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

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4answers
54 views

Finding the limit of the sequence $x(n) = (1+2/n)^n$ [duplicate]

What we are allowed to use - 1) The fact that limit of $(1+1/n)^n$ exists and assumed to be some real number $e$ 2) Subsequencial properties of limits of sequences 3) Basic properties of limit In the ...
1
vote
2answers
40 views

Solve equation with exponentials

I'm trying to obtain $x$ in the following equation: $$ 1= ae^{bx}+ce^{dx} $$ with a,b,c,d known. What I did was to take the ln of all the equation, so I have: $$ \log\frac{1/ac}{b+d} = x $$ ...
8
votes
3answers
123 views

General formula for the higher order derivatives of composition with exponential function

Suppose I have a function $x:\mathbb{R} \to \mathbb{R}$ and consider: $$g(t) = e^{x(t)}$$ When I start differentiating with respect to $t$ I obtain: \begin{align} g'&=e^xx'\\ g''&=e^x((x')^2+x'...
1
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1answer
54 views

How to prove the equality or inequality

Can anybody prove that the following equation is right or wrong? $$\int_0^te^{-t}(1-e^{-2x})^ke^x dx=\int_0^t2k(e^{-2x}-e^{-t-x})(1-e^{-2x})^{k-1}dx$$ where $t>0$ and $k$ is and integer. My small ...
3
votes
0answers
49 views

Proof that $|1 - e^{i \theta}| \geq \frac{2|\theta|}{\pi}$ for $-\pi \leq \theta \leq \pi$?

I would like to prove (geometrically if possible) the above result. Could someone help?
2
votes
4answers
95 views

Why is $\frac{(e^x+e^{-x})}{2}$ less than $e^\frac{x^2}{2}$?

I have read somewhere that this equality holds for all $x \in \mathbb {R}$. Is it true, and if so, why is that? $$\frac{(e^x+e^{-x})}{2} \leq e^\frac{x^2}{2}$$
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1answer
114 views

Find the limit of the sequence $(1-1/n)^n$

All that we have proven so far is that limit $(1+1/n)^n$ exists and considered to be a number 'e' which belongs to $(2,3)$ We haven't proven that 'e' is irrational or that lim $(1+(x/n))^n) = e^x$ ...
1
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1answer
29 views

exponentiating a matrix and sum of elements

$$ M= \begin{bmatrix} 1&1&0\\0&1&1\\0&0&1\\ \end{bmatrix} $$ Then the sum of all entries of $e^{M}$ i just don't know how to calculate this sum as this would be an infinite ...
4
votes
1answer
43 views

Bounding a function of norms on the unit cube

For a vector $v \in [0,1]^n$ and $p > 1$ we denote the p-norm of $v$ as: $||v||_p = (\sum_iv_i^p)^{\frac{1}{p}}$. where $v_i$ are the entries of $v$. Define the following (weird looking) function $...
1
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1answer
45 views

Find the original function by using convolution theorem

Seems like I don't know how to apply convolution theorem on this problem properly, I would appreciate some help and a brief explanation how did you solve it if you do it. \begin{equation}\frac{1}{((...
9
votes
5answers
697 views

Is there a way to prove this exponential inequality?

I came across this proposition while trying to prove that a function was injective: if $a>b$ then $a^a>b^b$, where $a$ and $b$ are real numbers bigger than 1 . Intuitively it (somehow) makes ...
2
votes
2answers
44 views

prove limit of exponential function without concept of logarithm

The question is, prove that if a real number $x>1$, then $\lim_{n\to\infty}x^n = \infty$, where $n \in \mathbb N$, without using the logarithmic concept. I came up with a proof, but I'm not so sure ...
0
votes
1answer
37 views

An easy way to define $\exp(x)$ - does it work?

$\exp(x)$ is usually defined in three different ways: 1) By its Taylor series: $\exp(x)=\sum_{k=0}^{\infty} \frac{x^k}{k!}$ 2) By its derivative: $\exp(x)'=\exp(x)$ 3) By the limit $\exp(x)=\lim_{N ...
7
votes
1answer
133 views

Solving $e^{\sin(z)}=1$ in the complex plane

I am trying to solve $e^{\sin(z)}=1$ in the complex plane. I know that this means that $\sin(z)=2k\pi i$ for some integer $k$. This is equivalent to saying that $$\frac{e^{iz}- e^{-iz}}{2i}=2k \pi i,$...
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2answers
36 views

Bounds on function $\exp(-\frac{1}{2}x^2)$

I have the following function : $$f(x)=\exp(-\frac{1}{2}x^2),$$ where $x >0$. I am looking for some tight bounds (upper bound and lower bounds) on $f(x)$. Any idea ? P.S.: The problem arises when ...
3
votes
4answers
54 views

Past exponential functions?

We have been taught that linear functions, usually expressed in the form $y=mx+b$, when given a input of 0,1,2,3, etc..., you can get from one output to the next by adding some constant (in this case, ...
1
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1answer
20 views

Volume Exponential Function

I should find the Volume received by rotating the region bounded by: $y = e^x $, $ y = 0 $,$ x = 0 $, $ x = 1 $ rotated around the x axis. I know how to find it by using the disc method but I could ...
8
votes
1answer
77 views

Solving an exponential equation with different bases

Solve the equation $2^x + 5^x = 3^x + 4^x$. I can figure out two special solutions $x=0$ and $x=1$, and I try to prove that they are the only two solutions. However, I find it hard to do so because I ...
0
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0answers
51 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 ...
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2answers
28 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
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0answers
36 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} = \int\...
0
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1answer
15 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 ...
1
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1answer
45 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
48 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
46 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
27 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
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0answers
58 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
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0answers
43 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 $\sum_{i=...
0
votes
1answer
22 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
69 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 ...
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4answers
90 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 this,...
4
votes
1answer
104 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
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0answers
28 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
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1answer
63 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}} + e^{-x}\...
0
votes
1answer
32 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: $min(x_1,...
-2
votes
2answers
60 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?
1
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2answers
43 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
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2answers
28 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
58 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
20 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
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2answers
51 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
52 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 $e^x≥x+...
2
votes
3answers
48 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
92 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 ...
2
votes
1answer
62 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
68 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
158 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 ...
3
votes
2answers
93 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
414 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
27 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 $$\lim_{r \...