Question

Differential Geometry

(6) Find the Frenet apparatus of α(t) = (e^{2t} cos(2t),
e^{2t} sin(3t), e^{2t} ).

Answer #1

Find the Laplace Transform of the following functions:
1. e^(-2t+1)
2. cos^2(2t)
3. sin^2(3t)

If u(t) = sin(6t), cos(2t), t and v(t) = t, cos(2t), sin(6t) ,
use Formula 4 of this theorem to find d dt u(t) · v(t) .

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.

Find a unit tangent vector to the curve r = 3 cos 3t
i + 3 sin 2t j at t =
π/6 .

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

r(t)=[cos(t),sin(t),cos(3t)]
r(t)=[tcos(t),tsin(t),t)
r(t)=[cos(t),sin(t),t2]
r(t)=[t2cos(t),t2sin(t),t]
r(t)=[cos(t),t,sin(t)]
Sketch the graphs.

Differential equations
Given that x1(t) = cos t is a solution of (sin t)x′′ − 2(cos
t)x′ − (sin t)x = 0, find a second linearly independent solution of
this equation.

Derive the Laplace transform of the following time domain
functions
A) 12 B) 3t sin(5t) u(t) C) 2t^2 cos(3t) u(t) D) 2e^-5t
sin(5t)
E) 8e^-3t cos(4t) F) (cost)&(t-pi/4)

Find the length of the curve
1) x=2sin t+2t, y=2cos t, 0≤t≤pi
2) x=6 cos t, y=6 sin t, 0≤t≤pi
3) x=7sin t- 7t cos t, y=7cos t+ 7 t sin t, 0≤t≤pi/4

Compute the Laplace transform of functions
a) f(t) = e^(−3t) sin(5t)
b) f(t) = (2t + 3)e^(−t)

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