Question

17.)Find the curvature of **r**(*t*) at the
point (1, 0, 0).

**r**(* t*) =

*e*^{t}
cos(* t*),

*κ* =

Answer #1

Find an arc length parametrization of
r(t) =
(et
sin(t),
et
cos(t),
10et )

Find the curvature of r(t) at the
point (3, 1, 1).
r(t) = <3t, t^2 , t^3>
k=

Consider the following vector function.
r(t) =
6t2, sin(t) − t cos(t), cos(t) + t sin(t)
, t > 0
(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) =

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

Find the unit tangent vector T(t) and the curvature κ(t) for the
curve r(t) = <6t^3 , t, −3t^2 >.

Given r(t) = (et cos(t) )i + (et sin(t) )j
+ 2k. Find
(i) unit tangent vector T.
(ii) principal unit normal vector N.

1.
(1 point)
Calculate κ(t)κ(t) when
r(t)=〈3t^(−1),5,1t〉
κ(t)=
2.
(1 point)
Find the arclength of the curve r(t)=〈−3sint,6t,−3cost〉, −9≤t≤9

Find the derivative r '(t) of the
vector function r(t).
<t cos 3t , t2, t sin 3t>

1.
(1 point) For the curve given by
r(t)=〈−7t,−4t,1+7t2〉r(t)=〈−7t,−4t,1+7t2〉,
Find the derivative
r′(t)=〈r′(t)=〈, , , 〉
Find the second derivative
r″(t)=〈r″(t)=〈
Find the curvature at t=1t=1
κ(1)=κ(1)=
2.
(1 point) Find the distance from the point (-1, -5, 3) to the
plane −4x+4y+0z=−3.

6) please show steps and explanation.
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the area of the triangle
PQR.

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