Homework Statement
Let {an} be a sequence with positive terms such that lim an = L > 0.
Prove lim (an)x = Lx.
Homework Equations
If x is a real number, there exists an increasing rational sequence {rn} with limit x.
A monotone sequence {an} is convergent if and only if {an} is...
Hi Dick,
Thank you very much for your help. I feel that I can prove the property of rationals that your described.
However, while I know that there are reals that don't have that property, such as \pi and the sqrt(1), I do not know how to prove that those two numbers do not have that...
Homework Statement
Prove that there is no isomorphism, \phi, from Q under addition to R under addition
Homework Equations
An isomorphism \phi:Q to R is a bijection such that \phi(x + y) = \phi(x) + \phi(y), where x,y are elements of Q
\phi(0) = 0.
\phi(-x) = -\phi(x)
The...
integrating both sides gives ln(y) = ln(x)/4 = ln(x^1/4)... so y = x^1/4.. as for the inital condition, my inclination would be to plug in (2,3) into the equation... so 3 = 2^1/4 + C.. However, based on what I think it should be, instead of adding C I should be multiplying by C. But I don't know...
[SOLVED] Integrating factors or separating the variables
Homework Statement
The following equation can be solved by intergrating factors or by separating the variables.
\frac{dy}{dx} - \frac{y}{4x} = 0
with the initial condition of y(2)=3
Homework Equations
The Attempt at a...
if x(t) = e^{-8t} and y(t) = e^{-2t} that would work right? so differential equations are just logic? at least at this level anyways..
but as for the constant \alpha you were talking about.. that would change my parametrizations to x(t) = e^{-8t} and y(t) = 4e^{-2t} if it is to go through the...
[SOLVED] Trajectory using gradient and differential equations
Homework Statement
A heat-seeking particle is located at the point P on a flat metal plate whose temperature
at a point (x, y) is T(x, y). Find parametric equations for the trajectory of the particle if
it moves continuously in...
oh ok...so
the derivative of \frac{dz}{du} = \frac{d^{2} z}{du^{2}} \frac{\partial u}{\partial x}
because we really are taking the derivative with respect to x.. so basically there was another chain that I didn't see..
is that right?
[SOLVED] Chain rule problem with partial derivatives
Homework Statement
Suppose that z = f(u) and u = g(x,y). Show that..
\frac{\partial^{2} z}{\partial x^{2}} = \frac{dz}{du} \frac{\partial^{2} u}{\partial x^{2}} + \frac{d^{2} z}{du^{2}} \frac{(\partial u)^{2}}{(\partial x)^{2}}...
[SOLVED] Partial differential with respect to y
Homework Statement
Given the equation f(x,y) = (x^{3} + y^{3})^{1/3}
Show that f_{y}(0,0) = 1
Homework Equations
Basic chain rule..
The Attempt at a Solution
Based on the chain rule...I believe that
f_{y} = 1/3(x^{3} +...