Show that at every point on the curve
r(t) = <(e^(t)*cos(t)), (e^(t)*sin(t)), e^t>
the angle between...
Show that at every point on the curve
r(t) = <(e^(t)*cos(t)), (e^(t)*sin(t)), e^t>
the angle between the unit tangent vector and the z-axis is the
same. Then show that the same result holds true for the unit normal
and binormal vectors.
Find the point of intersection of the tangent lines to the curve
r(t) = 5 sin(πt), 2 sin(πt), 6 cos(πt) at the points where t = 0
and t = 0.5. (x, y, z) =
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=
The plane curve represented by x(t) = t − sin(t), y(t) = 7 −
cos(t), is a cycloid.
(a) Find the slope of the tangent line to the cycloid for 0 <
t < 2π.
dy
dx
(b) Find an equation of the tangent line to the cycloid at t
=
π
3
(c) Find the length of the cycloid from t = 0 to t =
π
2
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)?
If u(t) = < sin(8t), cos(4t), t > and v(t) = < t,
cos(4t), sin(8t) >, use the formula below to find the given
derivative.
d/(dt)[u(t)* v(t)] =
u'(t)* v(t) +
u(t)* v'(t)
d/(dt)[u(t) x v(t)] =
<.______ , _________ , _______>
If u(t) = < sin(5t),
cos(5t), t > and
v(t) = < t, cos(5t),
sin(5t) >, use the formula below to find the given
derivative.
d/dt[ u(t) * v(t)] = u'(t) * v(t) + u(t)* v'(t)
d/dt [ u(t) x v(t)] = ?
Given the parametrized curve r(u) = a cos u(1 − cos u)ˆi + a sin
u(1 − cos u)ˆj, u ∈ [0, 2π [ , (with a being a constant)
i) Sketch the curve (e.g. by constructing a table of values or
some other method)
ii) Find the tangent vector r 0 (u). What is the tangent vector
at u = 0? And at u = 2π? Explain your result.
iii) Is the curve regularly parametrized? Motivate your answer
using...
show your work. Find the sin t and cos t for t
=150o.
You may need to draw the reference angle first, but you only
have to enter the sine and cosine values.