In: Math

Consider the helix
**r**(*t*)=(cos(2*t*),sin(2*t*),−3*t*)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=

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) =

Consider the spiral path x(t) = (cos^2t,sin^2t,t) for 0 ≤ t ≤
π/2. Evaluate the integral x dx−y dy + z^2 dz

Consider the following vector function. r(t) =<3t, 1/2 t2,
t2> (a) Find the unit tangent and unit normal vectors T(t) and
N(t).
(b). Find the curvature k(t).

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
.

3sin(t)+3sin(t)sin(2t)=3cos(t)-cos(3t).
I'm trying to solve for t.

(1 point) For the given position vectors r(t)r(t) compute the
unit tangent vector T(t)T(t) for the given value of tt .
A) Let r(t)=〈cos5t,sin5t〉
Then T(π4)〈
B) Let r(t)=〈t^2,t^3〉
Then T(4)=〈
C) Let r(t)=e^(5t)i+e^(−4t)j+tk
Then T(−5)=

compute the unit tangent vector T and the principal normal unit
vector N of the space curve R(t)=<2t, t^2, 1/3t^3> at the
point when t=1. Then find its length over the domain [0,2]

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)?

Let r(t) = (cos(πt), sin(πt), 3t). Calculate r'(t), T(t) and
evaluate T(1).

For this parametrized curve:
x = e^(2t) sin t , y = cos(4t)
find tangent line to curve when t=1

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