 0 ≤ t ≤ 2π)
Find the length of the given curve. (10 point)
2) In the circle of r = 6, the area
above the r = 3 cos (θ) line
Write the integral or integrals expressing the area of this
region by drawing. (10 point)

1. This question has several parts that must be completed
sequentially. If you omit a part of the question, you will not
receive any points for the omitted part, and you will not be able
to return to the omitted part. Tutorial exercise Find dy / dx y d 2
y / dx 2, and find the slope and concavity (if possible) at the
given value of the parameter. Parametric equations Point
x = 4 cos θ, y = 4...

1.Let y=6x^2. Find a parametrization of the
osculating circle at the point x=4.
2. Find the vector OQ−→− to the center of the
osculating circle, and its radius R at the point
indicated. r⃗
(t)=<2t−sin(t),
1−cos(t)>,t=π
3. Find the unit normal vector N⃗ (t)
of r⃗ (t)=<10t^2, 2t^3>
at t=1.
4. Find the normal vector to r⃗
(t)=<3⋅t,3⋅cos(t)> at
t=π4.
5. Evaluate the curvature of r⃗
(t)=<3−12t, e^(2t−24),
24t−t2> at the point t=12.
6. Calculate the curvature function for r⃗...

Your task will be to derive the equations describing the
velocity and acceleration in a polar coordinate
system and a rotating polar vector basis for an object in general
2D motion starting from a general
position vector. Then use these expressions to simplify to the case
of non-uniform circular motion, and
finally uniform circular motion.
Here's the time-dependent position vector in a Cartesian coordinate
system with a Cartesian vector
basis: ⃗r(t)=x (t)
̂
i+y(t)
̂
j where x(t) and y(t)...

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