Exponential Definition and 1000 Threads

Homework Statement Find the first and second derivative of the following function: F(x)=e4ex Homework Equations d/dx ex = ex d/dx ax = axln(a) The Attempt at a Solution I know the derivative of ex is just ex, but I'm not sure how to go about starting this one. I'm near certain I need to use...

This was just very basic, I have accepted it in just a heartbeat, but when I tried to chopped it and examined one by one, somethings fishy is happening, this just involved \int_{0}^{\infty}x e^{-x}dx=1. Well, when we do Integration by parts we will have let u = x du = dx dv = e^{-x}dx v =...

Homework Statement Hi, the problem is imply to show the following \lim_{n\rightarrow \infty} 10^n e^{-t} \sinh{10^{-n}t} = \lim_{n\rightarrow \infty} 10^n e^{-t} \sin{10^{-n}t} = te^{-t} How can I do this? Just a hint or a first step would be great, thanks :) Homework EquationsThe Attempt...

Homework Statement Solve the ODE: y''+x*y'-y=0 Homework EquationsThe Attempt at a Solution Since this is a variable coefficient ODE, I have used the method of reduction of order, and assumed the solution in the form: y=c1*y1+c2*y2 In this case: y1=x, and I have the reached the integral below...

If we have two square matrices of the same size P and Q, we can put one in the exponent of the other by: M = P^Q = e^{ln(P)Q} ln(P) may give multiple results R, which are square matrices the same size as P. So then we have: M = e^{RQ} which can be Taylor expanded to arrive at a final square...

Why is (5.302) an approximation instead of an equality? Let ##T## be the operator ##\frac{p_x^2}{2m}##. By the law of indices, we should have ##e^{-\frac{i}{\hbar}(T+V)\Delta t}=e^{-\frac{i}{\hbar}T\Delta t} e^{-\frac{i}{\hbar}V\Delta t}## exactly. Is it because ##T## and ##V## do not commute...

Homework Statement The probability that a particular computer chip fails after a hours of operation is given by 0.00005∫e^(-0.00005t)dt on the interval [a, ∝] i. find the probability that the computer chip fails after 15,000 hours of operation ii. of the chips that are still operating after...

Homework Statement There are initially two cells seen through a microscope. every hour, a computer was tasked with taking a cell count through the microscope. Taking the sum of all the cell counts after the 7th hour, the total cell count was 280 cells. Find the exponential growth rate...

Homework Statement I have been working on a lab report recently in which I took some data and then further used this data to gather results. I was plotting magnetic flux density as a function of current, this turned out to be an exponential saturation curve. I plotted this curve using a...

Homework Statement Hi! I've been interpolating a data set using Legendre polynomials and Newton's method and now I've been asked to, given a data set, approximate the function using the Least Squares Method with the following fitting function: ##g(r)=a+be^{cr}##, where ##a##, ##b## and ##c##...

Please help me understand the following step

Hi, I know that when you take this limit it is equal to e^-wo, but I was just wondering how you got there when taking the limit? lim w-->wo ((exp(w)-exp(wo))/(w-wo))^-1 = 1/e^wo w and wo are both two points within the same plane.

Homework Statement [/B] Calculate \widehat{Y^{2}} (i.e., the mean of the square of Y. Homework Equations Y=\sum_{k=0}^{N-1}y_{k} where y_{k}=e^{-\gamma t}e^{\gamma \tau k}G_{k} and t=N\tau The quantities y_{k} (or G_{k}) are statistically independent. The Attempt at a Solution...

Homework Statement The amount of time that a surveillance camera will run without having to be reset is a random variable having the exponential distribution with beta = 50 days. find the probabilities that such a camera will a) have to be reset in less than 20 days b) not have to be reset in...

I am multiplying a lognormal distribution by an function to scale it larger. While I know that scaling a lognormal distribution by a constant multiplier yields a lognormal distribution, in this case the multiplier is not a constant. Instead, smaller values from the lognormal distribution are...

I know this is probably the least of my worries at the moment but my quantum textbook solves ##\frac{\mathrm{d}\phi (t) }{\mathrm{d} t}=\frac{iC}{h}\phi (t) ## as ##\phi (t) = e^{-i(\frac{C}{h})t}##. Is this not off by a sign? Its really bugging me.

Homework Statement A particle in SHM is subject to a driving force F(t)= ma*e^(-jt). Initial position and speed equal 0. Find x(t). Homework Equations F = -kxdx = mvdv F(t) = F(0)*e^(iωt) x(t) = Acos (ωt +φ) The Attempt at a Solution I have no idea how to deal with the exponential term. I...

Hi! I know some constitutive models for elastic materials like Neo-Hooke or Mooney-Rivlin, which give a relation between elongation ##\lambda=y/y_o## (where ##y## and ##y_o## are the length of the elastic material in a uniaxial compression test in the direction of the compression at stress ##P##...

a) On January 1 2000, the park estimated that they had 500 deer on their land. Two years later, they estimated that there were 550 deer on the land. Assume that the number of deer was changing exponentially, i.e. P(t)=ae^bt where P is the number of deer at year t, and a and b are parameters.Find...

Homework Statement Find the action of the operator ## e^{\vec{a} . \vec{\nabla}} \big( f(\theta,\phi) . g(r) \big) ## where \nabla is the gradient operator given in spherical coordinates, f and g are respectively scalar functions of the angular part ## ( \theta, \phi) ## and the radial part ##...

Homework Statement The question is: Mr Snail learns 1 fact/day. When will he know as much as the hare? Homework Equations The hare's learning rate is given by the equation: Learning Rate=50(e^((15-t)/5))+100 Where t is the time in days. The Attempt at a Solution Basically I have tried to make...

Please help give references on solving the following integral: \int\frac{1}{c_{1}e^{ax}+c_{2}e^{bx}}dx where a\neq b Thanks a lot in advance.

what is the nature of singularity of the function f(x)=exp(-1/z) where z is a complex number? now i arrive at two different results by progressing in two different ways. 1) if we expand the series f(z)=1-1/z+1/2!(z^2)-... then i can say that z=0 is an essential singularity. 2) now again if i...

I've always thought that when people say exponential growth, they mean e^x. But other times I hear people just say that the quantity doubles 2^x every time. So which is it? And if it's just doubling, why don't we say that 3^x or 4^x is exponential growth as well, since that's what those...

How does one show that eAeB=eA+Be[A,B]/2 where A,B are operators and [ , ] is the commutator. The QM book I am using states it as a fact without proof, but I would like to see how it is proved. I've muddled around with the series expansion, but can't get farther than a few term by term products...

Hi, I read on a previous post that to calculate the exponential of a large negative number I use the formula: e-r=10(-rloge) This is just a quick question but it it meant to be a log 10 or natural log and also is it meant to be loge0 or loge1 Thanks in advance

Hello :) I don't get this integral (Peskin & Schroeder P. 27 ) ##\int {{{{d^3}p} \over {{{\left( {2\pi } \right)}^3}}}{1 \over {{E_{\bf{p}}}}}{e^{i{\bf{p}} \cdot {\bf{r}}}}} = {{2\pi } \over {{{\left( {2\pi } \right)}^3}}}\int\limits_0^\infty {dp{{{p^2}} \over {2{E_{\bf{p}}}}}{{{e^{ipr}} -...

Homework Statement A bacteria culture initially contanis 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420. a. Find an expression for the number of bacteria after t hours. b. find the number of bacteria after 3 hours c. find the rate of...

Show that, for all real p and m, e^{2mi\cot^{-1}(p)}\left(\dfrac{pi+1}{pi-1}\right)^m=1

Hello guys, How can I evaluate the following integral please? ##\int_{-\infty}^{+\infty}\frac{e^{i A x}}{\sqrt{x^2+a^2}}dx## Thank you

How can I find an Integral of an exponential with Polynomial argument with finite limits: \int_0^\pi \exp^{-a x^2 -b x^4} dx \\ \int_0^\pi \exp^{-a x^2 -b x^4} (x - x^3)dx

Directions: Use a log table to solve for x: ${2.884}^{x}=0.01439$ $x*\log\left({2.884}\right)=\log\left({0.01439}\right)$ $x=\frac{\log\left({0.01439}\right)}{\log\left({2.884}\right)}$ is the exact answer. The solution to the problem is -4.004 in the back of the book. To evaluate the...

I just learned how to derive the wave equation and now I have some questions about it. In my physics text (first year) it simply says (without reason) that the solution to the wave equation is y(x,t) = Acos(kx-wt), where A is the amplitude of displacement, k is the wave number and w is the...

What can be said about the arguments of the exponential functions and trig functions ? Can the argument be a vector or must it be a scalar ? If it can only be a scalar must it be dimensionless ?

Hi, I want to transform a complex exponential with quadratic phase to discrete form, in other words to a vector form. can anyone help me with that? Thanks

What are the assumptions of exponential smoothing? For example, should autocorrelation of the data be low in order for the exponential smoothing model to be valid, should the process be stationary, etc?

Hi guys, This question has me a bit stumped and I can't seem to find any methods on Google that would help me solve this: The last part of the question asks me to find the value of the scale factor, which in this equation I believe is 1.8. I know how to calculate percentage increases the...

Homework Statement "Show that the Hermite polynomials generated in the Taylor series expansion e(2ξt - t2) = ∑(Hn(ξ)/n!)tn (starting from n=0 to ∞) are the same as generated in 7.58*." 2. Homework Equations *7.58 is an equation in the book "Introductory Quantum Mechanics" by...

Hi. can you find value (m) from equation Y^m = X/Z Y,x,z= known m=? unknown i want value (m) as equation m= ...... thank a lot

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Please delete, got mixed up, apologies.

How do I write taylor expansion of a function of x,y,z (not at origin) as an exponential function? Please see the attached image. I need help with the cross terms. I don't know how to include them in the exponential function?

Hi, so I need to write a fortran code with 2, 2x2 matrices. These matrices are in the form of B=(1 exp(i)(theta) 0 0) and D=(0 0 exp(i)(theta) 1) where i is sqrt of -1 and theta is an angle between 0 and 2pi. I've expanded the exponential so it reads cos(theta)+isin(theta) and let theta=pi/2...

Homework Statement Solve the following system. 2. Homework Equations (2^(x+y))=16, (2^(x-y)) = 4 The Attempt at a Solution (2^(x+y))=16 (2^x)(2^y) = 16 EQ1 [(z)(q) = 16] 2^(x-y)) = 4 (2^x)/(2y) = 4 EQ2 [(z)/(q)=4] Let z = (2^x) Let q = (2^y) Isolate the "z" variable in EQ1 so it can be...

We know that Fourier Transform F(W) of function f(t) is summation from -infinity to +infinity product of f(t) and exp^{-j w t}Here, what does the exponential term mean?

Let $X$ = the time between two successive arrivals at the drive-up window of a local bank. $X$ has an exponential distribution with $\lambda = 2$. That is the probability density of $X$ is $f(X | \lambda) = \lambda e^{-\lambda x}, X > 0 $ with $\lambda = 2$. Compute the following: a) The...

y=CektA) First find k. [Hint:Use the given information of y=100 when t=2, and y=300 when t=4 to compute k.] B) Finally, find the value for C. [Hint use ine of the two pieces of information given in the problem to solve for C. in other words, use either y=100 when t=2 or use y=300 when 4=4 to...

I found the following identity in a paper: ## \sum_{l=1}^{\infty}exp(-\pi\alpha l^2)=(\frac{1}{2\sqrt{\alpha}}-\frac{1}{2})+\frac{1}{\sqrt{\alpha}}\sum_{l=1}^{\infty}exp(\frac{-\pi l^2}{\alpha}) ## Someone please let me give some hints on how to prove this.

Homework Statement Customers arrive in single server queue to be serviced according to Poisson process with intensity 5 customers an hour. (a) If the customers begin to arrive at 8am, find the probability that at least 4 customers arrived between 9am and 10am. (b) Find the probability that the...

I've just encountered the infinite series ## \sum_{n=0}^\infty n e^{-n\lambda} ##. I know that in general its not possible to evaluate an infinite series but because wolframalpha.com could evaluate it(which gave the result ## \frac{e^{\lambda}}{(e^\lambda-1)^2} ##), it seems to me that this one...