Question

Give an example to show that the curvature k(t) of a regular curve
might not be a smooth function of t without the assumption that
k>0.

Answer #1

Consider the curve r(t) = i + tj + e^(t)k
a) find the curvature k
b) Find the normal plane at the curve (1,0,1)

Find the curvature and radius of curvature for the curve swept
out by r(t) = 4cos(t) i + 4cos(t) j + t k. Use the formula K(t) = (
||r'(t)*r"(t)|| ) / ( ||r'(t)|| )^3

1) Find the curvature of the curve r(t)= 〈2cos(5t),2sin(5t),t〉
at the point t=0
Give your answer to two decimal places
2) Find the tangential and normal components of the acceleration
vector for the curve r(t)=〈 t,5t^2,−5t^5〉 at the point t=2
a(2)=? →T + →N

Give your answer to two decimal places
1) Find the curvature of the curve r(t)=〈 5+ 5cos t , −5 ,−5sin
t 〉 at the point t=11/12π
2) Find the curvature of the curve r(t)= 〈4+3t,5−5t,4+5t〉 the
point t=5.

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...

Problem 5. Find the curvature of the curve R(t)=(t-sinh(t),
4cosh(t/2)), t > 0.

Find the curvature of the curve r(t)= < et , t ,
t2 >.

Find the curvature of the curve r(t) = <etcos(t),
etsin(t), t> at the point (1,0,0)

Find the curvature, k(t), of the following:
r(t) = t i + t^2
j + e^t k

Why when the stock pays dividends. Give a numerical example
(choosing x, k, r, T−t,σ) in which it is obvious (without any
formulas) that American put price on
a nondividend paying stock is larger then the corresponding
European put price.

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