For question involving exponential functions and questions on exponential growth or decay.

learn more… | top users | synonyms (1)

0
votes
0answers
13 views

Expectation of an exponent of a random variable

Suppose that $X \geq 0$ is distributed according to some distribution $F$. What can be said about $E[e^{-r X}]$? I.e. is there a way to express this expectation only in terms of some characteristics ...
0
votes
2answers
24 views

Confusing result obtained taking second derivative of ye^y

I was doing my calculus homework, and one of the questions asked for the first and second derivative of $ye^y=x$, I did the computations and arrived at $-(x+1)^{-2}$, which was a lot neater and ...
0
votes
0answers
11 views

Need help with $\int\mathrm{exp}[-C(\frac{1}{(x -\frac{1}{2})^2 + (x + \frac{1}{2})^2} -1)^S ] \mathrm{dx}$ for a statistical mechanics problem

Can someone help solving this integral?: $$\int\limits_{-\frac{1}{2}}^{0} \mathrm{exp}\left[-C \left( \frac{1}{(x -\frac{1}{2})^2 + (x + \frac{1}{2})^2} -1 \right)^S \right] \mathrm{dx}$$ The ...
0
votes
1answer
11 views

Generalizing exponential moving average to n samples

Assume that we have a moving average like this: $E_t = a*S_{t-1}+(1-a)*E_{t-1}$ where $E$ would be an estimate we are interested in, and $S$ is a sample we take at each point in time. Now, if we ...
-2
votes
0answers
21 views

Exponential differential equation

I need help with this equation: $$ g( x ) \exp({f' ( x ) }) = 1-\exp\left(f\left(\frac{x}{2}\right)\right)$$ I can't see an applicable method for that one.
0
votes
1answer
33 views

Simplify exponential equation

I really need your help to solve this exponential equation. It looks so simple, but I haven't been able to find a solution so far: $$ {A_1 + A_2 \over 2} = A_1 \exp\left({-x^2 \over c_1^2}\right) + ...
0
votes
1answer
39 views

Infinite Sum Defined by $\int \frac{e^x}{x}dx$ vs. Exponential Function Taylor Series

Recently, when fiddling around with integration by parts, I noticed that it is possible to define infinite series that led to an integral. My calculus teacher noticed this, and told me to find $$ ...
0
votes
1answer
34 views

Supremum of $\cot(\pi z)$ where $z$ is on circle with radius $n+1/2$

I try to estimate the supremum of $|\cot(\pi z)|$ and where $z=(n+1/2) e^{i t}$, $n\in\mathbb N$ and $t\in[0,2\pi)$. I should be a constant. So far I did by wiriting it in exponential form and ...
0
votes
3answers
24 views

Why does $\frac{1}{6e^{2y}}=\frac{1}{2x-8}$ in this context?

This is the context: I tried substituting $y=3e^{2x}+4$ into $6e^{2y}$but I wasn't able to go any further. Does anyone what exactly is being done in the last step?
0
votes
0answers
48 views

Integration of $\frac{x^2}{2\left(e^x+1\right)}$

Let: $$f(x) = \int \frac{x^2}{2\left(e^x+1\right)}dx $$ Is there a way to find $f(x)$? I've tried through integration by parts, but that didn't work out. If substitution is the answer, I can't see ...
2
votes
0answers
34 views

The number $e$ approximated as sums of of $X \sim U(0,1)$. Why does it work?

In this post a computer simulation to approximate $e$ is based on the mathematical knowledge that $E[\xi]=e$, where $\xi$ is the random variable defined as the minimum number of $n$ such that ...
0
votes
1answer
16 views

Fitting a curve - trending line formula

In OSX Numbers I have a chart with these data points: 50 53 100 62 200 78 300 91 500 117 1000 192 2000 297 3000 412 5000 567 10000 990 Using the trending line ...
2
votes
1answer
57 views

Approximating the number $e$ through computer simulation - mathematical background

There is nothing original about this question. It was asked here. I am just curious about an answer that is beyond my mathematical level. In one of the simulations appearing in the comments to the ...
1
vote
4answers
73 views

Proving that the exponential inequality $e^x \ge x^e$ holds for all $x \ge 0$ [duplicate]

How does one prove that $$e^x \ge x^e$$ for all $x \ge 0$? I tried to do this by setting $f(x)=e^x-x^e$ Plotting this function shows this easily, as seen here. However, when I tried to prove ...
4
votes
1answer
68 views

How to solve $\ln(y)=\ln(x)e^{\ln(x+1)} $ for x?

I know that if I have had $y = x^{x+0} $ aka $y = x^x$ I could do $y = x^x$ // $x = e^{\ln(x)}$ $y=x^{e^{\ln(x)}}$ // $\ln$() $\ln(y) = \ln(x)e^{\ln(x)}$ then using Lambert's W function I ...
1
vote
0answers
26 views

Complex exponentials [duplicate]

How do I solve: $$ e^{4z}+e^{3z}+e^{2z}+e^z+1=0 $$ I'm getting lost on where to start. I tried using the definition $$ e^z=e^x(\cos(y) +i\sin(y)) $$ But that doesn't seem to do me any good. I also ...
1
vote
1answer
50 views

How do you differentiate the integral from $ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt$ [duplicate]

How do you differentiate the integral from $e^{-x}$ to $e^x$ of $\sqrt(1+t^2)$ with respect to t? $$ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt $$ I know the answer is $$ e^x\sqrt{1+e^{2x}} + ...
0
votes
1answer
21 views

Minimum of four exponential variables

Four accidents occur independently, with each accident following an exponential distribution with a mean of 22.5. What is the expected value of the minimum of the four accidents? Attempt: ...
-3
votes
2answers
38 views

find the value of $k$ in the term $2^{-k} = 1/n$

What is the value of $k$ if I have the following equation: $2^{-k} = \frac1n$? $$2^{-k} = \frac 1 n \implies n = 2^k \implies \log_{2} n = k$$ Is my solution correct?
2
votes
2answers
42 views

Limit of $\frac{e^{1/x}}{x^2}$ as x approaches 0 negatively

I know the following: $$\lim_{x\to 0^-} \frac{e^{1/x}}{x^2} =0$$ I cannot, however, see why. Is there a method that makes this result intuitively clear?
0
votes
2answers
25 views

Difficulty finding the sum of a hyperbolic function.

Can someone please point out where I am (If I am) going wrong during the solution process of the following question: I have been presented with the following : $$4sinh(2ln(2))-cosh(ln2)$$ and told ...
2
votes
3answers
54 views

Why Doesn't $2^{1/n}= 1/(2^n)$

Take $2^{1/n}$. Since $1/n$ can be simplified as $n^{-1}$, the original term can become $2^{n^{-1}}$. The exponents can then be multiplied to result in $2^{-n}$ which is $1/(2^n)$. However it is ...
0
votes
3answers
15 views

How to isolate X in ${A * B ^X = C * D ^ X}$

${A * B ^X = C * D ^ X}$ The idea is to find in how much time (X) a small (A) investment with a good tax (B) beats a big investment (C) with a bad tax (D). All values are nonzero and positive.
1
vote
2answers
43 views

taking the natural log of e^(2x) = (4/3)

I have been unable to answer the following question. I must solve for x: $$e^{2x} = (4/3)$$ I have been made aware that I must take the natural log of both sides, giving: $$ln(e^{2x}) = ln(4/3)$$ ...
3
votes
2answers
43 views

Proof of $a^x ≥ x+1 \; \forall x \in \Bbb R \implies a=e$

I'm trying to prove the following : Let $a>0$ a real number. Then : $\quad a^x ≥ x+1 \;\; \forall x \in \Bbb R \iff a=e$ I managed to prove the '$\Longleftarrow$' part : $x≥0$ then ...
2
votes
3answers
43 views

Solving inequlity with $e^x$

I'm studying differential calculus, but one of the questions involves solving an inequality: $$(x-2)e^x < 0$$ I intend to go deeper in solving inequalities later, but I just want to understand ...
1
vote
4answers
81 views

how to solve $x(e^{-{1\over x}}-1)=$ constant

As mentionned in the title, how to solve analytically the equation $x \cdot \left(e^{-\frac{c_1}{x}}-1\right)=c_2$ where $c_1$ and $c_2$ are known constants. I can easily find a solution ...
1
vote
1answer
50 views

Integral of $x^{-2}e^x$

This is the original problem. $$\int_2^1 \frac{x^2e^x - 2xe^x}{x^4}$$ My attempt at breaking it down $$\frac{x^2e^x}{x^4} - \frac{2xe^x}{x^4}$$ $$x^{-2}e^x - 2x^{-3}e^x$$ $$ ...
0
votes
2answers
42 views

Integral of exponential rational function

I'm asked to find $$\int_0^{\ln 2}{e^{2x}\over{e^{4x}+3}} \text{ d}x$$ I can't for the life of me figure out how to integrate this.
6
votes
6answers
147 views

why is the limit as n goes to infinity of $(1+\frac{1}{n}+\frac{200}{n^2})^n = e$?

I know that $$\lim_{n\to\infty}\left(1+\frac{1}n\right)^n = e .$$ But why does $$\lim_{n\to\infty}\left(1+\frac{1}n+\frac{a}{n^b}\right)^n = e ? \quad where\quad b\gt1$$ better yet, how can I ...
0
votes
0answers
24 views

integration of product two lower incomplete gamma function and exponential [closed]

i need helping to find the value of this integration : $$ \int_0^{\infty}e^{-\delta x}\gamma(\alpha,\theta x){\gamma(\beta,\theta x)\ }dx $$ where all parameters are positive. Can anyone help me how ...
2
votes
2answers
77 views

Integral of $x^2 e^{-x^2}$

Like the title says, I'm trying to find $$\int_0^r x^2 e^{-x^2}\,dx$$ Where $r$ is some finite value. I've done one step using integration by parts with $u=x^2$ and $dv=e^{-x^2}dx$, which has left ...
4
votes
3answers
257 views

Eigenvector and eigenvalue for exponential matrix

$X$ is a matrix. Let $v$ be an eigenvector of $e^{X}$ with corresponding eigenvalue $a$. Show that $v$ is also an eigenvector of $e^{X}$ with eigenvalue $e^{a}$ If $X$ is diagonalizable, then we can ...
0
votes
0answers
21 views

Let $f(z) = e^{z^2}$. Find $\theta \in (-\pi,\pi]$ such that $\lim_{r \to \infty} f(re^{i\theta})=0$

Let $f(z)=\exp(z^2)$, with $z=re^{i\theta}$. Find $\theta \in (-\pi,\pi]$ such that $\lim_{r \to \infty} f(re^{i\theta})=0$. With the identity $e^z=e^x(\cos(y)+i \sin(y))$, I found that ...
7
votes
2answers
496 views

Why doesn't this infinite exponential growth go beyond 2.5?

My calculus book says that with: $$a=x^{x^{x^{.^{.^{.}}}}}$$ (exponent tower goes on forever), then: $$x=a^\frac{1}{a}$$ I tried it out with $a=3$ so $x=3^\frac{1}{3}$ and then ran a python program ...
0
votes
3answers
41 views

Solving equation with infinite exponent tower

How to solve this equation for $x$ where $a>0$? The exponent tower goes on forever: $$a=x^{x^{x^{.^{.^{.}}}}}$$ My Calculus book gives the following reasoning: ...
0
votes
0answers
99 views

Mapping in the complex plane

I have the following two circles in the complex plane, $z = x + iy$, which bound a region, $R$. The equations for the circles and a sketch of the region is given as follows: $$ x^2 + (y-1)^2 = 1\\ x^2 ...
1
vote
2answers
17 views

Manipulating Complex Exponentials

I am trying to show that $$ sin(5\pi t) = \frac 12 e^{-j \frac {\pi}2}e^{j5\pi t} - \frac 12 e^{-j \frac {\pi}2}e^{-j5\pi t} $$ I am aware that $$ sin(\theta) = \frac {e^{j\theta} - ...
-2
votes
0answers
22 views

Exponential growth or exponential decay? [closed]

Can I get a step by step answer to this question? Write the function below in the form $P = P_0a^t$. Is this exponential growth or exponential decay? $$P = 2e^{-1.1 t}$$ Round the base of the ...
3
votes
1answer
81 views

Which is larger, $e^\pi$ or $\pi^e$? [duplicate]

I don't know how to approach this. I tried expanding $e^{\pi}$ using the power series but that was a dead end since I didn't know what to do with it. I tried estimating if $e \log({\pi})$ was ...
4
votes
3answers
81 views

Trying to understand the function $y = x^x$

I am trying to understand the function $y=x^x$: 1) Why is $0^0$ not defined? Why isn't it defined as $0^0=1$? The limit of the function for $x\to0$ also goes to $1$ 2) Why is it only defined for ...
5
votes
6answers
91 views

Finding the limit $\lim_{n\to \infty} \left({\frac{n+1}{n-2}}\right)^\sqrt n$

I have to find: $$\lim_{n\to \infty} \left({\frac{n+1}{n-2}}\right)^\sqrt n$$ But, to be honest, I haven't got a faintest idea how to even begin. Is there a way to evaluate this radical exponent?
1
vote
2answers
65 views

Prove that the exponential $\exp z$ is not zero for any $z \in \Bbb C$

How can the following been proved? $$ \exp(z)\neq0, z\in\mathbb{C} $$ I tried it a few times, but i failed. Probably it is extremly simple. If a draw the unit circle and then a complex number ...
3
votes
4answers
47 views

asymptotic behavior of the two sequences defining exponential function

There are two definitions of exponential function: $$e^x=\lim_{n\to\infty} S_n=\lim_{n\to \infty} a_n \text{ ,}$$ where $$S_n=1+x+\frac{x^2}{2!}+\dots+\frac{x^n}{n!}$$ and ...
2
votes
2answers
29 views

Common factor out from a sum of exponential functions

From the below equation, which is a sum of two exponential functions I would like to compute the common factor $n$ $$ d = \exp\left(\frac{-x}{n}\right)+\exp\left(\frac{-y}{n}\right)$$ Unfortunately, ...
0
votes
2answers
23 views

exponential regression for bacteria growth

I'm studying regression lines and curves, and I've learn the methods for working with curves of the types $ax^2+bx+c$ and $ax+b$ as well as $a\sin(x)+b\cos(x)$. Now I'm asked this: $$(0,32), ...
0
votes
0answers
11 views

What is a hurst exponent in simple terms and what is the relation of it to fractals.

So I read this blog/paper recently and it is talking about the hurst exponent and it mentions that it can be used as an indicator of the fact that the time series can be predicted or has some sort of ...
0
votes
2answers
64 views

Can you easily simplify these terms?

I need to simplify these terms step by step to prove they are equivalent $$(100^{(2n+1)}-1+99×10^{(4n+2)}-99×10^{(2(n+1)-1)})/(11×10^{(2n+1)}-11) $$ and ...
1
vote
1answer
87 views

prove that the following function is decreasing?

I am trying to prove that the following function is decreasing. \begin{align}&f(t)=\frac{1-g(t)}{\sqrt{1+e^t}}\cdot\exp\left(-\frac{te^t}{2(1-e^t)}\right)&t<0\end{align}where $ ...
1
vote
1answer
83 views

Which function to kill: Sine or Cos?

I got an equation which was a solution to some familiar Differential Equation I am solving, the solution takes the form of: $$V=Ce^{-ix}$$ but $$Ce^{-ix}=A\cos(x)+B\sin(x)$$ so ...