Question

Let y = x 2 + 3 be a curve in the plane.

(a) Give a vector-valued function ~r(t) for the curve y = x 2 + 3.

(b) Find the curvature (κ) of ~r(t) at the point (0, 3). [Hint: do not try to find the entire function for κ and then plug in t = 0. Instead, find |~v(0)| and dT~ dt (0) so that κ(0) = 1 |~v(0)| dT~ dt (0) .]

(c) Find the center and radius of the circle of curvature at the point (0, 3) on the curve.

(d) Write the equation for the circle of curvature at the point (0, 3) and sketch a picture of the curve and the circle of curvature together.

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. A plane curve has been parametrized with the following
vector-valued function, r(t) = (t + 2)i + (-2t2 + t + 1)j a.
Carefully make 2 sketches of the plane curve over the interval . (5
pts) b. Compute the velocity and acceleration vectors, v(t) and
a(t). (6 pts) c. On the 1st graph, sketch the position, velocity
and acceleration vectors at t=-1. (5 pts) d. Compute the unit
tangent and principal unit normal vectors, T and N at...

[6]
The object is on move along the curve (y = x^2) & (z = x^3)
with a vertical speed, which is constant (dz/dt = 2) find the
acceleration and velocity when the object is at point P (2, 4, 8)
[7] (a) A plane (z = 2+x) which does intersects the cone (z^2
= x^2 + y^2) in a parabola. Parameterize the parabola using (t =
y). [Find f(t)] & [h(t)] where as [r = f(t)i + (t) j...

Let F ( x , y ) = 〈 e^x + y^2 − 3 , − e ^(− y) + 2 x y + 4 y 〉.
a) Determine if F ( x , y ) is a conservative vector field and, if
so, find a potential function for it. b) Calculate ∫ C F ⋅ d r
where C is the curve parameterized by r ( t ) = 〈 2 t , 4 t + sin
π...

A space curve C is parametrically parametrically defined by
x(t)=e^t^(2) −10,
y(t)=2t^(3/2) +10,
z(t)=−π,
t∈[0,+∞).
(a) What is the vector representation r⃗(t) for C ?
(b) Is C a smooth curve? Justify your answer.
(c) Find a unit tangent vector to C .
(d) Let the vector-valued function v⃗ be defined by
v⃗(t)=dr⃗(t)/dt
Evaluate the following indefinite integral
∫(v⃗(t)×i^)dt. (cross product)

In calculus the curvature of a curve that is defined by a
function
y = f(x)
is defined as
κ =
y''
[1 + (y')2]3/2
.
Find
y = f(x)
for which
κ = 1.

Suppose you are looking at a field [m[x,y],n[x,y]] which has one
singualrity at a point P and no other singularities. While studying
the singulairty, you center a circle of radius r on the point P and
no other singularites. While studying the singularity, you center a
circle of radius r on the point P and parameterize this circle in
the counter-clockwise direction as Cr;{x[t],y[t]}. You calculate
(integralCr) -n[x[t],y[t]]x'[t] + m[x[t], y[t]]y'[t]dt and find
that it is equal to -1 +...

Find the unit tangent vector T(t) and the curvature κ(t) for the
curve r(t) = <6t^3 , t, −3t^2 >.

Consider the vector field F = <2 x
y^3 , 3 x^2
y^2+sin y>. Compute
the line integral of this vector field along the quarter-circle,
center at the origin, above the x axis, going from the point (1 ,
0) to the point (0 , 1). HINT: Is there a potential?

Let f(x, y) = −x 3 + y 2 . Show that (0, 0) is a saddle point.
Note that you cannot use the second derivative test for this
function. Hint: Find the curve of intersection of the graph of f
with the xz-plane.

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