Exponential Definition and 1000 Threads

Homework Statement The circuit in the diagram utilizes three identical diodes having I_S = 10−16 A. Find the value of the current I required to obtain an output voltage V_o = 2.4 V. If a current of 1 mA is drawn away from the output terminal by a load, what is the change in output...

Hello, I've been looking at the derivation of the exponential function, here http://www.statlect.com/ucdexp1.htm amongst other places, but I don't get how, why or what the o(delta t) really does. How does it help? It's really confusing me, and all the literature I've looked at just...

Homework Statement Consider the inner product $$\frac{1}{2\pi}\int_0^{2\pi} \left(\frac{3}{5 - 4\cos(x)}\right) e^{-ikx} dx, \quad k \in \mathbb{Z}, \quad x \in \mathbb{R}.$$ Homework Equations Is there a method to solve this without using the residue theorem, e.g. integration by parts...

So this is what I have tried so far playing around with exponent and radical properties to try and solve this textbook problem

Show that for $a >0$ and $b \ge 0$, $$\int_{0}^{\infty} e^{-a^{4}x^{2}(x^{2}-6b^{2})} \cos \Big(4a^{4}bx(x^{2}-b^{2}) \Big) \ dx = \frac{e^{a^{4}b^{4}}}{4a} \Gamma \left( \frac{1}{4} \right) $$

Homework Statement Part (a): Find the eigenvalues and eigenvectors of matrix A: \left( \begin{array}{cc} 2 & 0 & -1\\ 0 & 2 & -1\\ -1 & -1 & 3 \\ \end{array} \right) Part(b): Find the eigenvalues and eigenvectors of matrix ##B = e^{3A} + 5I##. Homework Equations The Attempt at a Solution...

Homework Statement Prove that y(x,t)=De^{-(Bx-Ct)^{2}} obeys the wave equation Homework Equations The wave equation: \frac{d^{2}y(x,t)}{dx^{2}}=\frac{1}{v^{2}}\frac{d^{2}y(x,t)}{dt^{2}} The Attempt at a Solution 1: y(x,t)=De^{-u^{2}}; \frac{du}{dx}=B; \frac{du}{dt}=-C 2...

hi I want to solve inflation problem for exponential potential. v(\phi) = v_0 exp(-\alpha \phi) (it's known as barrow or pawer law inflation ) we have 2 main equations: H^2 = 8π G / 3 (1/2 (\dot{\phi})^2 + v(\phi)) \ddot{\phi} + 3H \dot{\phi} + v(\phi)'=0 I must solve this 2 equ...

Homework Statement (2^(x+1)) / (5^x)) = 3 2.The attempt at a solution (2^(x+1)) / (5^x)) = 3 ((log 2 (x+1)) / (x log 5)) = log 3 (x log 2 + log 2) / (x log 5) = log 3 ( log (2^ (x+1))) / (log (5^x)) = log 3 log (base: 5^x) (number: 2 ^ (x+1)) = log 3 (5^x) ^ (log 3) =...

How do I compute the commutator [a,e^{-iHt}], knowing that [H,a]=-Ea? I tried by Taylor expanding the exponential, but I get -iEta to first order, which seems wrong.

Hi all, I'd just like to clear up something that's often confused me. In classes (particularly classical waves/QM) we've often seen the lecturer switch from describing a wave as (most commonly) Acos(kx-{\omega}t) to e^{i(kx-{\omega}t)} but doesn't the exponential representation include an...

I have to prove the inverse of the matrix e^A. we haven't studied exponential matrices in uni but he gave us the definition of it with the series e^A= I+A+A^2 ... A^k where A^k=0, even for numbers greater than k. I have tried to think of a way to prove it, but neither my classmates or I found...

We have a matrix with dimension NxN.For some m belongs to N,m0 we have A^m0=0.We consider the exponential matrix e^A=I+A+A^2/(2!)+A^2/(3!)+A^m/(m!).Find the inverse matrix of e^A. I tried to write the e^A=e^A(m0)+A^m/(m!) or (e^A)^(-1)=(...

Dear All, I would like to do an exponential function least-squares fitting, but having two or more exponents. For example the function looks like this: y (x) = A \exp (-x/a) + B \exp (-x/b) where A, a, B and b are the least-squares fitted parameters. My question is how to obtain the...

OK, I'm new to multi-variable calculus and I got this question in my exercises that asks me to integrate e^{-2(x+y)} over a diamond that is centered around the origin: \int\int_D e^{-2x-2y} dA where D=\{ (x,y): |x|+|y| \leq 1 \} I know that the region I'm integrating over is symmetric...

Homework Statement State whether (e^4-x) + 2 is an exponential growth function or an exponential decay function. Explain why. Homework Equations I want to use the formula f(x) = ae^kx where a>0, and k<0. The Attempt at a Solution I know it is an exponential decay formula...

Homework Statement (2^x - 2^-x)/3=4 Homework Equations Using log or exponential rules The Attempt at a Solution First multiply both sides by 3 so 2^x-2^-x=12 I thought I could take the log of both sides then condense the log, but that is not right. I also attempted to...

$$f(y) = \begin{cases} 0& \text{for }y< 0,\\ 2y& \text{for }0 ≤ y ≤ .5,\\ 6-6y& \text{for }0.5 < y ≤ 1, \\0& \text{for } y > 1\end{cases}$$ (1) Find cumulative distribution function, F(y) $$F(y) = \begin{cases} 0& \text{for }y< 0, \\\int_0^y 2t dt = y^2 & \text{for } 0 ≤ y ≤ .5,\\.5^2+...

$$f(y) = \begin{cases} \int_0^y\frac1\beta e^{\frac {-t}\beta}dt = -e^{\frac {-y}\beta}+1 & \text{for } 0 ≤ y < ∞,\\ 0& \text{for } elsewhere\end{cases}$$ P(Y>3) = 1 - P(Y ≤ 3) = 1 - (-e^{-3/beta}+1) = .1353 When I take log to both sides, I get 3.453. When I take ln to both sides, I get...

With mean = 2 with exponential distribution Calculate E(200 + 5Y^2 + 4Y^3) = 432 E(200) = 200 E(5Y^2) = 5E(Y^2) = 5(8) = 40 E(4Y^3) = 4E(Y^3) = 4(48) = 192 E(Y^2) = V(Y) + [E(Y)]^2 = 2^2+2^2= 8 E(Y^3) = m_Y^3(0) = 48(1-2(0))^{-4} = 48 is this right?

The amount of time to finish a operation has an exponential distribution with mean 2 hours Find the probability that the time to finish the operation is greater than 2 hours. My thinking is to integrate the exponential probability function. After integrating it, I got -e^{-y/2} + 1 , 0 ≤ y...

Is there a good example of a probability distribution where the support set does not depend on the parameters and is still not a member of the exponential family?

Hi, I'm having problems to solve this equation, pls help me: $$\left( 2+\sqrt {2}\right) ^{x}+\left( 2-\sqrt {2}\right) ^{x}=4$$

Hi, Is it possible to find a value of ##n## for the following expression other than by exhaustive search? $$e^{\frac{2n\pi i p}{q}}=e^{-\pi i p/q},\quad n=1,2,\cdots, q-1, \quad (p,q)\in \mathbb{N}\backslash 0$$ I can write a short program to search for the value of n, but that would...

Hello everyone, I was solving this problem: Justify that ln6= ln2+ln3 So: exp(ln2+ln3)=exp(ln2)*exp(ln3)= 2*3= 6 = exp(ln6) Till here, my work was okay. What I didn't understand is : why should we say that the exponential function is strictly increasing over R before being able to simplify the...

I confront an integration with the following form: \int d^2{\vec q} \exp(-a \vec{q}^{2}) \frac{\vec{k}^{2}-\vec{k}\cdot \vec{q}}{((\vec q-\vec k)^{2})(\vec{q}^{2}+b)} where a and b are some constants, \vec{q} = (q_1, q_2) and \vec{k} = (k_1, k_2) are two-components vectors. In the...

Please refer to the attached image.for part a) this is what i did: $G = k$, $k-1< X < k$ so I substituted $k-1$ and $k$ into the given exponential rv, this gave me $\lambda e^{-\lambda(k-1)}$ and $\lambda e^{-\lambda k}$ $= \lambda e^{-\lambda(k-1)} + \lambda e^{-\lambda k}$ But I...

No. of Real solution of the equation ##2^x\cdot \ln (2)+3^x\cdot \ln (3)+4^x\cdot \ln (4) = 2x## . The attempt at a solution Let ##f(x) = 2^x\cdot \ln (2)+3^x\cdot \ln (3)+4^x\cdot \ln (4) -2x## Here solution must be exists for ##x>0##, because for ##x\leq 0##, L.H.S>0 while R.H.S <0...

Suppose that there is initially x(not) grams of Kool-Aid powder in a glass of water. After 1 minute there are 3 grams remaining and after 3 minutes there is only 1 gram remaining. Find x(not) and the amount of Kool-Aid powder remaining after 5 minutes… So, i set up 2 equations… 3=x(not)e^-k(1)...

*solved

find dy/dx: exy+x2+y2= 5 at point (2,0) I'm confused with finding the derivative with respect to x of exy. this is what I did so far for just this part: exy*d(xy)/dx exy*(y+x*dy/dx) do I need to put the parentheses on here? I thought so because that is the part where I used the product rule...

no. of real solution of the equation $3^x+4^x+5^x = x^2$

hello friends, when i build the mathmatical model of robot,i face a new question that i ever seen before. i have a reverse kinematic lever as the leg and i want to use the tip position to get the relationship of fold angle and rotate angle reversely here is my equation: x*e^iθ - y*e^iθ *...

Hi, I have a quick question. Let R and S be two independent exponentially distributed random variables with rates λ and μ. How would I compute P{S < t < S + R}? I am a little bit confused because of the variables on either side of the inequalities. I have tried conditioning on both S and R...

Homework Statement The function T=190(1/2)^1/10t can be used to determine the length of time t, in hrs that milk will remain fresh. T is the storage temp. In Celsius How long will milk remain fresh at 22 degrees Celsius Homework Equations Bases have to be same then exponents will...

What does living in exponential times mean?

Homework Statement Suppose that the waiting time for the CTA Campus bus at the Reynolds Club stop is a continuous random variable Z (in hours) with an exponential distribution, with density f(z) = 6e–6z for z ≥ 0; f(z) = 0 for z < 0. (a) What is the expected waiting time in minutes (the...

Homework Statement I'm trying to solve trying logarithmic and exponential equations but for the life of me I can't seem to figure out what I'm doing wrong. Homework Equations Log equation: log3(x+2)+1 = log3x²+4x Exp equation: 2x + 2x+1 = 3 The Attempt at a Solution For the first...

Homework Statement I want to verify j^{-p}=e^{-j\frac{p\pi}{2}} Homework Equations e^{j\frac{\pi}{2}}=\cos(\frac{\pi}{2})+j\sin(\frac{\pi}{2})=j The Attempt at a Solution j^{-p}=(e^{j\frac{\pi}{2}})^{-p}=e^{-j\frac{p\pi}{2}} Am I correct? Thanks

Hi friends, i need some help for this number: By considering the integral $$\int_{\gamma(0;1)}\exp(z) \mathrm{d}z$$,show that $$\int_0^{2\pi}\exp(\theta)\cos(\theta+\sin(\theta)) \mathrm{d}\theta = 0$$ i know that since $$f(z)=\exp(z)$$ is holomorphic on and inside...

Homework Statement The problem says: u = 8[exp(y) - 1] Homework Equations The Attempt at a Solution Does this mean: 8ey-1 or 8ey-1

f(x)=x2ex the answer is f'(x)=(x2 + 2x)ex but I don't understand how to get there. Also I need to find g'(x) if g(x)=sqrtx(ex) would the answer for the second one be .5x-1/2ex?

So, if we start with logarithm ln(2) assuming a base of 10, how would I solve this without a calculator? ln(2) 10^x = 2 And I get stuck there without a calculator. I could .. ln(10^x) = ln(2) x ln(10) = ln(2) x = ln(2) But that gets me back to where I started. So, how would I go...

just want to confirm if i did set up my integral correctly and got a correct answer. $\displaystyle\int_0^a (e^{\frac{x}{a}}-e^{-\frac{x}{a}})$ using substitution for the first term in my integrand $\displaystyle u=\frac{x}{a}$ $\displaystyle du=\frac{1}{a}dx$; $\displaystyle dx=adu $ for...

[1] Total no. of real solution of the equation ##e^x = x^2## [2] Total no. of real solution of the equation ##e^x = x^3## My Solution:: [1] Let ##f(x) = e^x## and ##g(x) = x^2## Now we have use Camparasion Test for derivative So ##f^{'}(x) = e^x## which is ##>0\forall x\in...

Homework Statement I am to find the electric field for a charge distribution of $$ \rho(x)= e^{-\kappa \sqrt{x^2}} $$ Homework Equations I know that gauss law is $$ \int E \cdot da = \frac{q_{enc}}{\epsilon_0} $$ The Attempt at a Solution I am not sure what the charge...

Homework Statement t(s) = 1 15 30 45 60 75 90 105 120 135 N(counts) = 106 80 98 75 74 73 49 38 37 22 Consider a decaying radioactive source whose activity is measured at intervals of 15 seconds. the total counts during each period are given. What is...

Homework Statement How many oscillations occur before Mxy decays to approximately 1/3 of its initial value, for a Larmor frequency of 100 MHz and T2 of 100ms?Homework Equations I was learning about how NMR works and about transverse relaxation. According to what I learned, we can express...

Homework Statement A certain type of transistor has an exponentially distributed time of operation. After testing 400 transistors, it is observed that after one time unit, only 109 transistors are working. Estimate the expected time of operation. Homework Equations The Attempt...

Homework Statement Suppose that the velocity v(t) (in m/s) of a sky diver falling near the Earth's surface is given by the following exponential function, where time is measured in seconds. v(t) = 55 (1-e-0.18(t)) Find the initial velocity of the sky diver and the velocity after 6...