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=
3. Consider the parametric curve x = sin 2t, y = − cos 2t for
−π/4 ≤ t ≤ π/4.
(a) (2 pts) Find the Cartesian form of the curve.
(b) (3 pts) Sketch the curve. Label the starting point and
ending point, and draw an
arrow on the curve to indicate the direction of travel.
(c) (5 pts) Find an equation for the curve’s tangent line at the
point
√2/2, −√2/2
.
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)] =
<.______ , _________ , _______>
Find an equation of the tangent line to the curve cos ( x ) + 11
y ^2 = x y ^3 + 34 at the point ( 0 , √ 3 ) . Assume that y is a
function of x . Express all numbers in exact form and write the
equation of the tangent line in terms of x and y .
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