If u(t) = < sin(8t), cos(4t), t > and v(t) = < t,
cos(4t), sin(8t) >, use the formula below to find the given
derivative.
d/(dt)[u(t)* v(t)] =
u'(t)* v(t) +
u(t)* v'(t)
d/(dt)[u(t) x v(t)] =
<.______ , _________ , _______>
Let Zt = U sin(2*pi*t) + V cos(2*pi*t), where U and V are
independent random variables, each with
mean 0 and variance 1.
(a) Is Zt strictly stationary?
(b) Is Zt weakly stationary?
Consider a fruit fly flying a room with velocity v(t) =
< -sin(t), cos(t), 1 >
a. if the z = 1 + 2(pi) is the room's ceiling, where
will the fly hit the ceiling?
b. if the temperature in the room is T(z) = 65
+ (1/2)z2 how quickly is the temperature increasing for
the fly at time t = 2.
c. from the velocity, find the location of the fruit fly at time
t if at t =...
Given the parametrized curve r(u) = a cos u(1 − cos u)ˆi + a sin
u(1 − cos u)ˆj, u ∈ [0, 2π [ , (with a being a constant)
i) Sketch the curve (e.g. by constructing a table of values or
some other method)
ii) Find the tangent vector r 0 (u). What is the tangent vector
at u = 0? And at u = 2π? Explain your result.
iii) Is the curve regularly parametrized? Motivate your answer
using...
P(u,v)=(f(v)cos(u),f(v)sin(v),g(v))
Find formulas for the Christoffel symbols, the second
fundamental form, the shape operator, the Gaussian curvature and
the mean curvature.
. Given the
following non-periodic signal:
x(t) = 3 e-5t cos(12t)
u(t)
Find the Fourier transform expression X(ω) without
using Table.
Calculate the magnitude spectrum of X(ω) for ω = π/8, π/4, and
π/2
Let
f (t) =
{
7
0 ≤ t ≤ 2π
cos(5t)
2π < t ≤ 4π
e3(t−4π)
t > 4π
(a)
f (t) can be written in the form
g1(t) +
g2(t)U(t −
2π) +
g3(t)U(t −
4π)
where U(t) is the Heaviside function. Enter the
functions g1(t),
g2(t), and
g3(t), into the answer box below (in
that order), separated with commas.
(b)
Compute the Laplace transform of
f (t).
Calculus dictates that
(∂U/∂V) T,Ni = T(∂S/∂V)T,Ni – p = T(∂p/∂T)V,Ni – p
(a) Calculate (∂U/∂V) T,N for an ideal gas [ for which p = nRT/V
]
(b) Calculate (∂U/∂V) T,N for a van der Waals gas
[ for which p = nRT/(V–nb) – a (n/V)2 ]
(c) Give a physical explanation for the difference between the
two.
(Note: Since the mole number n is just the particle number N
divided by Avogadro’s number, holding one constant is equivalent...
Using Matlab, Write a script that validates the relationship
between sin u, cos u, and tan u by evaluating these functions at
suitably chosen values. Please screenshot Matlab screen. Thank
you!