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Questions tagged [exponential-function]

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

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How to prove that $f'(x) = e^{-x} g(x)$?

Hello guys can someone help me with that? I tried all the ways but it always leads me far away from the wanted result. Given $g(x)= e^{x}-x-1$ and $f(x)=x-1+(x+2)e^{-x}$, prove that $f'(x) = e^{...
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47 views

Explicit calculation of the given integral

Can we calculate this integral explicitly? $$\huge\int\frac{dx}{1+e^{-k\left(\sin^2\left(\frac{\Gamma(x)+p}{x+q}\right) - 1\right)}}$$
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What whole numbers with exponents when multiplied, equal 1350000? [on hold]

What whole numbers with exponents when multiplied, equal $1350000$? Example: $$X^3Y^4Z^5=1350000$$
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1answer
34 views

How do i solve this differential equation. [on hold]

I'm not really concerned with the answer, im just wondering how the last two lines are equal.
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34 views

Why does sigmoid function use e instead of another constant?

I know this has already been asked, but I'm just curious why the sigmoid function P(y) = 1 / ( 1 + e ^ -y ) does use e and not pi for instance, or 1. Does it ...
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4answers
44 views

How do you come up with $e^{\ln{x}}$

I know that $e^{\ln{x}}$ is just the inverse of the exponential function but I don’t get it how it is done to arrive with the form: $e^{\ln{x}}=x$. Let’s say this: $\ln{x} = a$ so $x = e^a$, but my ...
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1answer
18 views

Exponential decrease of amplitude with time

I was wondering about a particular math problem. It says that a particular trigonometric function, $10 \cos(2\pi x)$ models a bus going over a speed bump. They say that the amplitude decreases over ...
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2answers
59 views

How to prove that $\lim\limits_{\epsilon\rightarrow 0}\int_{-\infty}^{\infty}f_\epsilon(x)g(x)dx=g(0)$ (Dirac delta function))

I'm currently studying the Dirac delta function using a textbook which unfortunately provides only partial solutions to its explanations. Why does $\lim\limits_{\epsilon\rightarrow 0}\int_{-\infty}^{\...
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29 views

Does there exist an integer sequence that satisfies the following properties?

Does there exist an integer sequence $\{a_n\}_{n = 1}^\infty$ that satisfies the following properties: $\forall t > 1, n^t = o(a_n)$ $\forall p > 1, q > 0, a_n = o(p^{n^q})$ ? The only ...
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44 views

How to solve mixed power and exponential equation?

The exercise says that I should find "real zeros" of the following function: $$f(x)=7-2x^2+2^{-x}$$ I have tried several ways, but I keep getting stuck as I take log-s of both sides. Can I solve ...
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10 views

Pre-Calc word problem Exponential Growth and Decay [closed]

A bacteria culture starts with 200 bacteria and every 5 hours increases by 40 percent. a. Determine the continuous growth model for its size at time t. b. Determine the population after 24 hours. c....
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69 views

Underapproximating the exponential function from below [duplicate]

I think that for positive natural numbers t and n we have $$ \left(1+\frac{n}{t}\right)^t\ \le\ e^n\,. $$ Is this true? I have constructed a proof of it (which would probably take some lengthy ...
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53 views

Prove $(1+1/n)^n$ and $(1-1/n)^{-n}$ converge to same number?

Consider the functions: \begin{align} A(n) &= \left(1 + \frac1n \right)^n \\ B(n) &= \left(1 - \frac1n \right)^{-n} \\ C(n) &= 1+\sum_{m=1}^n \frac{1}{m!} \end{align} Is it possible to ...
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1answer
25 views

proof $1 < \frac{(1-1/n)^{-n}}{(1+1/n)^n} < \frac{n}{n-1}$ when n= 2,3,…

I am pretty sure this is somehow related to binomial expansion, but when I try to convert it into binomial series it becomes too complicated and can't proceed.
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3answers
46 views

Exponential Equations and Inequalities and logarithms

So this is one kind of task: $2\times 8^{x}-7\times4^{x}+7\times2^{x}-2=0$ And I don't get any of these tasks, so I am asking for some kind of literature with introduction about this and good ...
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1answer
32 views

Is a set of values of an exponential function uncountable?

E.G. $f(x) = 2^x$ If yes, is there an easy, informal proof that a layman could understand?
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4answers
59 views

How to algebraically solve this equation?

The original question is: How many solutions does a given equation have. I understand that we can quickly draw graphs of both equations and see how many times they cross, but how would you solve this ...
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1answer
38 views

Find convergence and limit of $\lim_{x\to0}\frac{e^{-x^2}-(1+ax^2)}{x^4}$

I have an exercise of finding a real number "a" and limit of $$\lim_{x\to0}\frac{e^{-x^2}-(1+ax^2)}{x^4}$$ I have no idea what should I do, can someone help me solve this step-by-step? Thank you in ...
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2answers
61 views

Prove that if $e<y<x$ then $x^{y}<y^{x}$

Prove that if $e<y<x$ then $x^{y}<y^{x}$ My try:I tried to use Taylor's theorem: $$y^{x}-x^{y}=e^{xlny}-e^{ylnx}=1+xlny+o(xlny)-1-ylnx-o(ylnx)=$$ $$=lny^{x}-lnx^{y}+o(xlny)-o(ylnx)=ln\frac{y^...
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2answers
71 views

Prove inequality using M.V.T.

For $x>2$, prove that $(x-1)e^{\tfrac{2}{x}}-(x-2)e^{\tfrac{1}{x}}<e$ using the mean value theorem. I can't find the function to applying the mean value theorem. Give some advice or hint. ...
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1answer
15 views

Intersection of two functions from different classes of functions

I am wondering that $e^x$ and $x^2$ intersect how many times? Is there any way to compute efficiently and is there any generalization of questions like that? For example, $e^x$ and $x^n$ or one is ...
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0answers
26 views

Exponential equation with top and bottom limits?

So I'm coding a interactive sliding bar that changes based on the value that is given. Currently the interaction between the value given and the bar is static and I would like to make it flow with the ...
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2answers
43 views

Are $C^\infty$ Functions with all derivatives positive on [a,$\infty$),a$\gt$0 always made of exponential?

Are there any $C^\infty$ real functions except the exponential family and gamma function family which has all the derivatives of same sign on an interval [a,$\infty$) with a$\gt$0 ? I speculate the ...
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48 views

Prove $f(x)= e^{(\frac{-1}{(x-a)})}$ is differentiable for all $x$

Prove that, $$f(x)= e^{(\frac{-1}{(x-a)})}$$ is differentiable for all $x$ and in particular $f′(a) = 0$ which implies $f(a)$ is continuous at $x = a$ too. Prove $f(x)$ is twice differentiable for ...
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2answers
40 views

Can a sum of a finite number of exponentially growing numbers be calculated as a function of the growth rate and the number of growths?

This is a question based on the exponential growth pattern of a game mechanic in World of Warcraft: The Heart of Azeroth item in the game has a level that can be increased by gathering Azerite, with ...
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0answers
20 views

Integral involving upper incomplete gamma function, exponential and rational funcitons

Related to these questions here and here, I found a different form of the integrals, which result in $$f(x)=\frac{(-1)^n \,2^n}{\pi\,\lambda^{n+1}} \,\mathrm{e}^{-\lambda\,x}\int_{-\infty}^{\infty}\...
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4answers
21 views

Exponential Equation answer

$a$ and $b$ are real numbers! $\frac{1}{a}+\frac{1}{b}=2$ $3^a=5^b=?$ Is there anyone have an idea?
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3answers
54 views

How to solve exponential equation of $x + e^{-x }= 3$?

I've got the following equation: $\lambda + e^{-\lambda} = 3$ $3 - \lambda = \frac{1}{e^{\lambda}}$ $\frac{1}{3-\lambda} = e^{\lambda}$ Now, I take the natural log of both sides: $ln(\frac{1}{...
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5answers
90 views

Solve for x when $e^{2x}-3e^x=4$

This is what i've gotten so far: \begin{array} he^{2x}-3e^x&=&4\\ ln(e^{2x}-3e^x)&=&ln(4) \\ \frac{ln(e^{2x})}{ln(3e^x)}&=&ln(4)\\ \frac{2x}{ln(3) + ln(e^x)}&=&ln(4) \\ ...
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1answer
29 views

Know any alternative function names?

I know $e^x$ can be written as $exp(x)$ and $ln(x)$ can be written as $log_e(x)$. I wanted to know whether there are any alternative names/syntax for $sinh(x)$, $cosh(x)$, $|x|$, $|Re(z) + iIm(z)|$, ...
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78 views

Prove $f$ is negative

I need help to prove the following function is negative for all $0<u<1$, $0<v<1$, $u\neq v$ and an integer $a\ge 2$, $$f(v,u,a)=\frac{(u-1) u^2 (1-v)^a+v (1-u)^a (v-u ((a-2) u-a v+v+2))}{(...
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1answer
9 views

$Ker(exp_G )$ is discrete

$G ⊂ GL_n (\Bbb R)$ is an abelian connected Lie group, $\mathfrak{g}$ its Lie algebra and $exp_G : \mathfrak{g} → G$ the exponential map. Prove that $Ker(exp_G )$ is discrete. My attempt: $Lie(Ker(...
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3answers
48 views

How to prove that $\lim_{n \to \infty}\Bigl (\frac{n^n}{(2n)!} \Bigr )=0$

I'm asked to prove that $$\lim_{n \to \infty}\Bigl (\frac{n^n}{(2n)!} \Bigr )=0$$ I thought about using Stirling's approximation since $n \rightarrow\infty$ and then L'hopital in order to prove it ...
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2answers
63 views

Solve $(5+2\sqrt{6})^{x^2-3}+(5-2\sqrt{6})^{x^2-3}=10$

Find the real values of $x$ which satisfy the equation $(5+2\sqrt{6})^{x^2-3}+(5-2\sqrt{6})^{x^2-3}=10$ My Attempt $$ e^{10}=e^{(5+2\sqrt{6})^{x^2-3}+(5-2\sqrt{6})^{x^2-3}}=e^{(5+2\sqrt{6})^{x^2-3}}\...
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1answer
19 views

question on expected value of exponential functions

When is the approximation $\mathbb{E}\left\{{e^{f(x)}}\right\} = e^{\mathbb{E}\left\{f(x)\right\}}$ tight? I mean for which function of f(x) and which conditions? there is any theorem on it?
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2answers
23 views

Estimate exponential equation from graph

Output y of a system is a non increasing function of an input x Based on my studies I have found that it can be estimated as ...
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0answers
32 views

Why is sum of shifted exponential gamma?

My book says the following Suppose $X_{i} \sim$ iid $Exp(1,\eta)$ Where $Exp(\theta,\eta)$ is the shifted exponential ie has density $$\frac{1}{\theta}e^\frac{-(x-\eta)}{\theta}$$ for $x \ge \eta$ ...
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1answer
13 views

having variable as linear factor and exponent,

I simplified an expression to the term : 1= x- e^(0.616+0.326*x) using wolfram alpha I get a solution that contains a function called Wn, that I have never heard of. Can someone maybe shed some ...
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1answer
69 views

How to show that ${\lim_{n\to\infty}{e^{-t\sqrt{n}}\bigg({\frac{1}{1-\frac{t\sqrt{n}}{n}}\bigg)}^n}\large}=e^{\frac{t}{2}}$?

How to show the following limit? $${\lim_{n\to\infty}{e^{-t\sqrt{n}}\bigg({\frac{1}{1-\frac{t\sqrt{n}}{n}}\bigg)}^n}\large}=e^{\frac{t}{2}}$$ I tried Taylor expansion and tried exponentiating. But ...
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1answer
50 views

Find and classify the critical points of $f(x) = \ln(e^{3x} −2e^{2x} + e^x + 2)$, then sketch its graph

Question : Find and classify the critical points of $$f(x) = \ln(e^{3x} −2e^{2x} + e^x + 2),$$ then sketch its graph I have found out that: first derivative = $( 3e^{3x} - 4e^{2x} + e^x )(e^{3x} −2e^...
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1answer
22 views

Transforming this exponential function?

Here's the problem: A biologist places agar, a gel made from seaweed, in a Petri dish and infects it with bacteria. She uses the measurement of the growth ring to estimate the number of ...
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0answers
67 views

define open sector and closed sector in complex plane

Let us define open sector and closed sector in complex plane respectively by $\{z : \alpha<arg(z)<\beta \}$ and $ \{z: \alpha\leq arg(z)\leq \beta \}$, where $ 0<\alpha-\beta\leq 2 \pi$. ...
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3answers
56 views

How to decelerate from velocity v to stop time t over distance d?

I'd be grateful for some help with this problem I am trying to solve. Let's say that I have an object travelling at a velocity v. I want that object to come to a halt in time t AND travel exactly ...
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2answers
62 views

Problems in solving $\cos^{2}z=-1$

I want to find the solutions of $$\cos^{2}z=-1$$ and I proceeded in a couple of different ways that are giving different result (so I'm questioning if this idea is correct). I defined $w=e^{iz}$ and ...
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3answers
52 views

How to solve this inequality $3> 2^x+2^{1-x}$

By hit and trial I get the answer as $x\in\mathbb(0,1)$ but I don't think about the actual process by which this inequality can solve. Plz tell any process by which I solve this ? I take log both ...
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2answers
54 views

$0<e^{-\sqrt x}<\frac{1}{x^a}$ inequality

How can we prove that $0<e^{-\sqrt x}<\frac{1}{x^a}$ is true for all $x>B(a)$ for some $B(a)$ and some fixed $a>0$. By the mean theorem I can only receive $\frac{1}{1+\sqrt x}>e^{-\...
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1answer
24 views

How do I solve this to represent the complex number in Cartesian coordinate

How do I solve this to represent the complex number in Cartesian coordinate $$z=\frac{e^{\frac{\pi}{3}i}}{e^{\frac{2}{3}\pi i}\cdot 2e^{\frac{\pi}{6}i}}$$ So I got this question as a homework and I ...
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0answers
96 views

Intuitive reason that $(1/n)^n$ is maximized for $n = 1/e$

Consider the function $f(n) = \Big( \dfrac{1}{n} \Big)^n$. By setting $f'(n) = 0$, we find that the maximum of $f(n)$ occurs at $n = \dfrac{1}{e}$. Going through the calculations, there doesn't seem ...
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0answers
16 views

upper bound to sum of exponentials

Let $k\geq 1$ and $$ \beta_j=\frac{20^{j-1}}{(\log\log T)^2} $$ for $1\leq j\leq \mathfrak{I}-1 $ where $$ \mathfrak{I}=1+\max\{j\mid \beta_j\leq e^{-1000k}\} $$ So $\beta_{\mathfrak{I}}$ is the ...
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0answers
18 views

How to distort a sigmoid / logistic function?

Please help. I need to move an object such that its distance / time profile resembles a sigmoid curve but in a non-symmetrical manner. 1- Let's say that I need to move 800cm in 3 seconds. How do I ...