Exponential Definition and 1000 Threads

Can the derivative of an exponential function be calculated with logs base something other than e? Like base 10 or 2?

Hello gurus, I've been trying to prove the following inequality for several days: \int_1^\infty \frac{\exp\left(-\frac{(x-1)^2}{2a^2}\right)}{x}dx > \ln(1+a)\quad \forall a>0. I've demonstrated by simulations that this inequality holds. I‘ve also proved that this inequality holds for large...

Homework Statement In the exercise 8.4 from Quarks and Leptons. An Introductory Course in Modern Particle Physics - F.Halzem,A.Martin we can see: if the charge distribution \rho(r) has an exponential form e^{-mr}, then: F(q) \propto (1 - \frac{q^2}{m^2})^{-2} where...

I'm having some trouble solving for t in the following exponential equation. $$ B = A_1 e^{-\lambda_1 t} + A_2 e^{-\lambda_2 t} $$ I can't divide out the leading coefficients A1 and A2 because they differ. I'm not really sure how to immediately take the natural logarithm of both sides...

Homework Statement Find the inverse of exp(y-x)+5 2. The attempt at a solution I think the solution is y-ln(x-5) but I can't think of how to solve it to get that. I don't know what to do about the x and the y being together.

Homework Statement prove that, for gamma distribution μ=αθ σ^2=αθ^2 for exponential distribution μ=θ σ^2=θ^2

I'm reviewing for my final exam so one of the practice problems is: 5= 3^(x+5) Here's my attempt at it: ln 5= x+5 ln 3 ln 5 / ln 3 = x+5 (ln 5 / ln 3)-5= x 1.46-5 ≈ x -3.54 ≈ x I checked my answer and I get 3^1.46 ≈ 4.97 so rounding it up gives me 5 since I rounded off 1.46...

What exactly is the difference between the mean waiting time and the median waiting time for an exponential distribution? I'm looking for a slightly intuitive understanding. I know the formulae, with the mean waiting time as 1/λ and the median as ln2/λ (which I notice is also the formula for...

How would one evaluate $$\Phi = \int_{-\infty}^{+\infty} e^{-(ax+bx^2)} dx$$. I was trying to change it into a product of an error function and a gamma function, but I needed an extra dx. Any other ideas?

Hi, I am working on a modeling exercise and I would like to know which is the correct general expression for a Exponential Function: y = abc(x+d)+e or y = abcx+d+e Thanks in advance, Peter G.

Homework Statement Consider the complex number z=(i^201+i^8)/(i^3(1+i)^2). (a) Show that z can be expressed in the Cartesian form 1/2+(1/2)i. (b) Find the modulus of 4z − 2z*. (z* meaning z-bar/complex conjugate of z) (c) Write 2z in polar form. (d) Write 8z^3 in polar exponential form...

Solve 2^(4x-3)=1/8, find the exact answer

I need to compute the matrix exponential of a real matrix that has no special structure (is not symmetric, positive definite, nilpotent, normal, etc...). Currently, this is the most time consuming step in my program; I need to be able to transform this matrix, I suppose, into something that can...

Can anyone check my work? I'm doubtful of my answer. Homework Statement The bacteria in a 4-liter container double every minute. After 60 minutes the container is full. How long did it take to fill half the container? Homework Equations I used: F = A × 260 (1/2)F = A × 2x F...

I need help calculating the exponential map of a general vector. Definition of the exponential map For a Lie group G with Lie algebra \mathfrak{g}, and a vector X \in \mathfrak{g} \equiv T_eG, let \hat{X} be the corresponding left-invariant vector field. Then let \gamma_X(t) be the maximal...

Homework Statement Find the MLE of θ = P (X≤ 2) in a random sample of size n selected from an exponential distribution EXP(λ) Homework Equations f(x, λ) = λ e^(-λx) F(x, λ) = 1 - e^(-λx) The Attempt at a Solution I know how to find the MLE of the mean of an exponential...

I'm a very conceptual person, and I've been reading about exponential and logarithmic growth but don't fully have the kind of conceptual grasp on the two and how they differ that I'd like, so I"m curious: What is the difference between exponential growth and logarithmic growth and what causes...

I would like to find derivations of exp(-ik0r) respect to k in order to calculate exp(-ik1r) by using Taylor expansion: exp(-ik1r) = (exp(-ik0r))(0) +(k1 -k0)(exp(-ik0r))(1)/1! + (k1 -k0)2(exp(-ik0r))(2)/2! + ... This expansion converges when the value of r is relative low (0.3 - 1.2)...

I've been asked to prove that the following distribution is a member of the generalised exponential family of distributions. f(y;β) = (ky2β(y+k))/((β+3)(y+2k)(y+1)1/2) I know that i have to transform the equation into the form f(y) = exp{(yθ-bθ)/a∅ +c(y,∅)} and that to do this i...

Homework Statement I try to solve this integral with with parameter x as a member of this scale:(-∞ , +∞) I=∫∏dx[i] exp(-0.5XAX + XB)=∫∏dx[i] exp( Ʃ-0.5x[i]a[i][j]x[j] +Ʃ x[i]b[i] ) In which a[i][j] and b[i] are components of telated matrix and vector and the first sum is on i and j ranges...

Homework Statement A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means 1 year, 1.5 years, and 3 years. What is the average length of time the boat can remain at sea?Homework Equations Density...

Homework Statement If we have the exponential distribution f_X(x)=\frac{1}{2}e^{-x/2} then show that the cumulative distribution function of Y=\sqrt{X} is given by F_Y(y)=1-e^{-y^2/2}Homework Equations F_Y(y)=f_X(x)\cdot\left| \frac{dx}{dy}\right| F_Y(y)=f_X(h^{-1}(y))\cdot\left|...

I'm taking the Fourier transform of a signal. This integral has bounds from -∞ to ∞, but since the signal is 0 for negative t, the bounds become 0 to ∞ doing the integration, the antiderivative I get is et*(-3-jω+2j) where j is sqrt(-1) Now I have to evaluate this at t=infinity (since it is a...

I was just thinking, shouldn't hooke's law be represented with an exponential equation? I am thinking that in class, the springs we use for the hooke's law experiment are small, but if they were very large, then the higher coils would be sustaining much more weight than the lower coils, and...

Homework Statement I am of the conclusion that, under any circumstance, the extended exponential rule can not be applied to (1+x)^{1/x}. Thus, there is no way for the extended exponential rule to arise when taking the derivative of: f(x) = (1+x)^{\frac{1}{x}}e^{x} For instance, if my first...

[SIZE="4"]Which of the following numbers can be added directly in exponential notation? 1. 3x10^-4 2. 600x10 3. 32.6x10^-3 4. 0.4x10^-4 _______________________________ a. 1 and 2 c. 1 and 4 d. 2 and 4 b. 1and3

Homework Statement The question is to evaluate the expression e^-iA, where A is a Hermitian operator whose eigenvalues are known (but not given) using bra-ket algebra. Homework Equations See above. The Attempt at a Solution I have been looking around, reading the textbook and...

Homework Statement This question was asked in the Indian National Maths Olympiad. The question is: Find all the possible real ordered pairs of (x,y) for equation 16^[(x^2) + y] + 16^[x + (y^2)]=1 Homework Equations That was the only equation. The Attempt at a Solution...

Homework Statement 6. Three disease-carrying organisms decay exponentially in seawater according to the following model: P(t) = Ae-1.5t + Be-0.3t + Ce-0.05t t 0.5, 1, 2, 3 , 4, 5, 6, 7, 9 p(t) 6, 4.4, 3.2, 2.7, 2, 1.9, 1.7, 1.4, 1.1 Estimate the initial concentration of each...

Homework Statement In the last couple threads, it has become apparent that I need to organize my understanding of some of the derivative rules, specifically as they relate to exponential functions, such as e^x. So I wrote out a couple possible ways of evaluating e^x. Could you tell...

Homework Statement y = sqrt(exp(x) - exp(-x)) Homework Equations dy/dx = dy/du.du/dx - chain rule d/dx(exp(x)) = exp(x) - derivative of exp(x) The Attempt at a Solution y = sqrt(exp(x) - exp(-x)) y' = (1/2).(1/sqrt(exp(x) - exp(-x))).[exp(x) - (exp(-x).-1)] y' = [exp(x) +...

now i have a function defined on Z+ that is, it is defined on all positive integers, and it is complex now i know that it is the sum of a few (perhaps 6) exponential functions in the form of A_i exp(k_i x) how can i decompose the original function into the exponential functions? i.e...

Homework Statement What is the derivative of y=x^(13/x^2) with respect to x? The Attempt at a Solution I went through multiple techniques to solve this, but all of them have failed so far ._. In my latest attempt, I took the natural log of both sides: lny= lnx^(13/x^2) I...

Is that any way to find a finite value which is not equal to zero using L'hopital's rule in limit(z=-ia) exp[-A/(z+ia)]/(z+ia)^2 i kept getting 0/0 after differentiation Thank you

Sum of reciprocal of some base (I just chose e as example) to prime power? Ʃ \frac{1}{e^{p}} = \frac{1}{e^2}+\frac{1}{e^3}+\frac{1}{e^5}+\frac{1}{e^7}+\frac{1}{e^{11}}+\frac{1}{e^{13}}+\frac{1}{e^{17}}+... p\inP Brute force simulation gives me ~0.19279118970439518 Is there an...

Hi guys I have a problem in finding my error in a calculation, I will be glad if you help me to find the error that I am doing ok the problem is basically about matrix exponentials, here we go: A, B, U, P are matrices n is a natural number t and T are rational numbers and T=n*t now in...

Homework Statement Compute the derivative of the following function (1-2x)e^{-x} Homework Equations Product Rule and Quotient rule The Attempt at a Solution My problem here is that I come up with two different answers when I use the quotient rule vs. the product rule. Trying it with the...

Homework Statement A biomedical company finds that a certain bacteria used for crop insect control will grow exponentially at the rate of 12% per hour. Starting with 1000 bacteria, how many will they have after 10h? Homework Equations Topic is application of differential equations...

1. Problem Statment: Sketch the sequence x(n)=\delta(n) + exp(j\theta)\delta(n-1) + exp(j2\theta)\delta(n-2) + ... 3. Attempt at the Solution: The angle theta is given in this case Can someone remind me of how to multiply a complex exponential by a delta function? This sequence represents...

Homework Statement http://img4.imageshack.us/img4/224/32665300.png The Attempt at a Solution http://img684.imageshack.us/img684/2920/scan0003xo.jpg I've uploaded my work so far since its much faster than typing and I'm stuck on the last line trying to solve the integral. The first...

I'm trying to prove eA eB = eA + B using the power series expansion eXt = \sum_{n=0}^{\infty}Xntn/n! and so eA eB = \sum_{n=0}^{\infty}An/n! \sum_{n=0}^{\infty}Bn/n! I think the binomial theorem is the way to go: (x + y)n = \displaystyle \binom{n}{k} xn - k yk = \displaystyle...

Homework Statement Digital filter analysis - this is just one part of a multi-part question I can't move forward with. It's supposed to be an auxilliary question and isn't the "meat" of the problem. Find b, such that maximum of the magnitude of the frequency response function...

Hello all, I am trying to figure out how to solve for the complex conjugate of the following: (-0.5)^abs(x) Thanks for your help. -Brian

I'm taking a signals and systems class and the textbook (Signals and systems by Oppenheim) says the CT complex exponential of the form x(t) = C eat with C and a complex is a periodic signal. I fail to see how. Let C = |C| ejα (exponential form of a complex number) and a = r + jω (rectangular...

Homework Statement Solve \sqrt{-e^{(i2\pi)/3}} Homework Equations The Attempt at a Solution I seem to be missing something simple, as I take: \sqrt{-1} = i then, e^{(1/2)*(i2\pi)/3} which comes out as: ie^{i\pi/3} however, the solution is: -ie^{i\pi/3}, and I can't seem to see where...

Homework Statement Determine the image of the line segment joining e^(i*2*pi/3) to -e^(-i*2*pi/3) under the mapping f = e^(1/2*Log(z)). Homework Equations The Attempt at a Solution The line joining the two points: {z | -0.5 < x 0.5, y = sqrt(3)/2} f = the principle branch of...

Homework Statement A bacteria culture starts with 840 bacteria and grows at a rate proportional to its size. After 3 hours there will be 2520 bacteria. How long will it take for the population to reach 1030 ? 2. The attempt at a solution dP/dt = k*P dP/P = k*dt *Integrated both...

Homework Statement prove that: 1-exp(-iwt)= 2i*sin(wt/2) Homework Equations exp(iwt)= cos (wt) + i*sin(wt) The Attempt at a Solution I attempted to express the exponential into sum of cos and sin and considering t=2*t/2 in order to obtain an argument like (t/2) (using...

Homework Statement I need to prove that (1+x_{1})·...·(1+x_{n})≥(1-Ʃ^{n}_{i=1}x_{i}^2)e^{Ʃ^{n}_{i=1}x_{i}} with all 0≤x_{i}≤1 I've already proven that (1+x_{1})·...·(1+x_{n})≤e^{Ʃ^{n}_{i=1}x_{i}} with all 0≤x_{i}≤1 and (1-x_{1})·...·(1-x_{i})≥1-Ʃ^{n}_{i=1}x_{i} with all 0≤x_{i}≤1 , but...

Homework Statement Big problem with exponential function proof assignment, need some help. let x≥0 and for every k\in N there is n_{k}\in N and x_{k1}≥...≥x_{k_{nk}} and x_{k1}+...+x_{k_{nk}}=x. Proof: if lim_{k→}∞ x_{k1}=0 then lim_{k→}∞ (1+x_{k1})·...·(1+x_{k_{nk}})=exp(x)=e^{x}