Question

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) .

Answer #1

Prove the identity
1) sin(u+v)/cos(u)cos(v)=tan(u)+tan(v)
2) sin(u+v)+sin(u-v)=2sin(u)cos(v)
3) (sin(theta)+cos(theta))^2=1+sin(2theta)

find t?
x= 2cos(t)+ sin(2t)
y=2sin(t)+cos(2t)
when x= 0, y= -3

The homogeneous solutions to an ODE are sin(2t) and cos(2t).
Suppose that the forcing function is 1.5 cos(2t) what is an
appropriate form of the general solution?
y(t)=Acos2t +Bsin2t + C t cos(2t+ᶲ)
, (b) y(t)=Acos2t +Bsin2t
+ C cos2t + Dsin2t
y(t)=Acos2t
+Bsin2t,
(d) y(t)=Acos2t +Bsin2t + C cos(2t+ᶲ)
What is the total number of linearly independent solutions that
the following ODE must have?
y" +5y'+6xy=sinx
Two (b) Four
(c) Three
(d) Five

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)

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) =

let r(t)=<cos(2t),sin(2t),3>
describe the shape of the path of motion of the object.
how far has the object travelled between time T= 0 and time T =
2pi?

Differential Geometry
(6) Find the Frenet apparatus of α(t) = (e2t cos(2t),
e2t sin(3t), e2t ).

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

Derive the Laplace transform for the following time
functions:
a. sin ωt u(t)
b. cos ωt u(t)

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

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