Exponential Definition and 1000 Threads

Homework Statement A radioactive substance has a half-life of 20 years. If 8 mg of the substance remains after 100 years, find how much of the substance was initially present. Homework Equations A=A0ekt The Attempt at a Solution I set the equation up so that 8=A0e100k and...

Homework Statement In the 2 following problems they use the term in the brackets differently, in one case its a percentage and in the other case i have no idea where they get the number from, this is what i would like to find out A cell loses 2% of its charge every day C is total charge t...

Homework Statement I have to evaluate P(t)=|<+,n|\exp{\frac{-iHt}{\hbar}}|+,n>|^2 where H=\hbar \omega_0 S_z + \hbar \omega a^+a+\hbar \lambda(a^+S_-+aS_+) and |+,n>=\left( \begin{array}{c} 1\\0 \end{array} \right) Homework Equations Eigenvalues of H are E_\pm =\hbar \omega (n...

Homework Statement i = 12.5(1-e-t/CR) Current = i Resistance = R Capacitance = C Time = t Firstly i had to calculate the current flowing after 0.5s, given a 30k\Omega resistance and a 20µF capacitance.. easily done. Then i had to work out how long it took to reach 10A, so...

I need to write 2 equations that represent the same exponential function with a y-intercept of 5 and an asymptote at y=3. I got y=2^(x+1) + 3 but I don't know how to find the second equation. Can someone please explain this to me. Thanks.

Does anyone know how to prove this? exp(jAt)+exp(jBt)= 2exp(j(A+B)/2)cos((A-B)/2)

I'm sure this integral is easy, but could someone perhaps show the working of: \int e^{i2t} dt between t and 0. I've tried it with trigonometric identities and keep getting lost! Cheers!

Homework Statement Let z=|z|e^{\alpha*i} Using the fact that z*w=|z||w|e^{i(\alpha+\beta)}, find all solutions to z^4 = -1 The Attempt at a Solution Not quite sure how to proceed, except for the obvious step i=z^2=|z*z|e^{i(2\alpha)}= |z*z|[cos(2\alpha)+isin(2\alpha)] Kinda stuck here :s...

Hi, I am having trouble with this exponential limit problem. Homework Statement \lim \ x \rightarrow 0 \ [ e^{-1/x}] Homework Equations Answer: 0 The Attempt at a Solution \lim x \rightarrow 0 e^(-x)^{-1} \lim x \rightarrow 0 e^{-1}^{x}^{-1} I don't think the second step is right...

Hi, I currently need to solve a problem which leads to an equation of the form a*exp(a*x)+b*exp(b*x)-c=0. The difference between a and b can often become very big, like a=1000 and b=1. Therefore numerical algorithms I tried so far, fail to solve this equation. Has anyone got any clues on a...

Hi, besides integration by parts, does anyone know of a simple integration trick to solve the integral (wrt x) of exp(x)*f'(x)?

Homework Statement Let's say A is a 7x7 matrix which is defined as [a b c 0 0 0 0; b a 0 d 0 0 0; c 0 a b e 0 0; 0 d b f 0 e 0; 0 0 d 0 f b g; 0 0 0 d b f h; 0 0 0 0 0 0 0] where semicolon (;) represent a new row and a space is a new column.Homework Equations If y = expm (A*t), where expm...

Homework Statement exp^\prime(0)B=B for all n by n matrices B. Homework Equations exp(A)= \sum_{k=0}^\infty A^k/k! The Attempt at a Solution Obviously I want to calculate the limit of some series, but I don't know what series to calculate. I wanted to try \lim_{h \to...

Homework Statement I am trying the find the derivative of the function: y= (2x-1)^(tan(3x)) Homework Equations Chain rule, in this case: f'(g(h(x))) * g'(h(x)) * h'(x) exponent rule: where (d/dx) a^x = (a^x) ln a The Attempt at a Solution I feel as if to apply the...

Homework Statement A bacteria culture starts with 3000 bacteria. After 3 h, the estimated counting is 48 000. What is the doubling period? The Attempt at a Solution I figured it would look like the half-life formula, so I wrote it as: B = B0(0.5)t/h Then I subbed numbers in: 48...

Homework Statement A colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 3 hours there are 10,000 bacteria. At the end of 5 hours there are 40,000. How many bacteria were present initially? Homework...

I am trying to compute the inverse Fourier transform numerically (using a DFT) for some complicated characteristic functions in order to compute their corresponding probability distribution functions. As a test case I thought I would invert the characteristic function for the simple exponential...

X1,X2,...,XN are independently identically exponentially distributed with expected value of 5. How can I compute X[bar]n when n=20 and N=1000? Then compute the proportion of values of X[bar]n that lie between 6.99 and 7.01. repeat the above question with n=100 My thoughts so basically...

how to compute the nonconstant matrix exponential؟ i.e. if there is a matrix A(t), how i can find exp(A(t)) ?

Homework Statement http://d.imagehost.org/view/0659/Capture Link to wolfram alfa:http://www.wolframalpha.com/input/?i=integrate%28cos%28e^x%29*e^x What i don't understand is why whey do it like this and why i can't integrate by parts in this case? Thanks for any replies!

I really appreciate if anyone could indicate me how to handle this inverse Laplace transformation (ILT): L-1[Exp(-c0*Sqrt(a(s)))/Sqrt(a(s))] where a(s)=(s2+c1s)/(c2s+c3) c0,c1,c2,c3 are all constants. I searched some literatures regarding the ILT of Exp funtions but no such form. I used...

Homework Statement If there is a quantity T comprised of two other quantities such that T=L+B, and quantity T and B are both increasing in every period at a fixed percent such that %growth T > %growth B, it will be true that %growth L > %growth T. It will also be true that as n approaches...

Hello, Something I've wondered about for some time is what happens to units once we pump them into a exponential or log function. For example in neutron attenuation I(x) = I_0 * exp(-Sigma * x) I feel like this something I should know, but I just don't get it. I suspect what's...

I am having difficulty trying to rearrange this equation to solve for one unknown variable. i=(io)exp(z*α*F*∆V)/(R*T) How would I rearrange this equation if i wanted to solve for io, also how would I rearrange this equation to solve for say α.

Homework Statement Show that e^{tA} = I - A + e^{t}A t \in T \ \ \ \ T \subset R R being the set of real numbers and T some interval. The matrix A is a projection matrix. i.e. A^2 = A The Attempt at a Solution First attempt at the problem involved showing that e^{tA}...

start point (1-Exp[-i x])^2, (i^2 = -1) finish point Sin^2(x)

I know how to do Frobenius on variable coefficient ODE's but only when the coefficients are powers of the independent variable. Can I do method of Frobenius on something like: y'' + e-xy = 0 ? What form would I assume a solution of? Just the regular y=sum(Akxk+r ? Thanks for the help!

Homework Statement Find the general solution to x'' + e^(-2t)x = 0, where '' = d2/dt2 Homework Equations - The Attempt at a Solution First I did a change of variables: Let u = e^(-t) Then du/dt = -e^(-t) dx/dt = dx/du*du/dt = -e^(-t)*dx/du d2x/dt2 = d/du(dx/dt)du/dt =...

I don not know whether I was right or not, please give me a hint. (R3,+) can be considered a Lie group. and its TG in 0 is still R3. suppose X as a infinitesimal generater, it can give a left-invariant vector field and also an one-parameter subgroup. but i think, this one-parameter...

Homework Statement I was given three plots of solutions for a forced exponential diffeq: y'[t]+1.85 y[t]=0.7t^2 with starter values on y[0] equal to -6, 0,and +7 The three plots eventually merge, how do I give the formula for this parabola? Homework Equations E^(-r t)...

I was given three plots of solutions for a forced exponential diffeq: y'[t]+1.85 y[t]=0.7t^2 with starter values on y[0] equal to -6, 0,and +7 The three plots eventually merge, how do I give the formula for this parabola?

Homework Statement Here is the question: solve y'[t]+0.2y[t]=Sin[t] with y[0]=15.3 How do I come up with a formula for y[t]? Thanks in advance! Homework Equations y[t]= starter+e^-rt The Attempt at a Solution I attempted to solve the equation as is. After doing some algebra...

Homework Statement Expand (1+e^(2x)), and (1+e^x)^2. Are they any different. I am confused please help Homework Equations The Attempt at a Solution

if i have a set of data, where as time increases, so does y, but is bounded by a number say y=c, how do i formulate my equation? how do i find the constants? i have y=c-a-x*b this is basically a transformation of y=b*ax+c

Homework Statement The problem is 3^2x+3^(x+1)-4=0 I know I had to solve it by substitution Homework Equations The Attempt at a Solution I had u = 3^x I put it in: (u )(u ), but I was not sure how to get the middle term 3^(x+1)

Does e^[(-infinity)+(i*w*infinity)] = 0 or 1? w = omega

Homework Statement Starting from the Gamma function: \Gamma (s) = \int^{\infty}_{0} dx \, x^{s-1} e^{-x} Make a change of variable to express it in the form: \Gamma (s) = f(s) \int^{\infty}_{0} dy \, \exp{\frac{-A(y)}{\zeta(s)}} And identify the functions f(s), A(y)...

Homework Statement For the exponential function exp(z) = \sum^{\infty}_{n=0} \frac{z^n}{n!}, we know that exp(z+w)=exp(z).exp(w), and that, if x \geq 0, then exp(x) \geq 1+x. Use these facts to prove that, if 1 \leq n \leq m, then |\prod^n_{j=1} (1+a_j) - \prod^m_{j=1} (1+a_j)| \leq...

Homework Statement So I have to prove that Ymin is an unbiased estimator for lambda from the distribution: (1/lambda)e^(-x/lambda) Homework Equations The Attempt at a Solution I kno to show that the estimator is unbiased requires that its expected value to equal the given...

e^2x = 5e^3x I understand that I need to take a natural log of both sides here, what I am thrown about is the constant "5". Can I bring that up as an exponent? So, e^2x = e^(3x)^5?

Homework Statement if z = -2 + 2i find r and θ The Attempt at a Solution our teacher told us that when we have z = a + bi r = sqrt(a^2 + b^2) and θ = tan^-1(b/a) so here it's supposed to be r = sqrt(8) and θ = - pi/4 but using wolfram alpha to see if the results are...

Homework Statement A concession stand serves customers with each customer starting as soon as the prior one finishes. The wait times are from an exponential distribution with mean = 2 minutes. Therefore the Total time is the summation of xi for i 1 to 4. Find the mean , variance and...

z_k=\sqrt[n]{u}=\sqrt[n]{r}e^{i\left(\frac{\phi+2k\pi}{n}\right)}, k=0,1,2,...,n-1 and z_k=\sqrt[n]{u}=\sqrt[n]{re}^\frac{\phi+2k\pi}{n}, k=0,1,2,...,n-1 Which one is incorrect (note that in the first, e is out of the root)?

Homework Statement According to the Inverse Function Theorem, for every z_0 \in C there exists r > 0 such that the exponential function f(z) = e^z maps D(z0; r) invertibly to an open set U = f(D(z_0; r)). (a) Find the largest value of r for which this statement holds, and (b) determine the...

Hello, I am trying to evaluate the series \sum{\frac{x^n}{n!}e^{cn^2}} where c is a constant. I think this problem is equivalent to find f(x) such that \frac{d^{n}f(0)}{dx^{n}} = \frac{e^{cn^{2}}}{n!} I believe this must be a modified exponential since for c=0, it reduces to...

Homework Statement I need to integrate the following function: y = \int e^{-\frac{n}{\omega}cos \omega t} dt Homework Equations How should I solve this? Will this lead me to Bessel function? 2. My attempt a=\frac{n}{\omega} u=-a cos (\omega t) du = a\omega sin...

Homework Statement Suppose X1,X2... are iid mean 1 exponential random variables. Use large deviation methodology to give a lower bound for the rate function R(a) for a>1 Homework Equations R(a) \leq \frac{-logP[Sn >n*a]}{n} The Attempt at a Solution I know that a sum of exponential random...

Homework Statement find three independent solutions using complex exponentials, but express answer in real form. d^3(f(t))/dt^3 - f(t) = 0 Homework Equations The Attempt at a Solution after taking the derivative of z = Ce^(rt) three times I put it in the following form...

Homework Statement write e^z in the form a +bi z = 4e^(i*pi/3) --------------------------------------- My guess: z = 4*(cos(pi/3) + i*sin(pi/3)) e^z = e^(4*(cos(pi/3) + i*sin(pi/3))) = e^(4*cos(pi/3))*(cos(4*sin(pi/3)) + i*sin(4*sin(pi/3))) but the solution says...

Hello, I am having difficulty solving the following integral: \int^{\infty}_{-\infty}e{-(a|x|+ikx)}dx I have tried to use an explicit form of the absolute, eg. -(a|x|+ikx) = \left\{\stackrel{-(ik+a)x\ x>0} {-(ik-a)\ x<0} Does this allow me to separate the integral into a sum of two...