Question

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.

Answer #1

1.
(1 point)
Find the distance the point P(1, -6, 7), is to the plane through
the three points
Q(-1, -1, 5), R(-5, 2, 6), and S(3, -4, 8).
2.
(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)=

Find T, N, and κ for the plane curve r(t) = (7t+2) i + (5 - t^7)
j

1) Find the curvature of the curve r(t)= 〈4+3t,5−5t,4+5t〉 the
point t=5.
2) Find a plane through the points (2,-3,8), (-3,-3,-6),
(-6,3,-7)

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

Let y = x 2 + 3 be a curve in the plane.
(a) Give a vector-valued function ~r(t) for the curve y = x 2 +
3.
(b) Find the curvature (κ) of ~r(t) at the point (0, 3). [Hint:
do not try to find the entire function for κ and then plug in t =
0. Instead, find |~v(0)| and dT~ dt (0) so that κ(0) = 1 |~v(0)|
dT~ dt (0) .]
(c) Find the center and...

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

17.)Find the curvature of r(t) at the
point (1, 0, 0).
r(t) =
et
cos(t),
et
sin(t), 3t
κ =

Find the curvature κ(t)κ(t) of the curve
r(t)=(−5sint)i+(−5sint)j+(−4cost)k

1) Find the curvature of the curve r(t)= 〈2cos(5t),2sin(5t),t〉
at the point t=0
Give your answer to two decimal places
2) Find the tangential and normal components of the acceleration
vector for the curve r(t)=〈 t,5t^2,−5t^5〉 at the point t=2
a(2)=? →T + →N

Give your answer to two decimal places
1) Find the curvature of the curve r(t)=〈 5+ 5cos t , −5 ,−5sin
t 〉 at the point t=11/12π
2) Find the curvature of the curve r(t)= 〈4+3t,5−5t,4+5t〉 the
point t=5.

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