Question

Determine D_t [r (t) x u (t)]

given

r(t) = cos9t) i + sen (t) j + tk

and

u(t) = j + tk

Answer #1

The position of an object at time t is given by: r(t)=e^−t i +
e^t j − t√2 k, 0≤ t<∞. (a) Determine the velocity v and the
speed of the object at time t. (b) Determine the acceleration of
the object at time t. (c) Find the distance that the object travels
during the time interval 0≤ t<ln3. Answers: (a) = velocity: v
=−e^−t i + e^t j − √2 k; speed: ||v||= e^t + e^−t, (b) =
acceleration:...

Find the work done by the force ﬁeld F(x,y,z) = yz i + xz j + xy
k acting along the curve given by r(t) = t3 i + t2 j + tk from the
point (1,1,1) to the point (8,4,2).

8. Find r(t) given the following information.
r''(t)= 8 i + 12t k, r'(0)=6 j , r(0)= -4 i

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

Find the velocity, acceleration, and speed of a particle with
the given position function.
(a) r(t) = e^t cos(t)i+e^t
sin(t)j+ te^tk, t = 0
(b) r(t) = 〈t^5 ,sin(t)+ t ^ cos(t),cos(t)+ t^2 sin(t)〉, t =
1

Suppose T: P2(R) ---> P2(R) by T(p(x)) = x^2 p''(x) + xp'(x)
and U : P2(R) --> R by U(p(x)) = p(0) + p'(0) + p''(0).
a. Calculate U composed of T(p(x)) without using matricies.
b. Assuming the standard bases for P2(R) and R find matrix
representations of T, U, and U composed of T.
c. Show through matrix multiplication that the matrix
representation of U composed of T equals the product of the matrix
representations of U and T.

A particle of mass 2.00 kg moves with position r(t) = x(t) i +
y(t) j where x(t) = 10t2 and y(t) = -3t + 2, with x and y in meters
and t in seconds.
(a) Find the momentum of the particle at time t = 1.00 s.
(b) Find the angular momentum about the origin at time t = 3.00
s.

A particle moves with position r(t) = x(t) i + y(t) j where
x(t) = 10t2 and y(t) = -3t + 2, with x and y in meters and t in
seconds.
(a) Find the average velocity for the time interval from 1.00
s to 3.00 s.
(b) Find the instantaneous velocity at t = 1.00 s.
(c) Find the average acceleration from 1.00 s to 3.00 s.
(d) Find the instantaneous acceleration at t = 1.00 s.

a. Let →u = (x, y, z) ∈ R^3 and define T : R^3 → R^3 as
T( →u ) = T(x, y, z) = (x + y, 2z − y, x − z)
Find the standard matrix for T and decide whether the map T is
invertible.
If yes then find the inverse transformation, if no, then explain
why.
b. Let (x, y, z) ∈ R^3 be given T : R^3 → R^2 by T(x, y, z) = (x...

Evaluate the following.
f(x, y) = x + y
S: r(u, v) = 5
cos(u) i + 5 sin(u)
j + v k, 0 ≤ u
≤ π/2, 0 ≤ v ≤ 3

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