Recent content by CaptainBlack

\(\log_{24}(48)=1+\log_{24}(2)\) But \(2^5 \gt 24\) so \(\log_{24}(2) \gt 1/5\) Also: \(\log_{12}(54)=1+\log_{12}(4.5)\), and \(4.5^5>12^3\) so \(\log_{12}(4.5)>3/5\) Hence: \[ (\log_{24}(48))^2 + (\log_{12}(54))^2 \gt 1.2^2+1.6^2 =4 \] CB

The question is slightly ambiguous in asking which charater you like most, and here changing that to who do you identify with. My family think I'm a cross between Leonard and Raj, I suppose I think I'm most like Leonard, but there was a time (a long time ago in a galaxy far far away) when I...

Your link is broken: Wikipedia Combinatorial Number System CB

Re: Ship problem , related time change Write the equations of the position of the two ships as a function of time. Now write the square of the distance between them as a function of time, and differentiate... CB

Re: Two Species Population Model These are one of the Lanchester models of combat, in this case you can solve it by differentiating one of the equations again to get: \[\frac{d^2A}{dt^2}=+k_1k_2A\] Which is a linear constant coefficient ODE and so the general solution can be found by the usual...

To say that \(f(n,k)\in O(n+k)\) means that there exists a \(C>0\) such that for \(n+k\) large enough: \[|f(n,k)| < C |n+k|\] That is they jointly define the bound on the growth of \(|f(x,k)|\) CB

In small time interval \(\Delta t\) all the vehicles less than a distance \(u(x,t)\Delta t\) down stream of \(x\) will pass \(x\). The number of vehicles in this stretch of road is \(u(x,t)\rho(x,t)\Delta t\), so \(u(x,t)\rho(x,t)\Delta t\) vehicles pass \(x\) in \(\Delta t\) so the vehicle flow...

Consider a road element between \(x\) and \(x+\Delta x\) the traffic flow into the element at \(x\) per unit time is \(u(\rho(x,t))\rho(x,t)\) and out at \(x+\Delta x\) is \(u(\rho(x+\Delta x,t))\rho(x+\Delta x,t)\) Therefore the rate of change of car numbers in the element is: \[\frac{\partial...

The next step is to write down the partial differential equation satisfied by the traffic density. This is derivable from a conservation of mass (or vehicle numbers) argument that you will have seen innumerable times. CB

For part (a) you need to observe that the flow rate \(f(x,t)\) in vehicles per unit time is \(u(x,t) \rho(x,t)\). Now you need to show that for (i) and (ii) that \(u(x,t)\le u_{sl}\), then as \(\rho(x,t) \le \rho_{max}\) you will have shown that the flow rate: \[f(x,t)\le u_{sl}\rho_{max}\]

A discrete random variable is one that can take values from a discrete set (one where each value is some how separated from its neighbours). A continuous random variable is one that can take any value in some interval of the real line. So what do you think for A,B,C and D CB

Even if that all works OK (and I am pretty sure that the probability of it being error free is as close to zero as I can imagine) you have the problem of proving the compiler/interpreter is correct, and even then you need to prove that the hardware is correct. (I will let you off the need to...

And who will prove the program correct? CB

(a) First sketch the curve. It obviously starts with slope \(2\) at \((0,0)\) and rises to a maximum of \(y=1\) at \(x=1/\sqrt(2)\) and then falls to \(y=0\) at \(x=1\). The area we want is the integral: \[I = \int_{x=0}^1 y(x) dx\] Use the substitution \(t=arcsin(x), x=sin(t)\). Then \(dx =...

"C4 question, please help.? the curve C has parametric equations x = sint , y = sin2t, 0<t<pi/2 a) find the area of the region bounded by C and the x-axis and, if this region is revolved through 2pi radians about the x-axis, b) find the volume of the solid formed How do you do this...