Consider the helix
r(t)=(cos(2t),sin(2t),−3t)r(t)=(cos(2t),sin(2t),−3t).
Compute, at t=π/6
A. The unit tangent vector T=T= ( , , )
B. The unit normal vector N=N= ( , , )
C. The unit binormal vector B=B= ( , , )
D. The curvature κ=κ=
Given r(t) = <2 cos(t), 2 sin(t), 2t>. • What is the arc
length of r(t) from t = 0 to t = 5. SET UP integral but DO NOT
evaluate • What is the curvature κ(t)?
Let r(t) = 2t ,4t2 ,2t be a position function for some
object.
(a) (2 pts) Find the position of the object at t = 1. (b) (6
pts) Find the velocity of the object at t = 1.
(c) (6 pts) Find the acceleration of the object at t = 1. (d) (6
pts) Find the speed of the object at t = 1.
(e) (15 pts) Find the curvature K of the graph C determined by
r(t) when...
Given the vector function r(t)=〈√t , 1/(t-1) ,e^2t 〉 a) Find: ∫
r(t)dt b) Calculate the definite integral of r(t) for 2 ≤ t ≤ 3
can you please provide a Matlab code?
Let r : R->R3 be a path with constant speed,
satisfying
r"(t) = (4 sin(2t))i + (-4 cos t)j + (4 cos(2t))k for all t
belongs to R:
Find the curvature w.r.t. t of r. (Hint: cos(2t), sin(2t), and
cos(t) are linearly independent. i.e. if c1cos(2t) +
c2sin(2t)+
c3cos(t) = 0 for all t belongs to R, then c1
= c2 = c3 = 0.)
How to find the unit vectors for the following equation: r(t) =
<e^t,2e^-t,2t>
A) Compute the unit Tangent Vector, unit Normal Vector, and unit
Binomial Vector.
B) Find a formula for k, the curvature.
C) Find the normal and osculating planes at t=0
Let L be the line parametrically by~r(t) = [1 + 2t,4 +t,2 + 3t]
and M be the line through the points P= (−5,2,−3) and
Q=(1,2,−6).
a) The lines L and M intersect; find the point of
intersection.
b) How many planes contain both lines?
c) Give a parametric equation for a plane Π that contains both
lines
Consider the vector function given below.
r(t) =
2t, 3 cos(t), 3 sin(t)
(a) Find the unit tangent and unit normal vectors T(t) and
N(t).
T(t) =
N(t) =
(b) Use this formula to find the curvature.
κ(t) =
There is an old drug for a certain disease. The cure rate of the
old drug (the proportion of patients cured) is 0.8.
Pharma Co has developed a new drug for this disease. The company
is conducting a small field trial (trial with real patients) of the
new drug and the old drug. Assume that patient outcomes are
independent. Also, assume the probability that any one patient will
be cured when they take the drug is p. For the
old drug,...