Question

Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at (−4,2), oriented counterclockwise. The point (1,2) should correspond to t=0. Use t as the parameter in your answer.

find r⃗ (t)=

Answer #1

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...

(1 point) Consider the paraboloid z=x2+y2. The plane 5x−3y+z−3=0
cuts the paraboloid, its intersection being a curve. Find "the
natural" parametrization of this curve. Hint: The curve which is
cut lies above a circle in the xy-plane which you should
parametrize as a function of the variable t so that the circle is
traversed counterclockwise exactly once as t goes from 0 to 2*pi,
and the paramterization starts at the point on the circle with
largest x coordinate. Using that...

Find a parametrization for the line perpendicular to
(4, −1, 1),
parallel to the plane
4x + y −
8z = 1,
and passing through the point
(1, 0, −7).
(Use the parameter t. Enter your answers as a
comma-separated list of equations.)

Find a parametrization for the line perpendicular to (2, −1, 1),
parallel to the plane 2x + y − 6z = 1, and passing through the
point (1, 0, −3). (Use the parameter t. Enter your answers as a
comma-separated list of equations.)

A particle in R^2 travels along a circle centered at (x, y) with
radius r > 0. Parametrize this circular path r(t) as a function
of the parameter variable t. Please prove that at all t values, the
tangent vector r'(t) is orthogonal to the vector r(t) - vector(x,
y)

particle in R2 travels along a circle centered at (h,k) with
radius a > 0. Parametrize this circular path r(t) as a function
of the parameter variable t. Please prove that at all t values, the
tangent vector r0(t) is orthogonal to the vector
r(t)−<h,k>

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).

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...

A circular ring of charge, with radius R,is placed in the
xy-plane and centered on the origin. The linear charge density of
the ring isλ=λ_o*cos^2(φ), where φ is the cylindrical polar
coordinate such that any point in space is indicated by (r, φ, z).
Find the electric potential anywhere on the z-axis as a function of
z . Using this electric potential find the electric field anywhere
on the z-axis also as a function of z

Find a path that traces the circle in the plane y=5 with radius
r=5 and center (2,5,0) with constant speed 25.
r1(s)

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