Question

The curvature of a circle centred at the origin with radius 1/3 is

Answer #1

Given question solved.

Demonstrate that the curve r=2cos(θ) is a circle centred at
(1,0) or radius 1. You will NOT receive any partial credit for a
correct sketch.

Demonstrate that the curve r=4sin(θ) is a circle centred at
(0,2)of radius 2.

Find the radius of the curvature of the following curve at the
given point. Then write the equation of the circle of curvature at
the point. The radius of curvature at a point P is given by 1/k,
where k is the curvature at P
y=ln 2x at x=1/2
The radius of curvature at x=1/2 is 1/k = ?
The equation of the circle of curvature at this point is ?

Evaluate ∫c(1−xy/2)dS where C is the upper half of the circle
centered at the origin with radius 1 with the clockwise rotation
followed by the line segment from (1,0)to (3,0) which in turn is
followed by the lower half of the circle centerd at the origin of
radius 3 with clockwise rotation.

A particle moves on a circle of radius 5 cm, centered at the
origin, in the ??-plane (? and ? measured in cen- timeters). It
starts at the point (10,0) and moves counter- clockwise, going once
around the circle in 8 seconds.
(a) Writeaparameterizationfortheparticle’smotion.
(b) What is the particle’s speed? Give units.

9a. Find a set of parametric equations for a circle with a
radius of 3 centered at the origin, oriented clockwise.
9b. Write the equation of the circle using polar coordinates if
the circle is now centered at (0,1).

Problem 2. Let C be the circle of radius 100, centered at the
origin and positively oriented. The goal of this problem is to
compute Z C 1 z 2 − 3z + 2 dz.
(i) Decompose 1 z 2−3z+2 into its partial fractions.
(ii) Compute R C1 1 z−1 dz and R C2 1 z−2 dz, where C1 is the
circle of radius 1/4, centered at 1 and positively oriented, and C2
is the circle of radius 1/4, centered...

Consider the unit circe C (the circle with center the origin in
the plane and radius 1). Let S = {α : 2α < (the circumference of
C)} . Show that S is bounded above. Let p be the least upper bound
of S. Say explicitly what the number p is. This exercise works in
the real number system, but not in the rational number system.
Why?

Consider the circle around the origin (0,0) with a radius of
r=2. You may need to recall the equation defining such a circle.
Our goal is to draw a rectangle inside the circle, but we'd like to
draw the largest possible rectangle. Find the dimensions of the
largest rectangle that would fit and make sure the dimensions give
the maximum area possible.

Let (X,Y) be chosen uniformly from inside the circle of radius
one centered at the origin. Let R denote the distance of the chosen
point from the origin. Determine the density of R. From there,
determine the density of the random variable R2 =
X2 + Y2.

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