Question

1. A plane curve has been parametrized with the following vector-valued function, r(t) = (t + 2)i + (-2t2 + t + 1)j a. Carefully make 2 sketches of the plane curve over the interval . (5 pts) b. Compute the velocity and acceleration vectors, v(t) and a(t). (6 pts) c. On the 1st graph, sketch the position, velocity and acceleration vectors at t=-1. (5 pts) d. Compute the unit tangent and principal unit normal vectors, T and N at t=-1. (6 pts) e. Compute the curvature, κ(t), at t=-1 using 3 different methods. (12 pts) f. Compute the normal and tangential components of the acceleration vector, and , at t=-1. Also verify the value of using an alternative method. (9 pts) g. On the 2nd graph, again sketch the acceleration vector at t=-1 as well as sketch the normal and tangential vector components of the acceleration vector, and . (6 pts)

Answer #1

Consider the following vector function.
r(t) = <9t,1/2(t)2,t2>
(a) Find the unit tangent and unit normal vectors
T(t) and
N(t).
(b) Use this formula to find the curvature.
κ(t) =

sketch the plane curve and sketch the unit tangent and the unit
normal vectors. r(t)= ti+1/t j, t=2

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

An object moves in a coordinate system with
velocity vector v(t) = < sqrt(1+2t), tcos(?t), tsin(?t) > for
t >= 0.
?=pi
a. Find the equation of the tangent line to
the objects path when it reached the
origin.
b. When the object reached the origin, with
what angle did it strike the xy-plane?
c. Give the (unit) tangent, (unit) normal, and
binormal directions for the object when it hits the
xy-plane.
d. Give a vector-valued function describing
the objects...

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

Letr(t)=(4t2+1)i+2tjfor−2≤t≤2.
(a) (10 pts) Draw a sketch of the curve C determined by
r(t).
(b) (5 pts) Plot r(0) and label its endpoint P.
(c) (10 pts) Plot the vector tangent to C at P .
(d) (10 pts) Find the equation of the line tangent to C at P
(you may give this parametrically
or not).
(e) (5 pts) Find the curvature K at P .
(f) (5 pts) Find the radius of curvature ρ at P.
(g) (5...

6. Given vector function r(t) = t2 − 2t, 1 + 3t, 1 3 t 3 + 1 2 t
2 i (a) Find r 0 (t) (b) Find the unit tangent vector to the space
curve of r(t) at t = 3. (c) Find the vector equation of the tangent
line to the curve at t = 3

Let r(t) = 1/2t ,4t^1/2 ,2t be a position function for some
object.
(a) (2 pts) Find the position of the object at t = 1. (b) (6
pts) Find the velocity of the object at t = 1.
(c) (6 pts) Find the acceleration of the object at t = 1. (d) (6
pts) Find the speed of the object at t = 1.
(e) (15 pts) Find the curvature K of the graph C determined by
r(t) when...

Let C be the curve parametrized by r(t)=(t2+2)i+(1+t)j+2t^2k,
with 0≤t≤. Consider the conservative vector field
F=yz2i+xz2j+2xyzk, Calculate ∫CF⋅dr

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