Hi,
I'm wondering if there is an exact solution to:
(2t+1)e^{-2t}=5
I've tried and have been unable to solve this for t. I can get an approximate numerical answer using the calculator but that's it.
Cheers
If I take F(x)=\sqrt{1+x^2}, then the derivative is always less than one so this is a contraction mapping from R to R, right?
But there is no fixed point where F(x)=x, where the contraction mapping theorem says there should be.
So where have I gone wrong?
Cheers
Homework Statement
When considering conservation of energy and momentum in the collision between a photon and an electron (in Compton scattering for example), is it reasonable to worry about 'stationary' electrons?
The Attempt at a Solution
From what I can recall the derivation of the...
Hi,
The problem is to determine whether or not relativistic effects are relevant for an electron accelerated to an energy of a) 100MeV and b) 100GeV. So I need to find the gamma factor of the electron in each of these cases.
I have used E = \gamma m_0 c^2 and solved for gamma = 195 in the...
Homework Statement
Suppose \Pi \subset \mathbb{R}^3 is a plane, and that P is a point not on \Pi . Assume that Q \in \Pi is a point on \Pi whose distance to P is minimal. Show that the vector PQ is orthogonal to \Pi . Define a differentiable vector function r(t) with r(t) \in \Pi and...
Homework Statement
The running time of quick sort can be improved in practice by taking advantage of the fast running time of insertion sort when its input is nearly sorted.
When quicksort is called on a subarray with fewer than k elements, let it simply return without sorting the subarray...
Thanks. So do you mean: view the transformation associated with matrix A in a basis of {eigenvector, orthogonal to eigenvector} and find the matrix for the transformation in this basis?
I'm not sure what the significance of the orthogonal vector is here.
Homework Statement
Let A be a 2x2 real matrix which cannot be diagonalized by any matrix P (real or complex). Prove there is an invertible real 2x2 matrix P such that
P^{-1}AP = \left( \begin{array}{cc} \lambda & 1 \\ 0 & \lambda \end{array} \right)
I know how to diagonalize a matrix...
An acid solution flows at a constant rate of 6L/min into a tank which initially holds 200L of a 0.5% acid solution. The solution in the tank is kept well mixed and flows out of the tank at 8L/min. If the solution entering the tank is 20% acid then determine the volume of acid in the tank after t...
Ok, thanks. The other part of the question goes:
A is a 4x3 matrix
C is a 3x4 matrix such that CA = I
Suppose, for some given b in R4 that Ax=b has at least one solution. Show that this solution is unique.
Can I just say x = Cb which implies that there is only one solution for x? I'm thinking...
Suppose A is a 3 x 4 matrix and there exists a 4 x 3 matrix C such that AC = I (the 3x3 identity matrix). Let b be an arbitrary vector in R3. Produce a solution of Ax=b.
I'm not quite sure what the question is asking. I think I just need someone to point me in the right direction.
Thanks
Homework Statement
Find the angle between the line x=7 and the catenary y=20\cosh{(\frac{x}{20})}-15
The Attempt at a Solution
I found the tangent has gradient \sinh{(\frac{7}{20})}
Then I used \tan{\theta}=|\frac{1}{m}|
where m=\sinh{(\frac{7}{20})}
And evaluated using inverse...
Thanks for the replies. I had another idea, I could show (by taking the lower sum of small interval slices) that the area under the graph of 1/x between 1 and 3 has to be greater than 1, therefore 3 > e.
Thanks for the reply. Thats what I was thinking, the difference gets larger as x gets larger. But then I thought, when you have a function like x^2 + x, it behaves more and more like x^2 as x gets larger. This function behaves more like x as x gets larger, doesn't it?
I am trying to draw the graph of
y = x + \sqrt{|x|}
Can I say that as x approaches infinity, y approaches x? That would mean that the function has an oblique asymptote at the line y=x but I'm not sure.
Thanks for any help!
Find the Taylor polynomial of degree 9 of
f(x) = e^x
about x=0 and hence approximate the value of e. Estimate the error in the approximation.
I have written the taylor polynomial and evaluated for x=1 to give an approximation of e.
Its just the error that is confusing me. I have:
R_n(x) =...
If it had a load on the wheel (like a generator) am I right to assume that this increases the moment of inertia? It would seem that the top speed of the wheel should be slower though?
So let's say the radius was 1m and the speed of the hamster was 10m/s
1 revolution = 2pi = 6.28 metres
time for 1 rev = 2pi/10 = 0.628 seconds
angular velocity = 0.628^-1 = 1.59 rev/s = 10radians/s
Correct?
Still confused, if there was no friction wouldn't the angular velocity increase forever...
Lol I guess so. But isn't it the weight force of the hamster that causes the torque? If its just the speed then how would I work out the angular velocity given the running speed of the animal?
If a hamster runs in a wheel, how do I figure out the rotational velocity that the wheel achieves (after it stops accelerating)? Is it to do with the mass of the hamster or the speed at which it runs?
I'm familiar with the basic concepts of rotational motion (moment of inertia, torque etc) but...
Hi
I'm trying to integrate
\int (-40x^3 + 38.4x^2 - 13.288x + 1.98072)\sqrt{14400x^4 - 18432x^3 + 9087.36x^2 - 2041.0368x+ 177.570944}dx
The way I thought I could do it was express the first part (the cubic) in terms of the derivative of the second and do it by substitution. Unfortunately...
Hi
I am trying to integrate:
\sqrt{1+u^2}
It looks simple but it's causing me a lot of problems. I've tried substitution, and by parts but can't get it. Thanks for any help!
I have a question about RMS voltage. I know Vrms is Vpeak/sqrt(2) if the sine wave is centered around 0V, but what if the voltage has a DC offset? Shouldn't this mean the RMS voltage is more?
Homework Statement
Use Gauss' law to find the charge density on a Van Der Graff dome (r=40cm) if it is charged to 100kV. What is the electric field strength at r=25cm?
I understand gauss's law and I know that I need to use it to find the total charge enclosed on the dome. I can then work out...
The circuit I'm working on has a number of capacitors that can be switched in parallel with a resistor to give different frequency response depending on which one you select.
I reworked the above as:
\frac {1}{z_{eq}} = \frac {1}{R} + j \omega C
which gives
|z_{eq}| = \frac {R}{\sqrt{1 + (...
ahhh i meant parallel, sorry
I got it like this:
\frac {1}{Z_eq} = \frac {1}{R} + \frac {1}{j \omega C}
Z_eq = \frac {R \omega C}{R + j \omega C}
|Z_eq| = \frac {R \omega C}{\sqrt{R^2 + ( \omega C )^2}}
Thanks for your help with this.
edit: should the second term in...
I know the impedance of a capacitor is,
\frac {1}{j \omega C}
so in an audio circuit it let's more high frequency energy through, which is obvious looking at the equation. When a capacitor is in parallel with a fixed resistor, I worked out the magnitude of the impedance of the pair to be...
I need to find three unknown resistances given the following three equations:
\frac{1}{R_2+R_3} + \frac{1}{R_1} = \frac{1}{670}
\frac{1}{R_1+R_3} + \frac{1}{R_2} = \frac{1}{679}
\frac{1}{R_1+R_2} + \frac{1}{R_3} = \frac{1}{1349}
I figured since there are three unknowns...
I have the following function:
h(x)= \frac{x^2}{x-1.5}
and I want to know what kind of equation you would call this. It looks like a hyperbola when I graphed it but it has no y^2 term.
Thanks heaps. I think they put them in the kit so I can solder the connecting wires to the board after I have mounted the board (ie can't get underneath it anymore).
Homework Statement
Expand and simplify the product of two quaternions:
(3 + 2i + 3j + 4k)(3 + 3i + 2j + 5k)
Justify your response.
The Attempt at a Solution
I have done this by expanding brackets normally, keeping the ijk's in the same order because the multiplication is not...
Thanks for the link. It says "When the underlying ring is commutative, for example, the real or complex number field, the two multiplications are the same. However, if the ring is not commutative, such as the quaternions, they may be different."
Lol, I am actually working with quaternions. The...
I have a question about commutativity.
I have two matrices X and Y and a constant k. I want to calculate X * kY. Can I bring k out the front to give k(X*Y)?